Restoration of the Box-Ironbark Forests and Woodlands of Victoria, Australia

Incorporating uncertainty into expert models for management of Box-Ironbark forests and woodlands in Victoria, Australia

Christina Anne Czembor

Submitted in total fulfillment of the requirements of the degree of Masters of Science by Research

June 2009

School of Botany The University of Melbourne

ABSTRACT

Anthropogenic utilization of forest and woodland ecosystems can cause declines in flora and fauna species. It is imperative to restore these ecosystems to mitigate further declines. In this thesis, I focused on a highly degraded region, the Box-Ironbark forests and woodlands of Victoria, Australia. Rather than mature stands with large trees, stands are currently dominated by high densities of small stems. This change has resulted in reduced populations of many flora and fauna species dependent on older-growth forests and woodlands. Managers are interested in restoring mature Box-Ironbark forests and woodlands through three alternative management strategies: allocating land to National Parks and allowing stands to develop naturally without harvesting, modifying timber harvesting regimes to retain more medium and large trees, or a new ecological thinning technique that retains target habitat trees and removes competing trees to encourage growth of retained stems.

The effects of each management strategy are not easy to predict due to complex interactions between intervention and stochastic natural processes. Forest simulation models are often employed to overcome this problem. I constructed state-and-transition simulation models (STSMs) to predict the effects of alternative management actions and natural disturbances on vegetation structure. Due to a lack of empirical data, I relied on the knowledge of experts in Box-Ironbark ecology and management to construct STSMs. Models predicted that the development of mature woodlands under all strategies was minimal over the next 150 years, and neither current harvesting nor ecological thinning is likely to expedite the development of mature stands relative to growth and natural disturbances. However, differences in experts’ opinions led to widely diverging model predictions.

Uncertainty must be acknowledged in model construction because it can affect model predictions. I quantified uncertainty due to four sources – between-expert variation, imperfect expert knowledge, natural stochasticity, and model parameterization – to determine which source caused the most variance in model predictions. I found that models were very uncertain and between-expert uncertainty contributed the majority of

i Abstract variance in model predictions. This brings into question the use of consensus methods in forest management where differences between experts are ignored.

Using uncertain model predictions to make management decisions is problematic because any given action can have many plausible outcomes. I applied several decision criteria to uncertain STSM predictions using a formal decision-making framework to determine the optimal management action in Box-Ironbark forests and woodlands. I found that natural development is the most risk-averse option, while ecological thinning is the most risky option because there is a small likelihood that it will greatly expedite the development of mature woodlands. Rather than selecting one option, managers could rely on a risk-spreading approach where the majority of land is allocated to no-cutting National Parks and a small amount of land is allocated to the other two harvesting strategies. This would allow managers to collect monitoring data for all management strategies in order to learn about effects of harvesting and update model predictions through time using adaptive management.

DECLARATION

This is to certify that

1. The thesis comprises only my original work except where indicated in the Preface;

2. Due acknowledgment has been made in the text to all other material used;

3. The thesis is 33,500 words in length, inclusive of footnotes, but exclusive of tables, maps, appendices and bibliographies.

Christina Anne Czembor

iii

PREFACE

The work presented herein was completed solely for this thesis and is primarily my own work. The work presented in Chapter Two has contributed to the following publication:

C.A. Czembor, P.A. Vesk. Incorporating between-expert uncertainty into state-and- transition simulation models for forest restoration (submitted to Forest Ecology and Management).

ACKNOWLEDGEMENTS

I am extremely grateful to my friend and supervisor, Dr. Peter Vesk, for his many hours of assistance, endless clever suggestions, and generous support.

I would also like to thank Dr. Terry Walshe, Dr. Brendan Wintle, and Dr. Mick McCarthy for their input throughout this project. Heaps of thanks go to Laura Shirley and William Morris for sharing invaluable technical knowledge with me and continuously lending me a helping hand; you guys are the best.

I am very grateful to the experts who participated in this research: Andrew Bennett (Deakin University), Jon Cuddy (Department of Sustainability and Environment), Ron Hateley (University of Melbourne), Marie Keatley (Parks Victoria), John Kellas (Green Triangle Regional Plantation Committee), Rob Price (Department of Sustainability and Environment), Arn Tolsma (Arthur Rylah Institute), and Les Vearing (Department of Sustainability and Environment).

Generous thanks to Kori Blankenship, Leonardo Frid, James Merzenich, Jim Smith, and Ayn Shlisky for their assistance with modelling software. I also appreciate the time Yung En Chee, Libby Rumpff, Joslin Moore, and Yacov Salomon contributed towards this thesis. Thank you to Todd Gretton for access to DSE fire data, to Cindy Hauser for assistance with Visual Basic, and to Yung and Amy Hahs for assistance with ArcGIS.

I acknowledge the funds provided through the Australian Postgraduate Award, the Ethel McLennon Scholarship, the Ecological Society of Australia, the School of Botany, and the Plant Ecology Lab’s Special Postgraduate Studentship.

Thanks also to my good friends in Botany – Russ Johnson, Stuart Gardner, Lesleigh Kraft, Warwick McCallum, Kyatt Dixon, Emily McQualter, Victor Kabay, and Rob Piccinin. We’ve had a lot of fun times, stayed at uni far too late together, and been on some fantastic road trips.

v Acknowledgements

Finally, sincere thanks to my friends and family, here and in Canada. Most heartfelt thanks go to my parents, Harold and Liz Czembor, for their support and encouragement, and for instilling in me the love of forests and learning that brought me here (even though I came all the way to Australia and barely even went out into the forest!). Thanks to my family, Lisa Bains, Preet Bains, Judy Miller, Peter Gardner, and Margaret Gardner, for listening to me and feeding me over the years. Finally, thanks to my good friends: Dana (for everything), Bill (for making me smile), Judy (for baking and conversation), Laura (for always going for a drink when I needed a drink), and Reza (for tea, tunes, and encouragement).

TABLE OF CONTENTS

ABSTRACT ______I DECLARATION ______III PREFACE______IV ACKNOWLEDGEMENTS______V TABLE OF CONTENTS ______VII LIST OF TABLES ______IX LIST OF FIGURES ______X ABBREVIATIONS ______XII CHAPTER ONE: INTRODUCTION AND LITERATURE REVIEW ______1 1.1 INTRODUCTION ______3 1.1.1 Background ______3 1.1.2 Study objectives ______5 1.2 LITERATURE REVIEW ______7 1.2.1 Introduction______7 1.2.2 Forest restoration and simulation models ______8 1.2.3 Expert Opinion in Modelling ______15 1.2.4 Incorporating uncertainty into forest simulation models______19 1.2.5 Decision-making and adaptive management ______25 1.2.6 Conclusion______28 CHAPTER TWO: INCORPORATING BETWEEN-EXPERT UNCERTAINTY INTO STATE-AND-TRANSITION SIMULATION MODELS FOR FOREST RESTORATION ______31 2.1 ABSTRACT______33 2.2 INTRODUCTION ______34 2.2.1 Study area ______37 2.3 METHODS ______38 2.3.1 Defining vegetation states and transition agents ______40 2.3.2 Disturbance and management scenarios ______42 2.3.3 Model construction and simulation ______42 2.4 RESULTS ______43 2.4.1 Defining vegetation states ______43 2.4.2 Describing transition agents ______44 2.4.3 Results from each management scenario ______48 2.5 DISCUSSION ______51 2.5.1 Management scenarios ______51 2.5.2 Model uncertainty ______53 2.5.3 Limitations ______54 2.6 CONCLUSIONS ______57 CHAPTER THREE: ASSESSING SOURCES OF UNCERTAINTY IN EXPERT STATE- AND-TRANSITION SIMULATION MODELS______63 3.1 ABSTRACT______65 3.2 INTRODUCTION ______66 3.3 METHODS ______68

vii Table of Contents

3.3.1 Modelling uncertainty ______69 3.3.2 VDDT Model simulation ______72 3.3.3 Data analysis ______73 3.4 RESULTS ______76 3.4.1 State-and-transition simulation model predictions ______76 3.4.2 Variance parameter estimates ______77 3.4.3 Effects of initial conditions ______78 3.4.4 Sensitivity analyses ______78 3.5 DISCUSSION ______79 3.5.1 Incorporating uncertainty ______79 3.5.2 Limitations ______82 3.6 CONCLUSIONS ______83 CHAPTER FOUR: INCORPORATING UNCERTAINTY INTO DECISIONS ABOUT FOREST RESTORATION; TOWARDS ADAPTIVE MANAGEMENT ______97 4.1 ABSTRACT______99 4.2 INTRODUCTION ______100 4.3 METHODS ______103 4.3.1 Modelling the effects of management ______103 4.3.2 Decision criteria______104 4.4 RESULTS ______109 4.4.1 Vegetation dynamics according to management scenarios______109 4.4.2 Deciding between management scenarios ______110 4.5 DISCUSSION ______114 4.5.1 Criteria for decisions under uncertainty and risk ______114 4.5.2 Managing Box-Ironbark forests and woodlands ______117 4.5.3 Limitations ______117 4.5.4 Conclusions ______118 CHAPTER FIVE: SYNTHESIS AND CONCLUSIONS ______131 5.1 INTRODUCTION ______133 5.2 A SUMMARY OF FINDINGS______133 5.3 IMPLICATIONS ______135 5.4 FUTURE RESEARCH ______136 5.5 CONCLUSIONS ______137 REFERENCES ______139 APPENDIX A. INTERVIEW PROCEDURE. ______159 APPENDIX B. ALL TRANSITION AGENTS SPECIFIED BY EACH EXPERT. ______165 APPENDIX C. THE R CODE USED FOR DATA ANALYSES IN CHAPTER THREE. __ 170 APPENDIX D. THE AMOUNTS OF LAND IN EACH VEGETATION STATE BETWEEN INITIAL CONDITIONS OVER TIME. ______172 APPENDIX E. THE WINBUGS AND R CODE USED FOR DATA ANALYSES IN CHAPTER FOUR. ______173

viii

LIST OF TABLES

TABLE 2.1. ALL TRANSITION AGENTS THAT CAUSE TRANSITIONS BETWEEN VEGETATION STATES AND THE MANAGEMENT SCENARIO THEY ARE INCLUDED IN. ______58 TABLE 3.1. THE PERCENT OF CELLS IN EACH VEGETATION STATE AT THE BEGINNING OF SIMULATIONS FOR EACH OF THE THREE INITIAL CONDITIONS. AVERAGE REPRESENTS THE CURRENT ESTIMATED PROPORTION OF VEGETATION IN EACH STATE, WHILE OPEN AND DENSE REPRESENT TWO POSSIBLE HISTORIC CONDITIONS. ______85 TABLE 3.2. THE MEAN VARIANCE ESTIMATES AND 95% BAYESIAN CREDIBLE INTERVALS (IN SQUARE BRACKETS) FOR THE VCA PARAMETERS, BETWEEN-EXPERT, WITHIN-EXPERT, AND MONTE CARLO/RESIDUAL, DERIVED FROM POSTERIOR PROBABILITY DENSITY DISTRIBUTIONS. VALUES ARE PROVIDED FOR EACH TRANSITION AGENT CONFIDENCE INTERVAL, 80%, 90%, AND 95%.______86 TABLE 3.3. THE THREE TRANSITION AGENTS FOR EACH EXPERT WITH THE STEEPEST SLOPE FOR THE REGRESSION LINE BETWEEN TRANSITION AGENT PARAMETER VALUES AND THE PERCENT OF CELLS IN LOW DENSITY MATURE AT 150 YEARS. RESULTS ARE SORTED IN DESCENDING ORDER BY ABSOLUTE VALUE OF THE SLOPE. ______87 TABLE 4.1. THE MEAN PERCENT OF LAND IN HIGH AND LOW DENSITY MATURE, WITH UPPER AND LOWER 95% BAYESIAN CREDIBLE INTERVALS (IN SQUARE BRACKETS), AND STANDARD DEVIATIONS DERIVED FROM BAYESIAN POSTERIOR PDFS WHEN ALL OF THE LANDSCAPE IS ALLOCATED TO ONE MANAGEMENT SCENARIO, (A) NATURAL DISTURBANCE, (B) CURRENT HARVESTING, OR (C) ECOLOGICAL THINNING. RESULTS ARE AT 50 YEARS.______120

LIST OF FIGURES

FIGURE. 1.1. CLASSIFICATION OF FOREST GROWTH MODELS. ______30 FIGURE. 2.1. A MAP OF VICTORIA, AUSTRALIA, SHOWING THE HISTORIC EXTENT OF THE BOX- IRONBARK REGION (IN GREY) AND AREAS OF REMAINING FORESTS AND WOODLANDS (IN GREEN). ______59 FIGURE. 2.2. A SCHEMATIC OF THE VDDT SIMULATION ALGORITHM.______60 FIGURE. 2.3. THE FOUR VEGETATION STATES USED IN STATE-AND-TRANSITION SIMULATION MODELS ACCORDING TO THE DENSITY AND DBH OF EUCALYPTUS SPECIES. ______61 FIGURE. 2.4. THE MEAN PERCENT OF LAND IN EACH VEGETATION STATE OVER TIME ACCORDING TO EXPERTS (E1-E5) AND ARRANGED BY MANAGEMENT SCENARIO, (A) NATURAL DISTURBANCE, (B) CURRENT HARVESTING, AND (C) ECOLOGICAL THINNING. ______62 FIGURE. 3.1. PROBABILITY DENSITY FUNCTIONS FOR PARAMETER VALUES DRAWN RANDOMLY FROM BETA DISTRIBUTIONS FOR TRANSITION AGENT: ‘GROWTH FROM HIGH DENSITY REGROWTH TO HIGH DENSITY MATURE.’ ______88 FIGURE. 3.2. PROBABILITY DENSITY FUNCTIONS FOR PARAMETER VALUES DRAWN RANDOMLY FROM BETA DISTRIBUTIONS FOR TRANSITION AGENT: (A): ‘SELF-THIN FROM HIGH DENSITY MATURE TO LOW DENSITY MATURE,’ (B): ‘GROWTH FROM LOW DENSITY REGROWTH TO LOW DENSITY MATURE,’ (C): ‘WILDFIRE,’ AND (D): ‘WINDTHROW.’ ___ 89 FIGURE. 3.3. THE MEAN PERCENT OF LAND IN LOW DENSITY MATURE OVER TIME FOR EACH EXPERT. ______90 FIGURE. 3.4. BARPLOTS SHOWING THE TOTAL VARIANCE DUE TO BETWEEN-EXPERT UNCERTAINTY, WITHIN-EXPERT UNCERTAINTY, AND VDDT MONTE CARLO SIMULATIONS / RESIDUALS EVERY 10 YEARS FOR 150 YEARS. ______91 FIGURE. 3.5. BAYESIAN POSTERIOR PROBABILITY DENSITY FUNCTION FOR (A) ALL VARIANCE PARAMETERS, (B), BETWEEN-EXPERT VARIANCE, (C) WITHIN-EXPERT VARIANCE, AND (D) MONTE CARLO / RESIDUAL VARIANCE. ______92 FIGURE. 3.6. THE DIFFERENCES IN AMOUNTS OF LAND IN LOW DENSITY MATURE BETWEEN INITIAL CONDITIONS OVER TIME: (A) OPEN AND (B) DENSE ARE COMPARED TO AVERAGE. ______93 FIGURE. 3.7. TOTAL VARIANCE DUE TO BETWEEN-EXPERT UNCERTAINTY, WITHIN-EXPERT UNCERTAINTY, AND VDDT MONTE CARLO / RESIDUALS FOR EACH INITIAL CONDITION. RESULTS ARE FOR THE 90% CONFIDENCE INTERVAL AT YEAR 150.______94

x List of Figures

FIGURE. 3.8. THE MEAN PERCENT OF LAND IN LOW DENSITY MATURE AT 150 YEARS FOR EACH EXPERT AND THE (A) AVERAGE INITIAL CONDITION, (B) OPEN INITIAL CONDITION, AND (C) DENSE INITIAL CONDITION. ______95 FIGURE. 4.1. THE MEDIAN PERCENT OF LAND IN LOW DENSITY MATURE OVER TIME. EACH DOT REPRESENTS ONE OF THE 66 REPLICATE MODELS WHERE A PROPORTION OF CELLS WAS ALLOCATED TO THE THREE MANAGEMENT SCENARIOS, AS INDICATED BY GRAPH AXES. 121 FIGURE. 4.2. THE MEDIAN PERCENT OF LAND IN HIGH DENSITY MATURE OVER TIME. EACH DOT REPRESENTS ONE OF THE 66 REPLICATE MODELS WHERE A PROPORTION OF CELLS WAS ALLOCATED TO THE THREE MANAGEMENT SCENARIOS, AS INDICATED BY GRAPH AXES. 122 FIGURE. 4.3. THE MEDIAN PERCENT OF LAND IN HIGH AND LOW DENSITY MATURE OVER TIME. EACH DOT REPRESENTS ONE OF THE 66 REPLICATE MODELS WHERE A PROPORTION OF CELLS WAS ALLOCATED TO THE THREE MANAGEMENT SCENARIOS, AS INDICATED BY GRAPH AXES. ______123 FIGURE. 4.4. BOX AND WHISKER PLOTS SHOWING THE PERCENT OF LAND IN HIGH AND LOW DENSITY MATURE FOR THE THREE REPLICATES WHERE THE ENTIRE LANDSCAPE IS ALLOCATED TO ONLY ONE OF THE MANAGEMENT SCENARIOS. 124 FIGURE. 4.5. THE PERCENT OF LAND IN HIGH AND LOW DENSITY MATURE IN 50 YEARS. EACH DOT REPRESENTS ONE OF THE 66 REPLICATE MODELS WHERE A PROPORTION OF CELLS WAS ALLOCATED TO THE THREE MANAGEMENT SCENARIOS, AS INDICATED BY GRAPH AXES. ______125 FIGURE. 4.6. BAYESIAN POSTERIOR PROBABILITY DENSITY FUNCTIONS SHOWING THE PREDICTED PERCENT OF LAND IN HIGH AND LOW DENSTY MATURE IN 50 YEARS WHEN THE ENTIRE LANDSCAPE IS ALLOCATED TO ONLY ONE ACTION, (A) NATURAL DISTURBANCE, (B), CURRENT HARVESTING, OR (C) ECOLOGICAL THINNING.______126 FIGURE. 4.7. CUMULATIVE DENSITY FUNCTIONS ILLUSTRATING THE APPLICATION OF AN OBJECTIVE FUNCTION AND FIRST ORDER STOCHASTIC DOMINANCE. CDFS INDICATE THE PROBABILITY OF ACHIEVING A GIVEN PERCENT OF LAND, OR LESS, IN HIGH AND LOW DENSITY MATURE IN 50 YEARS WHEN THE ENTIRE LANDSCAPE IS ALLOCATED TO ONLY ON MANAGEMENT SCENARIO. ______127 FIGURE. 4.8. AREAS UNDER THE CURVES OF THE CUMULATIVE DENSITY FUNCTIONS FOR NATURAL DISTURBANCE AND ECOLOGICAL THINNING TO ILLUSTRATE SECOND ORDER STOCHASTIC DOMINANCE. ______128 FIGURE. 4.9. AN APPLICATION OF RISK-SPREADING INDICATING THE PROPORTION OF TOTAL LAND THAT SHOULD BE ALLOCATED TO EACH MANAGEMENT SCENARIO DEPENDING ON THE CRITICAL LEVEL (LCRIT). ______129

ABBREVIATIONS

AM – adaptive management cdf – cumulative density function DBH – diameter at breast height, 130cm GIS – geographic information system MCMC – Markov Chain Monte Carlo pdf – probability density function ReML – Restricted Maximum Likelihood STSM – state-and-transition simulation model VDDT – Vegetation Development Dynamics Tool software

xii CHAPTER ONE

Introduction and literature review

2 Chapter One

1.1 Introduction

1.1.1 Background

Timber harvesting, mining, grazing, and other forms of forest utilization substantially alter vegetation structure and habitat quality. Often these changes detrimentally affect flora and fauna species and ecosystem function (e.g. Scherer et al. 2000; Peterman and Semlitsch 2009). When ecosystems have been impaired because of forest utilization, there are strong imperatives to restore forests. Restoration aims to return the altered function, composition, and structure of impaired forest ecosystems to more natural conditions (Allen et al. 2002; Chazdon 2008). This may be done using passive actions, such as the cessation of detrimental activities, or active actions, such as planting, weeding, or thinning (McIver and Starr 2001). However, effective restoration of forest ecosystems can be difficult because of the broad areas and long time horizons involved (Tester et al. 1997; Rieman et al. 2001; Kangas and Kangas 2004) and the complex interactions between restoration actions and natural, stochastic processes (McCarthy and Burgman 1995).

Forest simulation models can benefit restoration activities in several ways. First, simulation models can incorporate complex and interacting processes (e.g. Larson et al. 2004). Simulation models also assist restoration by predicting the potential outcomes of restoration activities over large areas and extended time periods (Tester et al. 1997; Peng 2000). In addition, if there are multiple possible management actions available for restoration of a system, then simulation models can be used to predict the potential outcomes of alternative actions (e.g. Tester et al. 1997; Forbis et al. 2006; Rüger et al. 2007). Using an explicit framework also facilitates interrogation of inconsistent assumptions about ecosystems (Rieman et al. 2001). Furthermore, by providing an explicit framework to assemble the available knowledge of an ecosystem, simulation models help to identify gaps in ecological knowledge.

The lack of comprehensive empirical data to identify key drivers of vegetation change and estimate parameter values is a common problem in ecosystem modelling (Regan et al. 2002; Forbis et al. 2006). This frequently occurs because data collection is costly (Starfield and Bleloch 1991) and managers may prefer to spend time and money on

3 Chapter One management actions rather than collecting monitoring data (Duncan and Wintle 2008). In addition, empirical data may not be available if appropriate field sites do not exist (Starfield and Bleloch 1991). Also, there is typically little information regarding conditions prior to natural disturbances or anthropogenic changes. When empirical data are limited, expert opinion is needed to confidently construct ecological models of system responses to management actions (e.g. Kangas et al. 2000; Martin et al. 2005; Forbis et al. 2006). Expert knowledge can also be used to speculate about historical rates and synthesize anecdotal evidence (Forbis et al. 2006).

While forest simulation models can assist in restoration, they are imperfect approximations of real systems (Starfield and Bleloch 1991) and, as such, contain inherent uncertainty. In addition, constructing models with expert opinion contributes to model uncertainty (Regan et al. 2002; Kangas and Kangas 2004) because experts will often disagree about how ecosystems respond to natural processes and management (Morgan et al. 2001). Furthermore, forest ecosystems exhibit variation in structure and function subject to stochastic variation in the rates and spatial distribution of natural disturbances. Uncertainty arising from model error (Regan et al. 2002), differences between experts (e.g. Pellikka et al. 2005), natural variation (McCarthy and Burgman 1995), and other sources needs to be transparently accounted for in model construction because it can affect model predictions. In addition, if uncertainty is not acknowledged, there will be an increased chance of making the wrong management decisions. While there are many examples of forest models that incorporate different sources of uncertainty (McCarthy and Burgman 1995; Kangas and Kangas 2004), there is a dearth of examples of forest simulation models that incorporate between-expert uncertainty.

When uncertainty is incorporated into forest simulation models, deciding between alternative restoration actions can be difficult because there are multiple plausible outcomes for a given action. To assist in making decisions under uncertainty, it is recommended that managers use a formal decision analysis framework (Clemen 1996; Peterman and Anderson 1999; Possingham et al. 2001). Adaptive management is one decision-making tool that relies on several explicit steps to make decisions under uncertainty (Holling 1978). These steps are common to many decision analysis frameworks and include stating clear management objectives, identifying possible management actions, constructing models that predict the effects of management

4 Chapter One actions, assigning measures of confidence in the models, and commencing a monitoring program to validate and update model predictions over time (Nichols and Williams 2006; Duncan and Wintle 2008). Adaptive management is an iterative process that aims to decrease uncertainty in biological processes through time by integrating monitoring data and current quantitative models in order to optimize management decisions (Nichols et al. 1995; Nichols and Williams 2006; Duncan and Wintle 2008). By completing each of these steps, decisions about restoration actions will be based on unambiguous criteria and, through collecting data and learning about the effects of management actions, can be more certain through time.

1.1.2 Study objectives

This thesis investigated the potential of using forest simulation models, specifically state-and-transition simulation models, to predict the outcomes of alternative restoration actions in a degraded forest ecosystem. Due to a dearth of empirical data for the forest system in question, models were constructed and parameterized using the knowledge of experts in the field. Uncertainty that arises from several sources, including the use of expert knowledge, the specification of initial conditions, and natural stochasticity, was incorporated into models and the effect of uncertainty on model predictions was investigated. I also explored the ability to make restoration decisions based on uncertain model predictions using a structured decision analysis framework and explicit management objectives.

I focused on a case study of the Box-Ironbark forests and woodlands of Victoria, Australia. Due to decades of timber extraction, mining, and grazing, these forests and woodlands are highly degraded and lack mature stands (Kellas 1991; Soderquist 1999). This ecosystem has recently received the attention of managers seeking to restore ecological structure and function. Specifically, managers are interested in expediting the development of large, hollow-bearing trees because they provide primary nesting and foraging sites for many fauna species (Bennett et al. 1994; Alexander 1997; van der Ree et al. 2006), which are now in decline due to a lack of appropriate habitat (Calder and Calder 1994; Alexander 1997; Soderquist 1999; Environment Conservation Council 2001). To assist in expediting large tree development in Box-Ironbark forests and woodlands, managers are relying on three alternative management actions: ecological

5 Chapter One thinning, current timber harvesting regulations, and growth without timber removal (Department of Natural Resources and Environment 1998; Sutton 2000; Parks Victoria 2004). Ecological thinning is currently being trialled as an intensive harvesting method that removes all stems within a certain distance around candidate habitat trees to reduce competition and allow retained trees to grow faster than those without thinning. Current harvesting aims to retain increased numbers of medium and large trees in harvesting coupes to allow development of hollows, and growth without timber harvesting relies on stand maturation and natural disturbances to develop large, hollow- bearing trees.

The primary objectives of this project were:

1. using expert opinion, to characterize Box-Ironbark vegetation structure based on unique states and to identify how natural processes and management actions will cause transitions between these states;

2. to construct and run a set of simulation models, one for each expert, that integrated the states and transitions identified in step one in order to predict changes in vegetation structure over time;

3. to investigate sources of model uncertainty; specifically, to assess the importance of initial condition specification, stochastic variation, within-expert uncertainty, and between-expert uncertainty on model predictions; and,

4. based on uncertain model predictions, to explore decision analyses for restoration of Box-Ironbark forests and woodlands using a structured decision framework and explicit decision criteria.

The following section reviews the application of simulation models for forest restoration, the usefulness of expert opinion in ecological modelling, the importance of acknowledging uncertainty in models, and the ability to make optimal decisions using uncertain model predictions. Chapter Two describes the construction of simulation models based on expert opinion and begins to investigate the effects of between-expert uncertainty on model predictions. Chapter Three quantifies the importance of various

6 Chapter One sources of uncertainty on model predictions, and Chapter Four explores the ability to make decisions for Box-Ironbark forests and woodlands using the steps of an adaptive management framework, with special focus on explicit management objectives. The final chapter integrates the results and discusses their implications.

1.2 Literature Review

1.2.1 Introduction

Restoration is often required when a forest ecosystem has been damaged by utilization. Due to the complexity involved in restoring and managing forests, managers may utilize simulation models to predict the effects of alternative restoration actions on vegetation structure. Models are, of necessity, simplified representations of complex systems subject to processes that occur stochastically over broad areas and long timeframes (Starfield et al. 1995). Use of expert opinion in model construction and parameterization is common in ecological contexts, particularly when there is a lack of relevant empirical data. Simulation models constructed with expert opinion ought to account for the fact that experts disagree and have imperfect knowledge. While there are examples of studies that incorporate uncertainty, there is a dearth of research that incorporates the effects of disagreement between experts and uncertainty within experts. When uncertainty is incorporated into models, deciding between restoration actions becomes difficult because there are multiple plausible outcomes for a single action. To assist in making decisions under uncertainty, it is wise to apply a structured decision analysis framework, such as adaptive management, that integrates models, explicit restoration objectives, and monitoring data to learn about alternative management actions and decrease the uncertainty in management over time.

In the following sections, I review the use of simulation models for forest restoration and investigate the application of expert knowledge to model construction. I also investigate the importance of acknowledging various sources of uncertainty in models. Finally, I consider the application of a formal framework to decision-making under uncertainty.

7 Chapter One

1.2.2 Forest restoration and simulation models

Forest alteration and restoration

The majority of anthropogenic changes to forest and woodland ecosystems have been due to agriculture (e.g. Yates and Hobbs 1997; Prober and Thiele 2005), changes in disturbance regimes (e.g. Shlisky et al. 2005; Zald et al. 2008), and resource extraction, including mining (e.g. Grant 2006) and timber harvesting (Fox 2000). Internationally, there is strong research interest in restoring degraded forest ecosystems to conditions prior to alteration (Boerner et al. 2008; Chazdon 2008; Laughlin et al. 2008) or at least to recover lost ecosystem functions (Prober and Thiele 2005). Interest in restoration is due to various social, biological, and economic factors. For example, there is strong motivation to incorporate social aesthetic values into forest management (Gobster 1999). There is also recognition that forests control vital ecosystem services, such as water flows, CO2 exchanges, and macroclimate (Ehrlich and Mooney 1983); ecosystem services that are estimated to be worth trillions of dollars per year (Costanza et al. 1997). Furthermore, our ability to restore degraded ecosystems will affect the survival of many species (Hobbs and Harris 2001).

Forest and woodland ecosystems affected by resource extraction, such as timber harvesting, typically display changes in species composition (Laughlin et al. 2008), ecological function (Prober and Thiele 2005), and vegetation structure (Lindenmayer and Franklin 1997) relative to pre-alteration. In terms of vegetation structure, altered forests tend to lack downed wood (Mac Nally et al. 2002b), have higher densities of small, middle seral-stage overstory trees (Hemstrom et al. 2001), and possess few large, older trees (Lindenmayer and Franklin 1997), which can impact habitat quality for a fauna species. Managers have relied on various strategies to restore degraded forest ecosystems. For example, researchers found that coarse woody debris loads were well below historical rates in Australia’s southeastern floodplains (Mac Nally et al. 2002b) and that this likely had detrimental effects on fauna (Mac Nally et al. 2001). To help restore this system, researchers redistributed large quantities of coarse woody debris and found that the bird species they monitored quickly increased in numbers as a result (Mac Nally et al. 2002a). Researchers monitored sites again three years after the addition of downed wood. They found that species increases were sustained and responses were not transient (Mac Nally 2006). In addition, managers restoring stands

8 Chapter One with high densities of smaller trees may apply mechanical thinning and controlled, low- intensity fires to restore forests (e.g. Allen et al. 2002; Boerner et al. 2008; Laughlin et al. 2008). Mechanical thinning is a promising technique often used to restore forest structure and improve habitat quality (e.g. Hayes et al. 2003; Hagar et al. 2004; Homyack et al. 2004) because it provides space for retained trees and, through reducing competition, should result in increased growth rates (Kellas et al. 1972; Marquis and Ernst 1991). This leads to an accumulation of old-growth characters, such as large trees and structural diversity, faster than stands that are not thinned (Sullivan et al. 2001; Homyack et al. 2004).

Long-term monitoring indicates that thinning can be beneficial for various fauna species. In their 10-year study of the effects of thinning on small mammals, Sullivan et al. (2001) found that species diversity and richness increased in second-growth lodgepole pine (Pinus contorta) forests when subjected to high- and medium-intensity thinning. Bird species have also been shown to respond to thinning (Hayes et al. 2003; Hagar et al. 2004; de la Montaña et al. 2006). Four years after thinning Douglas-fir (Pseudotsuga menziesii) forests, Hagar et al. (2004) recorded increases in species richness and bird density relative to pre-thinning. The researchers noted that the few bird species with adverse responses to thinning might respond positively over the long-term due to stand structure development (Hagar et al. 2004). Soon after thinning Douglas-fir forests, Hayes et al. (2003) found that some bird species’ abundances increased, while the abundances of others were unchanged or slightly decreased after thinning. However, they found that after five years, bird responses became more pronounced and thinning was highly beneficial for many rare bird species (Hayes et al. 2003). In a similar study, de la Montaña et al. (2006) completed a large-scale thinning trial in maquis forest and found that two years after thinning, there was a significant increase in bird species richness and an increase in the density of bird species with high conservation concern.

Unfortunately, long-term monitoring data are often not available to predict the effects of forest restoration actions. This frequently occurs because collecting data may not be possible if appropriate sites are not currently available (Starfield and Bleloch 1991). Also, due to the slow growth of overstory tree species, monitoring data likely cannot be collected by the time management decisions must be made (Hansen et al. 1993; Whalley 1994). In addition, the effects of management on forests are complicated by

9 Chapter One natural, stochastic processes (McCarthy and Burgman 1995) as well as long timeframes and broad areas (Tester et al. 1997; Rieman et al. 2001; Kangas and Kangas 2004). As a result, managers often rely on simulation models to predict the effects of management and restoration actions.

Forest simulation models

Explicit models are useful for management because, by utilizing formal frameworks to incorporate information from conceptual models, models can be interrogated for logic and internal inconsistencies (Rieman et al. 2001). Also, once a conceptual model has been used to construct a quantitative model, the parameters can be examined to determine how to generalize to other situations, and can be communicated to managers, researchers, and stakeholders (Starfield and Bleloch 1991). Simulation models are particularly useful for restoration because they can incorporate available information and conceptual models to predict the complex effects of management actions (Peng 2000).

Conceptual models have a long history in forest research and management, and have been implemented as simulation models since access to computers has increased. Many modelling frameworks have been developed independently and, consequently, there are various contradicting classification schemes (e.g. Munro 1974; Shugart and West 1980; Liu and Ashton 1995; Peng 2000; Porté and Bartelink 2002). To understand the applications and usefulness of forest models, it is helpful to understand the differences between the common modelling frameworks. I review some of the major forest models following the classification scheme of Porté and Bartelink (2002), who group models based on three criteria: the smallest unit of the model, spatial dependence, and whether forest heterogeneity is incorporated (Fig. 1.1). Modelling systems that are distance dependent (i.e. including spatial relationships) are not segregated according to the third criterion, forest heterogeneity, because by incorporating spatial relationships models inherently include heterogeneity. The classification system of Porté and Bartelink (2002) does not directly distinguish between empirical models, such as growth and yield tables, and mechanistic/process models that simulate photosynthesis, respiration, and nutrient cycles, which is a common method of distinguishing models (Peng 2000). However, I selected this classification scheme because it is logical and comprehensive.

10 Chapter One

The first criterion considered by Porté and Bartelink is the unit of the model; tree or stand. Tree models track individual trees and are split into distance-dependent (i.e. spatial) and distance-independent (i.e. nonspatial) models. Distance-dependent tree models track individual trees and their locations and can be empirical or mechanistic (Porté and Bartelink 2002). These spatially explicit models either incorporate solely the effects of growth (e.g. Pukkala et al. 1998) or growth, mortality, and recruitment, such as the SORTIE model (Pacala et al. 1993).

Distance-independent tree models, the second type of individual-based models, do not track the spatial location of trees. They can be discriminated into two forms based on the incorporation of forest heterogeneity: non-gap models or gap models. Distance- independent non-gap tree models typically consist of empirical growth-yield approaches, such as the PROGNOSIS model (Bare and Opalack 1987), and are reasonably rare (Porté and Bartelink 2002). They are typically used to assist timber production on managed forests (Liu and Ashton 1995).

Distance-independent tree gap models, on the other hand, are a very common modelling framework dating from the 1970’s (Shugart and West 1980; Liu and Ashton 1995) that reflect the landscape as a collection of patches of different ages and successional stages (Bugmann 2001). The patches are internally identical, meaning that there is no differentiation between individual trees within patches, and each patch has its own set of successional processes such that the patches do not interact (Bugmann 2001). The successional processes modelled include recruitment, growth, and mortality, but do not explicitly include processes such as photosynthesis, respiration, or carbon allocation (Hickler et al. 2004). Distance-independent tree gap models have been used to explore ecological theories about forest landscape functioning and dynamics (Liu and Ashton 1995) and to reconstruct the historical composition of forests (Shugart 2002). Some well-known examples of distance-independent tree gap models include JABOWA (Botkin et al. 1972), FORET (Shugart and West 1977), BRIND (Shugart and Noble 1981), and ZELIG (Smith and Urban 1988).

The second major group of forest models includes the stand-based forest models (Porté and Bartelink 2002). Stand models track groups of trees as the smallest unit, such that a

11 Chapter One single unit or cell is comprised of multiple individuals, and can again be differentiated into distance-dependent and distance-independent models. Distance-dependent stand models are reasonably uncommon and depict the forest as spatially-explicit mosaic of sub-forests. The sub-forests are defined by discrete states, and transitions between states depend on the state of neighboring cells. Distance-dependent stand models do not model the growth within cells and are not differentiated according to the third criterion, forest heterogeneity (Porté and Bartelink 2002). Two examples of density- dependent stand models include Riéra et al.’s (1998) spatially explicit mosaic model that describes how different diameter stems move between structural types, and Rupp et al.’s (2000) spatially-explicit frame-based model that partitions vegetation into frames and simulates processes that cause transitions between frames.

Distance-independent stand models are the other type of forest models based on groups of trees. These models describe forests as discrete units without spatial information and can be classified as average tree models, where the stand is the sum of N average or identical trees (no forest heterogeneity included), or as distribution models, where the stand is divided into size classes with each class containing average trees (forest heterogeneity included). Average tree models are essentially yield tables that model stand basal area and volume using large data sets based on permanent plots or chronosequences (Porté and Bartelink 2002).

The second group of distance-independent stand models incorporates forest heterogeneity and are called distribution models. Distribution models simulate recruitment, growth, and mortality over discrete time periods. Distance-independent stand distribution models include Markov models, which are stochastic and simulate the probability of changing from one discrete state to another, and state-transition matrix models, which are deterministic and simulate the fraction of trees that grow into the next class in one timestep (Porté and Bartelink 2002). Also, while not specifically included in published classification schemes, non-spatial frame-based models would also be placed in the classification (e.g. Tester et al. 1997; Hahn et al. 1999). Distance- independent stand distribution models are of particular interest as they have been used extensively to predict forest dynamics and assist in management. Because distribution models have been broadly applied in much forest management research, different types of these models will be considered in more detail below.

12 Chapter One

Frame-based model simulations are a type of distribution model that describe the key successional stages of the forest or stand of interest; these are the frames. In each frame, there is a simple model of the important biotic and abiotic mechanisms that cause a patch to shift from one frame to another. These mechanisms include seed set, tree growth, competition, fires, herbivory, and harvesting. When a patch shifts to another frame, the simplified model of that new frame takes over to determine the dynamics of the patch. The frame-based model is a collection of smaller models that each focus on the key processes that affect each successional stage (Tester et al. 1997; Hahn et al. 1999). For example, Tester et al.’s (1997) frame-based models of the dynamics of a white pine (Pinus strobus) forest in Minnesota, U.S.A, predict the effects of natural processes, such as seed germination, herbivory, and fire, and management for a 1000- year period. The researchers found that models play a key role in management as they can be easy to construct and can incorporate current knowledge to describe the effects of different management actions (Tester et al. 1997).

Another type of distribution model that has been widely used in forest management is the Markov model. Markov models are referred to in the literature by a variety of names, including Markov chain, population, demographic, usher matrix, Leslie matrix, and transition models (Porté and Bartelink 2002). The defining characteristic of Markov models is that they are based on the Markov property, which claims that future states of the system are dependent only on the current state and are independent of past states. In this way, it is assumed that the present state encapsulates all relevant information that could influence the future evolution of the system. Markov models have been used to model vegetation dynamics (e.g. Balzter 2000) and to manage near- to-nature forests (e.g. Moser et al. 2003). Transition models, or state-and-transition models as they will be referred to throughout this manuscript (Westoby et al. 1989), are based on Markov models (Scanlan 1994) and have been used extensively to model the effects of management and restoration actions on woodland and forest systems (e.g. Perry and Enright 2002a; Forbis et al. 2006; Spooner and Allcock 2006; Hemstrom et al. 2007; Wales et al. 2007). I focus now on state-and-transition models as they have been shown to provide support to forest management and, consequently, were the simulation modelling technique used in this thesis.

13 Chapter One

State-and-transition models

State-and-transition models were originally created as conceptual models to describe the vegetation dynamics of rangelands (Westoby et al. 1989). Managers of these systems began to notice that the vegetation did not respond as expected according to the traditional Clementsian or equilibrium view of rangeland succession (Clements 1916; Westoby et al. 1989). Instead managers found that rangelands often did not progress along specified succession trajectories to a single climax community, such that stable states were not achieved, but instead formed alternative stable states that changed dependent upon the application of transitions, such as grazing or fire. As a result, state- and-transition models were conceived because they could accommodate multiple stable states (Westoby et al. 1989; Ash et al. 1994; Bellamy and Brown 1994; Whalley 1994). These models segregate vegetation into discrete states that reflect the dominant vegetation component (e.g. McIvor and Scanlan 1994; Orr et al. 1994) and assign a list of possible causes of transitions between states, called transition agents (Westoby et al. 1989). These transition agents describe natural disturbances and management actions (Westoby et al. 1989) and can reflect gradual changes in vegetation as a result of growth, as well as rapid, episodic events (Grice and Macleod 1994). Since their inception as conceptual rangeland models, state-and-transition models have been adapted to other ecosystems, such as riparian habitats (e.g. Wondzell et al. 2007b), savannas or woodlands (e.g. Plant and Vayssieres 2000; Hill et al. 2005; Prober and Thiele 2005) and forests (e.g. McIver and Starr 2001; Grant 2006; Hemstrom et al. 2007).

The state-and-transition framework possesses various advantages for describing forest change. For example, the ability to integrate a variety of ecosystem states with the mechanisms that cause state changes promotes the organization of information and provides a useful framework for decision-making (Prober and Thiele 2005). Furthermore, in order to make predictions about the future characteristics of a landscape, conceptual models can be converted to simulation models by assigning a likelihood or rate of transitions occurring (Whalley 1994). In addition, state-and- transition models can easily combine both expert knowledge and empirical data (Spooner and Allcock 2006) and are relatively easy to parameterize (Westoby et al. 1989).

14 Chapter One

The information required to populate state-and-transition models can be acquired from various sources. For example, transition probabilities can be quantified or calibrated through monitoring of current management activities (Ash et al. 1994; Hill et al. 2005). In addition, aerial photographs taken at two distinct time periods can be utilized to calculate the probabilities of transitions between various states (e.g. Scanlan 1994). State-and-transition models are, however, more commonly based on expert opinion, either exclusively (e.g. Hill et al. 2003; McIntosh et al. 2003; Forbis et al. 2006) or supplemented by ancillary data such as published literature and empirical data (e.g. Hemstrom et al. 2002; Spooner and Allcock 2006; Hemstrom et al. 2007).

1.2.3 Expert Opinion in Modelling

Parameterizing ecosystem models can often be difficult because of a paucity of appropriate empirical data (Regan et al. 2002; Forbis et al. 2006). This happens because data collection is costly (Starfield and Bleloch 1991), managers may prefer to allocate resources to management actions rather than collecting data (Duncan and Wintle 2008), and data reflecting conditions prior to disturbances or anthropogenic changes are rarely collected. As a result, expert opinion is needed to confidently construct ecological models of system responses to management actions (e.g. Kangas et al. 2000; Yamada et al. 2003; Martin et al. 2005; Forbis et al. 2006). Expert opinion incorporates personal experience, data regarding ecosystems of interest, and the results of previous management actions (Forbis et al. 2006). As such, expert opinion can assist with decisions about restoration activities that must be made before data can be collected (Starfield and Bleloch 1991; Kangas et al. 2000; Dorazio and Johnson 2003). While expert opinion is not a direct substitute for empirical data, it can be a preliminary means to provide much-needed insight while more conclusive data are collected (Morgan et al. 2001).

Expert opinion has always been involved in decision-making (Ayyub 2001), and is ubiquitous in forest management planning and design. However expert opinion is often only implicitly employed, for example in validating existing ecological predictions (Kuhnert et al. 2005) and decision-making (Fried and Gilless 1989). Recently, there has been an effort to formally and explicitly incorporate expert opinion into habitat

15 Chapter One modelling (e.g. Pearce et al. 2001; Yamada et al. 2003) and forest management (e.g. Alho and Kangas 1997; Kangas et al. 2000; Morgan et al. 2001). Pearce et al. (2001) found that predictions of fauna distributions for forest reserve design based on expert opinion alone were less accurate than those based on survey data. However, the expert predictions were needed for species with insufficient data so that initial recommendations for reserve placement could be made. Morgan et al. (2001) asked experts to predict the effects of climate change on forests. They found that there was agreement between experts regarding most effects, but disagreement arose when experts attributed differing levels of importance to variables affecting climate change (Morgan et al. 2001). This disagreement was likely in areas where research was limited or inconclusive. These analyses using expert opinion are vital for providing preliminary or baseline information for management (Pearce et al. 2001; Yamada et al. 2003), and for identifying the state of current knowledge and future research needs (Morgan et al. 2001). These studies also illustrate that expert opinion can be extremely advantageous for restoration and management actions. In situations where expert opinion are available but field data are lacking, employing expert opinion is a cost-effective and rapid way to assist in making informed decisions (e.g. Garthwaite 1998; Kangas et al. 2000; Martin et al. 2005; McCarthy and Masters 2005).

One way to explicitly incorporate expert opinion into forestry and ecological studies is by treating expert opinion as a Bayesian prior probability distribution that is updated with data using Bayes theorem to produce a weighted average, known as the posterior distribution (Ellison 2004). Studies using expert opinion in this way have illustrated that expert priors will be altered in the posterior distribution to reflect the data when data are ample or precise, but will remain relatively unchanged when data are minimal or highly variable (Crome et al. 1996; Kuhnert et al. 2005). However, if the expert prior is uninformative due to a lack of knowledge or disagreement between experts, the posterior result will simply reflect the influence of the data (Crome et al. 1996; Kuhnert et al. 2005; Martin et al. 2005). These studies have also shown that expert opinion can be useful for predicting the possible effects of management; effects that would not be observed with limited data alone (Crome et al. 1996; Martin et al. 2005). Thus, expert opinion can provide an immediate, albeit preliminary, means of informing management decisions (Garthwaite 1998; McCarthy and Masters 2005).

16 Chapter One

In cases where no appropriate data are available, expert opinion alone has been employed to assist in forest and wildlife management (e.g. Morgan et al. 2001; Pellikka et al. 2005). Morgan et al. (2001) asked experts to provide subjective probability distributions of expected forest biomass under different climate change scenarios. Pellikka et al. (2005) used Bayesian Belief Networks parameterized with expert opinion to assess the effect of wildlife management on game populations. In addition to these studies of biological responses, expert opinion has also been utilized for simulation models. For example, as previously mentioned, state-and-transition simulation models (STSMs) have often been parameterized with the assistance of experts because empirical data were lacking (e.g. Hemstrom et al. 2002; Hill et al. 2003; McIntosh et al. 2003; Forbis et al. 2006; Spooner and Allcock 2006; Hemstrom et al. 2007).

Criticisms of using expert opinion in ecological research

There have been several criticisms of the use of expert opinion. A common complaint regarding expert elicitation revolves around its subjectivity (Crome et al. 1996; Dennis 1996). By utilizing subjective opinion in the formation of models, it is possible that those people expressing their opinions may knowingly provide a more extreme answer than what they believe is true in order to influence the results of a study (Dennis 1996). One way to avoid this bias is to conduct elicitations from a selection of experts in order to include different opinions (Crome et al. 1996). Another method to correct for this bias is to update expert opinions with empirical data as they become available using a Bayesian framework, which will generate a posterior distribution that reflects both the opinions and data (Ellison 2004).

Further criticisms of expert opinion relate to the cognitive techniques that experts use to recall and synthesize their knowledge; these are known as heuristics. There are four heuristics that can lead to biases in expert opinion: availability, anchoring, representativeness, and control (Morgan and Henrion 1990; Cooke 1991; Ayyub 2001). In addition, there are two other common biases in subjective estimation: the base-rate fallacy and overconfidence (Morgan and Henrion 1990; Cooke 1991; Ayyub 2001).

17 Chapter One

Availability biases judgments because people assess the frequency or probability of an event based on the ease that it is recalled (Morgan and Henrion 1990; Cooke 1991). As a result, recently experienced, dramatic, or plausible scenarios may be given a higher probability of occurrence than is accurate (Morgan and Henrion 1990), such as large wildfires or extinctions of characteristic species. However, if the expert has a great deal of appropriate personal experience, then this heuristic can be highly accurate (Morgan and Henrion 1990).

Anchoring occurs when an expert makes an initial estimate and then extrapolates from this estimate for all further estimates (Cooke 1991). Typically the adjustment from the initial value, or anchor, is inadequate and all future estimates are biased towards the anchor (Morgan and Henrion 1990). One way to account for anchoring is to elicit estimates for the best and worst case scenarios or the maximum and minimum probability before asking the subject to estimate the central or best estimate (Morgan and Henrion 1990).

Another heuristic that leads to biases is representativeness. This bias affects conditional probability assessments because subjects typically estimate that the probability of two events occurring is based on the similarity between the two events (Cooke 1991; Ayyub 2001). Subjects also bias probability assessments when they believe they have control over the outcome of the situation (Cooke 1991; Ayyub 2001).

The base-rate fallacy is another bias in subjective estimates. It occurs when a subject relies on recent information and ignores the historic rates for an event (Cooke 1991; Ayyub 2001). Similar to the availability heuristic, if a subject has extensive experience in the field in question, then the base-rate is often integrated into their subjective estimate (Cooke 1991). It is also possible to correct for this by updating the inaccurate subjective estimate based on data showing the historical values for the parameter (Ayyub 2001).

The final, and possibly most pervasive, bias in subjective estimate is overconfidence. Often, experts will estimate intervals to be much narrower than their true values (Ayyub 2001; Teigen and Jørgensen 2005). This is problematic because it underestimates the range of possible outcomes. To help alleviate this problem, it is possible to adjust

18 Chapter One subjects’ responses by providing feedback during interviews (Ayyub 2001). Several studies have also investigated the ability to use statistical analyses to correct for overconfidence (Soll and Klayman 2004; Teigen and Jørgensen 2005).

There are two other challenges to using expert opinion. These relate to imperfect knowledge and differences between experts. When eliciting information from experts, there is an expectation that experts will have imperfect or partial knowledge relating to the parameter of interest. This is considered within-expert uncertainty and arises when the true value of a parameter may exist, for example the mean fire return interval for an ecosystem, but the expert is unsure of its value. To accommodate for within-expert uncertainty, it is recommended that experts provide a range of possible values.

Another disadvantage of using expert opinion is that experts will often disagree (e.g. Morgan et al. 2001; Pellikka et al. 2005). Disagreement between experts may reflect a current lack of knowledge regarding flora and fauna responses and ecosystem responses. Additionally, disagreement may indicate complex or site-specific responses (Martin et al. 2005). Often studies rely on methods to reduce disagreements between experts, such as forced consensus and Delphi methods (Ayyub 2001; Burgman 2005). However, forced consensus estimates are susceptible to psychological frailties, including when participants dominate discussions and try to convince others of their opinions, when participants hold opinions that they will not explain, and when participants feel pressured to conform to the group majority (Clemen and Winkler 1999; Burgman 2005). Moreover forced consensus methods can lead to over-confidence (Cooke 1991). As a result, in addition to ignoring differences between experts, forced consensus can lead to incorrect estimates. The incorporation of these two types of uncertainty, as well as other common sources of uncertainty, into models is considered in the following section.

1.2.4 Incorporating uncertainty into forest simulation models

Uncertainty is inherent in forest management and restoration (McCarthy and Burgman 1995; Ducey 2001; Kangas and Kangas 2004), and there have been various attempts to identify and classify sources of uncertainty (e.g. Morgan and Henrion 1990; Burgman

19 Chapter One et al. 2001; Regan et al. 2002). An intuitive classification scheme by Regan et al. (2002) segregates sources of uncertainty into two categories: linguistic (relating to language) and epistemic (relating to determinate facts), which will be discussed in detail below.

Linguistic uncertainty

Linguistic uncertainty is the result of vague, context-specific, ambiguous, underspecific, or indeterminate vocabulary (Regan et al. 2002). Vague language leads to borderline cases. For example, according to criterion D of the IUCN classification (IUCN 2001), a species is “endangered” if the population has 50 mature individuals, but is considered “critically endangered” if the species has 49 mature individuals. It is unsatisfactory for a taxon to receive substantially more resources based on one individual; resources which could include the time and money spent on establishing and maintaining forest conservation areas. Vagueness can be dealt by employing fuzzy sets, which use a range of values, such as 40-60 mature individuals, to acknowledge uncertainty in definitions (Ayyub 2001).

Context dependence is a failure to specify the context. For example, when a species is said to have a “large” range, it is unknown if the range is large for bears, which travel many hundreds of kilometres (Nilsen et al. 2009) or is large for ants, which likely only travel several metres in search of food (LeBrun et al. 2007). The uncertainty associated with this example could have substantial impacts on allocating land to forest reserves, but can be alleviated by identifying the context, such as “the species has a large range for a vertebrate carnivore.”

Ambiguity arises when words have multiple meanings. For example, in many forest vegetation quality assessments, it is necessary to measure plant “cover” (Parkes et al. 2003), but “cover” can refer to projective foliage cover, which is the proportion of ground covered by aerial, photosynthetic parts of the plant excluding gaps in the canopy (Kershaw 1964), or canopy cover, which is the area encompassed by the perimeter of the shrub or tree crown including gaps in the canopy (Philip and Blyth 1994). These two measures of cover will have very different values for the same individual plant. This source of uncertainty can be corrected by being very clear about

20 Chapter One the sense in which the word is intended, such as defining the cover assessment should be foliage projective cover in a particular vegetation quality assessment.

Underspecificity occurs when there is unwanted generality. This could occur if a researcher were using historical survey records to model the distribution of a now- endangered species prior to forest land-clearing, but when the record was collected it was appropriate to imprecisely specify “east of Melbourne.” This source of uncertainty can only be dealt with by providing the narrowest possible bounds on estimates, which in this example would mean that any new survey locations should be specified according to a Global Positioning System with accuracy to a few metres.

Finally, indeterminacy arises when the meanings of words change through time. For example, if a species undergoes a taxonomic revision that splits a single species into several different species, including a new rare species, there will be indeterminacy in the usage of the species name before the split. As a result, it will be difficult to determine details about the locations or sizes of the rare species’ populations prior to the taxonomic revision. This source of uncertainty can only be dealt with by making conscious decisions about the future usage of terms, which is a very nontrivial issue (Regan et al. 2002).

Epistemic uncertainty

The second major source of uncertainty is epistemic uncertainty, which involves uncertainty in the knowledge of the state of a system (Regan et al. 2002). It includes six main types: inherent randomness, measurement error, systematic error, natural variation, model uncertainty, and subjective judgement. Inherent randomness occurs when a system cannot be reduced to a deterministic one. This source of uncertainty may be rare in forest ecology; complex systems like weather patterns and community assembly are not inherently random, but are, for convention, considered inherently random because the deterministic processes controlling them are not entirely described.

Measurement error arises due to imperfections in measuring equipment and operator error, and causes apparent random variation in the measurement. This source of

21 Chapter One uncertainty can be dealt with by reporting measurement bounds. For example, regrowth tree heights are often measured after harvesting to assess site productivity (Ryan and Yoder 1997). There will be uncertainty in trees’ true heights due to measurement error. To deal with this source of uncertainty when using laser range finders, for example, each instrument typically comes with accuracy specifications, i.e. ± 1.5 mm.

Systematic error arises from biases in the sampling procedure or measuring equipment, such as erroneous calibration of equipment, and can only be corrected for by identifying the bias and removing it. This might occur over long-term forest monitoring, where the measured diameter of stems appears to decrease over time because measuring tapes stretch over repeated use and provide underestimates. This source of uncertainty can be alleviated by calibrating the measuring tape to an accurate measurement and adjusting data accordingly.

The other three sources of epistemic uncertainty, natural variation, model uncertainty, and subjective judgement, are of particular importance to simulation modelling and will be discussed in more detail in the context of state-and-transition simulation models (STSMs).

Natural variation arises because ecological systems and processes change stochastically in space and time. Specifically, the occurrence of disturbances in time and space is random and unknown, and the true rates of disturbances change through time (McCarthy and Burgman 1995). In STSMs, the incorporation of natural variation will depend on the software used. The Vegetation Development Dynamics Tool software program (VDDT, ESSA Technologies Ltd. 2007) has been used extensively to model forest management and restoration (e.g. Kurz et al. 1999; Merzenich et al. 2003; Forbis et al. 2006; Hemstrom et al. 2007). VDDT incorporates natural variation by applying Monte Carlo methods that randomize the order of ecological processes, such as wildfires, insect attack, and windthrow, through time. In this way, VDDT simulates the stochastic occurrence of disturbances through time. In addition, it is possible to program broad inter-annual variation in processes by specifying that a certain proportion of years in a simulation will have, for example, low, normal, or high probabilities of disturbance (e.g. Merzenich et al. 2003; Forbis et al. 2006). However,

22 Chapter One in VDDT, disturbance probabilities are fixed values (e.g. Hemstrom et al. 2007; Wales et al. 2007); there are no known examples of STSMs in the literature that give a range of probability values to disturbances in order to reflect how parameter values change through time. As a result, there is a need to investigate how incorporating variation in parameter values will affect model predictions.

Another important source of uncertainty in forest modelling is model uncertainty. Model uncertainty arises when there are multiple plausible model structures (Wintle et al. 2003) and when the effects of initial model conditions on predictions are unknown (e.g. Fan et al. 2000; Keane et al. 2002). Model uncertainty is essentially impossible to eliminate, but it is possible to test how appropriate a model is for prediction through validation studies (Regan et al. 2002). STSM validation is difficult and sometimes impossible to accomplish (e.g. Bellamy and Lowes 1999). However, those studies able to validate model predictions have found that their models were reasonably accurate and could be used to formulate broad restoration objectives (Shlisky et al. 2005). There are, however, some aspects of model uncertainty that have not previously been incorporated in STSMs, such as multiple plausible models and the effects of initial conditions. When transition agents, i.e. processes that cause transitions between vegetation states, are identified, they are typically specified to cause a single transition from one state to another (e.g. McIntosh et al. 2003; Vavra et al. 2007). In poorly studied systems, there may be uncertainty in the effects of processes, such that a single transition agent might have multiple plausible effects. In addition, there are no known previous studies that investigate the effects of different initial conditions on model dynamics and predictions. These two sources of uncertainty may substantially affect model predictions and, as a result, require further investigation.

The final source of epistemic uncertainty is due to subjective judgement, or expert opinion (Regan et al. 2002). As noted earlier in this review, there are often insufficient empirical data to describe systems and, consequently, researchers use expert opinion to predict biological responses (e.g. Morgan et al. 2001; Martin et al. 2005) and parameterize models (e.g. Plant and Vayssieres 2000; Yamada et al. 2003; Forbis et al. 2006). While expert opinion is based on observations and experience, uncertainty arises as a result of interpreting data (Regan et al. 2002). Because experts observe different things and interpret data differently, there are often differences in opinions between

23 Chapter One experts, which leads to between-expert uncertainty (e.g. Alho and Kangas 1997; Morgan et al. 2001; Martin et al. 2005; Pellikka et al. 2005). In addition, because cognitive heuristics lead to errors in recalling information and because it is unlikely that one expert will have a complete understanding of a system, experts possess partial or imperfect knowledge; this leads to within-expert uncertainty.

In STSMs that utilize expert opinion, there has been little acknowledgement of uncertainty due to subjective judgement. Researchers typically ignore between-expert uncertainty by relying on forced consensus with groups of experts (Burgman 2005) to provide one model of transition agents’ probabilities and how transition agents will affect vegetation states (e.g. Forbis et al. 2006; Hemstrom et al. 2007; Vavra et al. 2007). As there are many disadvantages in using forced consensus methods, it is unsatisfactory to ignore variation between experts. To incorporate between-expert uncertainty, Pellikka et al. (2005) created a Bayesian Belief Network for each expert to predict the effects of game management on future populations. This approach could also be applied to STSMs, although it has not yet been attempted. In terms of within- expert uncertainty, there has been no previous research using STSMs based on subjective knowledge that acknowledges imperfect expert knowledge. Studies using expert-based STSMs present only the average value for parameters, which does not account for within-expert uncertainty (Forbis et al. 2006). In order to incorporate within-expert uncertainty, researchers can ask experts to provide minimum and maximum bounds on parameter estimates (Walker et al. 2001; Regan et al. 2002).

In summary, there are several sources of uncertainty that have not previously been acknowledged or incorporated into STSMs. These include natural variation in the rates of natural processes, the effects of initial conditions, between-expert uncertainty, and within-expert uncertainty. Further research using STSMs could attempt to incorporate these sources of uncertainty to determine their effects on model predictions. However, by incorporating uncertainty into models, it is likely that the predicted effects of management actions will also be uncertain. If managers are attempting to make restoration and management decisions based on uncertain model predictions, there will be some difficulty in selecting restoration actions because their exact effects on vegetation will be unknown. As a result, managers are advised to rely on an explicit

24 Chapter One management framework to decide between uncertain model predictions (Shea et al. 1998; Possingham et al. 2001).

1.2.5 Decision-making and adaptive management

Managers must often decide between multiple forest restoration or management actions available for implementation (Alho and Kangas 1997; Rüger et al. 2007; Boerner et al. 2008). Because the effects of these actions on the system in question are uncertain, it is wise to carry out decision-making using a formal decision analysis framework (Shea et al. 1998; Peterman and Anderson 1999; Prato 2000). Decision analysis tools were primarily designed for use in economics, but are becoming more commonly applied in ecological management research (e.g. Drechsler and Burgman 2004; Halpern et al. 2006). Various different decision analysis methods have already been applied to make forest management decisions under uncertainty, including the Dempster-Shafer theory of evidence (e.g. Ducey 2001), Bayesian Belief Networks (e.g. Bacon et al. 2002; Ames et al. 2005), Information-gap theory (McCarthy and Lindenmayer 2007), decision trees (e.g. Moser et al. 2003; Olofsson and Blennow 2005), and adaptive management (e.g. Dorazio and Johnson 2003; Chazdon 2008). While it is beyond the scope of this chapter to review the history and applications of all structured decision analysis frameworks, I will review the elements common to most decision analysis frameworks using adaptive management as an example because it shows considerable promise as a method to reduce uncertainty in management and improve decision-making through time (Holling 1978; Johnson 1999).

Adaptive management (AM) is a structured, iterative decision analysis process that aims to learn about and reduce uncertainty in management actions through time (Holling 1978; Nichols and Williams 2006). This is done by integrating models that predict the effects of management actions with monitoring data collected over time and adjusting models to reflect new knowledge (e.g. Nichols et al. 2007). AM is beneficial because it facilitates learning and provides transparency and accountability to decision-making (Duncan and Wintle 2008). As with any structured decision analysis framework (Peterman and Anderson 1999; Possingham et al. 2001), there are a selection of steps required to carry out AM. These steps include describing explicit objectives, identifying

25 Chapter One potential management actions, constructing models the describe the response of the system to alternative management actions, assigning measures of confidence in each model, and commencing a monitoring program to inform management of the system state (Holling 1978; Nichols and Williams 2006). These steps will be considered using the study of white pine (Pinus strobus) forest management in Minnesota (Tester et al. 1997).

The first step in adaptive management is to define explicit management objectives. Objectives are often social judgments based on the values of the community or stakeholders involved (Nichols and Williams 2006). Management objectives must be clear, unambiguous, and measurable in order to select between management alternatives and gauge management performance (Nichols et al. 1995; Peterman and Anderson 1999; Possingham et al. 2001). Unfortunately, objectives are often vague (Peterman and Anderson 1999) and the process of defining and clarifying objectives is not trivial (Shea et al. 1998), which occurs in when there are multiple stakeholders with differing views (Kangas and Kangas 2004). Tester et al. (1997) began their research into white pine management by identifying the management challenge; old-growth white pine forests were dominant prior to European settlement and are currently a major attraction in state parks. However, due to reduced frequencies of fires and increased herbivore browsing, white pines are not regenerating and are being permanently replaced by hardwoods. To deal with this challenge the researchers set a simple, but clear management objective: to maintain white pine forests.

The second step in AM is the specification of management options. This step involves both social values and scientific input (Nichols and Williams 2006; Duncan and Wintle 2008). Managers are encouraged to consider the full range of possible actions (Peterman and Anderson 1999) and ensure they are dissimilar enough to produce discernable differences in the system being studied (Nichols et al. 1995). Often it is not possible to find mutually exclusive management actions and, out of necessity, actions will be selected because they are economically feasible rather than because they reflect the full range of possible actions. In their study of white pine forests, Tester et al. (1997) identified several management options, including various combinations of prescribed ground fires, natural wildfire suppression, and clear cutting as possible ways to maintain white pine forests.

26 Chapter One

The third step in AM involves constructing models to predict the effects of management actions. All currently available knowledge can be incorporated into these models and, as a result, uncertainty in the models will reflect uncertainty in the appropriateness of alternative management actions (Nichols et al. 1995). Uncertainty in models may also reflect different views about how species will respond to management (Duncan and Wintle 2008). Tester et al. (1997) constructed frame-based simulation models to predict how seedling recruitment, tree growth, competition, herbivory, fires, wind, and alternative management actions would affect development and maintenance of white pine forests.

Collecting monitoring data and attributing confidence or weights to each model are the final steps of AM. These two steps form an iterative feedback loop where information collected from monitoring is directly applied to model predictions to learn about management and to weight more plausible models (Holling 1978). The collection of data that reflect the effects of management can be used to estimate system states (Nichols and Williams 2006) and discern between management actions (Duncan and Wintle 2008). Without this information, it would not be possible to learn about uncertain management actions (Nichols and Williams 2006), which is why it is important that monitoring programs collect data that can be used to compare competing models (Shea et al. 1998). Unfortunately, monitoring can be costly and managers may wish to spend money on implementing management actions rather than collecting data (Duncan and Wintle 2008), such that managers will have to decide how to allocate monitoring to optimize cost versus learning (Hauser et al. 2006). In addition, monitoring data are often collected because they are available and convenient, rather than because they are being used to assess management models. It is also difficult to collect appropriate data in forest management due to the slow growth of species, which means that AM is a very long-term learning process. While Tester et al. (1997) did not collect monitoring data to validate or update models over time, they were able to compare model results to forest sites, which provided confidence in the models’ results.

While AM provides a clear framework to make management decisions, there are disadvantages to this method, some of which have been touched upon already. Foremost, there are still few examples of AM (Duncan and Wintle 2008), even though

27 Chapter One it has received much attention as a conceptual model of management for many years (Lee 1999). This is because each step of AM is non-trivial. For example, selecting appropriate models can be problematic and may require significant effort to construct (Nichols et al. 1995). In addition, AM is time-consuming and expensive due to the need to collect monitoring data, and can provide inaccurate results when important parameters are ignored (Prato 2000). Furthermore, if a manager is constrained in terms of time and money, it may be difficult to encourage the large-scale shift in policy required to allocate limited resources to a process of long-term learning (Walters and Holling 1990), rather than to management actions themselves.

While these disadvantages of AM may be substantial, there is a need to apply structured decision-making protocols in order to make optimal decisions under uncertainty (Possingham et al. 2001). Furthermore, in the case study of the Box-Ironbark forests and woodlands of Victoria, managers are attempting to implement an adaptive management framework to learn about the potential benefits of restoration actions (Parks Victoria 2007). As a result, when STSMs with uncertain predictions about the effects of alternative management actions are available, it would be useful to begin by applying the first three steps of AM to model predictions - describing management objectives, choosing management actions, and constructing models to differentiate between actions - in order to decide between management actions.

1.2.6 Conclusion

Forest simulation models are a vital tool for modelling the complex effects of restoration actions. In particular, state-and-transition simulation models (STSMs) have a long history of predicting the effects of alternative management actions in support of restoration. Many STSMs constructed to assist forest management were parameterized with expert opinion because empirical ecological data have been lacking or unavailable. And, while some sources of uncertainty have previously been incorporated into STSMs, there are some sources that have not yet been considered. These include variation in the rates of natural processes, the effects of initial conditions, within-expert uncertainty, and between-expert uncertainty. Once uncertainty has been incorporated into STSMs, it becomes problematic to decide between management actions because any given

28 Chapter One action has multiple potential outcomes. To deal with this, management decisions can be applied using a formal decision analysis framework. It is the aim of this study to construct STSMs using expert opinion in order to predict the ability of management actions to restore a degraded forest system, the Box-Ironbark forests and woodlands of Victoria, Australia. Uncertainty due to natural variation, the effects of initial conditions, imperfect expert knowledge, and disagreements between experts will be incorporated into STSMs and their effects on model predictions will be investigated. Finally, I will apply steps of a formal decision analysis framework, adaptive management, and explicit management objectives to uncertain model predictions to investigate the ability to decide between management actions.

29 30

Stand models Tree models

Distance dependent Distance independent Distance dependent Distance independent

- spatial STSMs - SORTIE (Pacala et al., 1993)

Average tree models Distribution models Non-gap models Gap models - stand yield tables - matrix models - PROGNOSIS - JABOWA - Markov models (Bare and Opalack, 1987) (Botkin et al., 1972) - non-spatial STSMs - FORET (Shugart and West, 1977) - ZELIG (Smith and Urban, 1988) Chapter One

Fig. 1.1. Classification of forest growth models. Adapted from the classification of Porté and Bartelink (2002). CHAPTER TWO

Incorporating between-expert uncertainty into state-and- transition simulation models for forest restoration

32 Chapter Two

2.1 Abstract

Forest utilization has the potential to change vegetation structure and detrimentally affect flora and fauna. Managers may wish to restore such degraded forests with the assistance of simulation models. Forest simulation models can be used to predict the effects of alternative management actions on vegetation structure in an effort to optimize restoration. However, the lack of empirical ecological data to parameterize models often necessitates the use of expert opinion. Differences in opinion between experts may be large and can lead to uncertainty in model predictions, but this is rarely acknowledged in forest simulation models. In this paper, I constructed state-and- transition simulation models (STSMs) based on expert opinion to predict whether management will expedite the development of mature stands with large, old trees – a habitat resource considered critical to forest-dependent biodiversity – in the Box- Ironbark forests and woodlands of Victoria, Australia. The three candidate management actions are modified timber harvesting regulations, establishing no-cutting National Parks, or a new ecological thinning technique. I also investigated the importance of uncertainty caused by differences between experts when constructing STSMs. Model results predicted that mature woodlands with large trees will develop slowly. Also, mature woodlands will likely not develop sooner with either ecological thinning or modified harvesting, which may lead to a loss of desired vegetation. In addition, differences in model predictions caused by between-expert variation are substantial and can be used to identify biological processes requiring better understanding in this ecosystem.

33 Chapter Two

2.2 Introduction

Forest utilization can have considerable impacts on vegetation structure and habitat quality, which may consequently lead to detrimental effects on flora and fauna (Scherer et al. 2000; Rüger et al. 2007). Forest restoration and management can assist in recovering impaired ecosystems (McIver and Starr 2001), but effective restoration requires a good understanding of the system being managed. Forest simulation models may assist restoration by predicting the potential effects of alternative management actions on vegetation structure over time (e.g. Hansen et al. 1993; Pausas et al. 1997; Perry and Enright 2002b). While ecological models are abstractions of very complex systems, they provide an explicit and examinable framework for assessing the outcomes and goals of restoration and management (Starfield and Bleloch 1991).

The lack of comprehensive ecological data to identify important drivers of stand structure change and estimate parameter values is a common problem in ecosystem modelling (Regan et al. 2002; Forbis et al. 2006). This frequently occurs because the collection of data is costly (Starfield and Bleloch 1991) and managers may prefer to spend time and money on management actions rather than collecting monitoring data (Duncan and Wintle 2008). In addition, collecting data may not be possible if appropriate sites are not currently available (Starfield and Bleloch 1991). Typically, there is also a paucity of data reflecting conditions prior to disturbances or anthropogenic changes. When there are limited historical data, expert opinion is needed to construct ecological models of system responses to management actions (e.g. Kangas et al. 2000; Pellikka et al. 2005; Forbis et al. 2006).

While ecological models are inherently uncertain, constructing models with expert knowledge contributes to model uncertainty (Ducey 2001; Regan et al. 2002; Kangas and Kangas 2004) because experts will often have conflicting ideas about how ecosystems respond to natural processes and management (Morgan et al. 2001). This type of uncertainty must be transparently accounted for because it can affect model predictions (e.g. Pellikka et al. 2005), and can lead to increased chances of making the wrong management decisions. The majority of studies using expert knowledge to parameterize ecological simulation models have not addressed this form of uncertainty.

34 Chapter Two

Instead, data are often elicited from one or few experts or are assembled using a forced consensus method, which does not account for large uncertainty arising from disagreements between experts (Morgan and Henrion 1990; Cooke 1991). While there are examples of studies that acknowledge differences between experts (e.g. Morgan et al. 2001; Martin et al. 2005), they are typically restricted to studies of ecosystem responses rather than simulation models. In this chapter I explore the effects of variation between experts on simulation models for restoration of a degraded forest ecosystem; the Box- Ironbark forests and woodlands of Victoria, Australia.

Box-Ironbark forests and woodlands have a history of utilization dating from the 1830s to the present day (Newman 1961; Environment Conservation Council 2001). Historically, utilization consisted of mining and timber harvesting primarily for railway construction and urban development (Kellas 1991). The most intensive forest utilization occurred between approximately 1850 and 1920 (Forests Commission of Victoria 1928; Kellas 1991), but these actions still continue on a smaller scale in many of Victoria’s Box-Ironbark forests and woodlands (Department of Natural Resources and Environment 1998; Environment Conservation Council 2001). Pre-settlement old- growth forests and woodlands are thought to be relatively open, grassy systems with at least 30 large (i.e. with a diameter at breast height or DBH of 120-150 cm), hollow- bearing trees per hectare (Newman 1961; Kellas 1991). In contrast, as a result of timber harvesting, these forests and woodlands now have a relatively homogeneous structure (Bennett et al. 1994) dominated by dense stands of small coppice stems (Environment Conservation Council 2001). Current stands also typically lack old-growth characteristics such as large, hollow-bearing trees (Kellas 1991; Environment Conservation Council 2001), which are important for many Box-Ironbark fauna species (Alexander 1997; Soderquist 1999). These changes in vegetation structure and habitat have contributed to either regional or national extinction of several fauna and flora species (Muir et al. 1995; Environment Conservation Council 2001) and caused population declines in several other species (Calder and Calder 1994; Alexander 1997).

Restoration activities are required for Victoria’s Box-Ironbark forests and woodlands to help mitigate further decline or loss of species (Alexander 1997; Soderquist 1999). In 1996, the Victorian State Government initiated an investigation into the extent, condition, and potential uses of Box-Ironbark forests and woodlands, with the hopes of

35 Chapter Two making recommendations for the balanced use of these areas (Environment Conservation Council 2001). At roughly the same time that this investigation began, timber harvesting regulations for this area were modified to align forest practices with conservation concerns by specifying that hollow-bearing and large trees >60 cm DBH must be retained and that higher densities of medium-sized trees must be left for recruitment to larger size classes (Department of Natural Resources and Environment 1998; Sutton 2000). After the Box-Ironbark forests and woodlands investigation was complete, one of the first restoration actions initiated by the Victorian State Government in 2002 was the assignment of various Box-Ironbark forests and woodlands to national parks and other conservation areas (Parks Victoria 2002, 2003). In 2003, the state authority responsible for management of these conservation areas, Parks Victoria, initiated a second recommendation; an Ecological Thinning Trial (Environment Conservation Council 2001; Parks Victoria 2004). The trial aimed to modify the stem density within Box-Ironbark forests and woodlands to resemble the density of an older stand, which would enable an increase in the growth rates of remaining trees and accelerate the development of large trees (Environment Conservation Council 2001). It was anticipated that this would enhance the habitat value of these stands (Bennett 2002).

Due to the slow growth of overstory Eucalyptus species found in these ecosystems (Department of Natural Resources and Environment 1998) and the lack of historical data, the effects of the altered timber harvesting regulations and ecological thinning on vegetation structure are not likely to be conclusive within the next 30-40 years (Parks Victoria 2007). In addition, there are no previous examples of simulation models that describe the effects of management actions and natural disturbances on vegetation structure in this ecosystem. Due to the uncertainty in vegetation dynamics, I constructed forest simulation models that predict how the structure of overstory Eucalyptus trees will change as a result of natural disturbances, newly altered timber harvesting regulations, and ecological thinning. In addition, due to the lack of data regarding ecosystem dynamics, I relied on expert opinion to parameterize models.

State-and-transition simulation models (STSMs) are particularly useful for predicting the dynamics of very uncertain systems and incorporating expert knowledge because they are relatively straightforward to parameterize. STSMs function by segregating the

36 Chapter Two landscape into states based on vegetation structure. These states are linked by various transition agents, each with annual probabilities of occurrence that indicate the effects of vegetation development, natural disturbances, and management actions on vegetation states (Hemstrom et al. 2007). By modelling multiple simulations with different transition agents representing alternative management actions, STSMs are ideal for predicting future vegetation structure resulting from alternative management strategies (e.g. Wales et al. 2007; Wondzell et al. 2007a). STSMs incorporating expert opinion have been used extensively to predict changes in vegetation structure in response to management actions, natural disturbances, and growth (e.g. Plant and Vayssieres 2000; Hemstrom et al. 2002; Forbis et al. 2006). However, when experts were consulted for STSM parameterization in previous studies, data were compiled using consensus methods and differences between experts were not acknowledged (e.g. Forbis et al. 2006; Hemstrom et al. 2007; Vavra et al. 2007).

The purpose of this chapter is to investigate the validity of undertaking ecological thinning or new timber harvesting techniques to accelerate the development of old- growth forests and woodlands, in contrast to letting stands develop without timber harvesting in National Parks and conservation reserves. I constructed a series of state- and-transition simulation models (STSMs) that predict the effects of these alternative management actions on the long-term structural dynamics of Victoria, Australia’s, Box- Ironbark forests and woodlands. Given the extent by which this vegetation type has been reduced and altered as a result of a long history of utilization, there is a lack of appropriate data to parameterize models. Consequently, I use data elicited from experts to parameterize models. In addition, by incorporating uncertainty arising from differing opinions between experts, I could determine how between-expert variation affects model predictions.

2.2.1 Study area

The analyses focus on the 250,000 ha of Box-Ironbark forests and woodlands that currently remain in Victoria, Australia (Fig. 2.1). These forests and woodlands are dominated by Box and Ironbark eucalypt species, including Grey Box (Eucalyptus microcarpa Maiden), Red Box (E. polyanthemos Schauer), Yellow Box (E. melliodora A. Cunn. ex Shauer), Red Ironbark (E. tricarpa (L.A.S. Johnson) L.A.S. Johnson and K.D.

37 Chapter Two

Hill), and White Ironbark/Yellow Gum (E. leucoxylon F. Muell.)(Kellas 1991). The Box-Ironbark region spans an elevation range of 150 m to 400 m, extending from the low hills of the Great Dividing Range to the Northern Plains (Muir et al. 1995). This area experiences annual rainfall averaging between 380 to 510 mm (Kellas 1991). The region contains 25 floristic communities (Muir et al. 1995), which incorporate both forests and woodlands, but hereafter will collectively be referred to as woodlands for ease. Ordivician sedimentary rocks and granite outcrops are widespread in the area (Muir et al. 1995), and woodlands typically grow on sandy to clay-loam soils, low in organic matter, phosphorus, sulphur, and nitrogen relative to adjacent agricultural areas (Kellas 1991).

2.3 Methods

I investigated the effects of natural disturbances and management on overstory vegetation structure using a state-and-transition modelling framework. Under this framework, vegetation structure is used to define discrete states, which are connected by transition agents, such as growth, natural disturbances, or management actions, which cause transitions between vegetation states (Westoby et al. 1989). State-and-transition simulation models (STSMs) are built upon transition matrix methods, where vegetation development is described using a set of transition agents between various vegetation states (Hemstrom et al. 2007), and resemble simple Markov models, which utilize a set of discrete states, transition probabilities, and initial conditions to predict the future state of a given ecosystem (Scanlan 1994).

I used the Vegetation Development Dynamics Tool (VDDT) to implement STSMs (ESSA Technologies Ltd.). This software has been used extensively by managers and researchers to predict the effects of natural disturbances and management actions on vegetation structure and composition over time (e.g. Kurz et al. 1999; Beukema and Pinkham 2001; Merzenich and Frid 2005; Wales et al. 2007). VDDT is a non-spatial model that allocates independent cells, which represent 1 hectare of forest, into states based on vegetation structure. These states are linked through multiple transition agents, each with a specified annual probability of occurrence. Transition agents can be deterministic, such as growth in the absence of other disturbances, or stochastic, such as

38 Chapter Two a wildfire, insect outbreak, or grazing. Transition agents can be programmed to cause a transition only if a cell is of a certain age (i.e. a cell must stay in a state for a specified number of timesteps (years) in order for an agent to cause a transition). Transition agents can also be programmed to alter the age of a cell, such that growth is delayed or progressed. Age in this context refers to the number of timesteps a cell has remained in the same state (ESSA Technologies Ltd. 2007).

Natural stochasticity arises because disturbances occur randomly in space and time (e.g. McCarthy and Burgman 1995) and their rates vary through time (e.g. Raulier et al. 2003; Kangas and Kangas 2004). VDDT incorporates natural stochasticity by performing Monte Carlo simulations. Monte Carlo methods have been used extensively in ecological studies to account for uncertainty (e.g. Kangas 1998; O'Hara et al. 2002) by randomly sampling parameter values from a specified distribution (Vose 1996). In VDDT, natural stochasticity is mimicked by varying the order that transitions occur over multiple Monte Carlo simulations (Fig. 2.2). The list of possible transition agents with their attendant probabilities is specified according to a cell’s current vegetation state. At one timestep and for a given Monte Carlo simulation, VDDT randomizes the list of possible transition agents and arrays their probabilities consecutively from 0, such that each probability covers a distinct interval. VDDT then sums the probabilities of the possible transition agents and draws a random number from a uniform distribution between 0 and 1. If that random number is greater than the summed probabilities of transition agents possible in that state, then no transition occurs. If the random number is less than the summed probabilities, then the transition agent whose distinct probability interval corresponds to the random number is selected and a transition occurs. Cells’ ages are also adjusted if specified by the transition agent (ESSA Technologies Ltd. 2007). With this process, no more than one transition agent may be selected for a cell at one timestep. In addition, only a portion of natural stochasticity is modelled because the order of transitions varies randomly between Monte Carlo simulations but the parameter values do not change between Monte Carlo simulations – i.e. stochastic ordering in space and time is incorporated, but stochasticity in the frequency or rates of disturbances is not included.

39 Chapter Two

The probability distribution that describes the number of cells in a given state that undergo transitions at one timestep is the binomial distribution (Equation 2.1)(McCarthy 2007):

n! kxP )( == k − pp −kn ,)1( − knk )!(! (2.1) where n is the total number of cells, p is the probability of a transition, and k is the observed number of cells undergoing a transition (ESSA Technologies Ltd. 2007). The mean, μ , of this distribution is np and the variance, σ 2 , is np(1-p).

2.3.1 Defining vegetation states and transition agents

All data were in the form of elicited expert opinion and were collected with the use of electronic surveys and structured elicitation interviews. I adapted interview procedures from Morgan and Henrion’s (1990) interview protocol. Experts included scientists with experience in Box-Ironbark wildlife, landscape, fire, and/or plant ecologies and managers with experience in Box-Ironbark fire and forest management. I located participants through literature searches, Parks Victoria contacts, and recommendations from other researchers / participants. A full transcript of the elicitation procedure can be found in Appendix A.

All experts initially completed an electronic survey to determine which vegetation resources would be used to define vegetation states. Experts ranked habitat resources provided by vegetation according to their importance as elements of a healthy forest and experts specified the quantity of each resource they felt was required in a healthy forest. Forests were considered “healthy” if they were a functioning and biodiverse ecosystem. I focused on resources provided by vegetation because vegetation forms habitat – i.e. shelter, nesting sites, and food - for many vertebrate and arthropod species, and the aim of Box-Ironbark restoration is to enhance habitat quality through altering vegetation structure. Experts were given a list of eight vegetation resources: downed wood/coarse woody debris, coppice stems, fine litter, ground/understory flora, large overstory trees, small overstory trees, standing dead overstory trees, and shrubs. I chose these eight

40 Chapter Two vegetation components after reviewing the literature and consulting with ecologists and Parks Victoria managers. These components often provide vital habitat resources for fauna and are important indicators of forest health (e.g. Alexander 1997; Soderquist 1999; Soderquist and Mac Nally 2000; Brown 2001; Mac Nally et al. 2001). In addition to the eight resources I specified, several blank spaces were provided for experts to nominate supplementary vegetation structures.

After experts completed electronic surveys, I carried out individual elicitation interviews to parameterize STSMs. I provided experts with a structurally identical STSM based on the same vegetation states and a set of possible transition agents (Table 2.1). Transition agents were selected through preliminary interviews with experts. Experts estimated how each transition agent would change vegetation states and the probability or rate of occurrence of each transition agent. They also specified the proportion of vegetation that would undergo a transition or the probability that the state change would occur if the transition agent occurred. If an expert did not believe that a transition agent would lead to a transition, it was not included in that particular expert’s model. Occasionally experts felt it was pertinent to specify combinations of transition agents, due to the likely covariance of stochastic events, such as the effect of an intense overstory wildfire following a windthrow event.

Experts’ estimates for wildfire return intervals were validated with geographic information system (GIS) data collected by the State Government between 1980 and 2006. I used a binomial distribution to estimate the annual probability of fire. Fire return intervals and intensity vary substantially between ridges and gullies or streambeds (Tolsma et al. 2007), but because vegetation states were grouped regardless of topography, this probability remained constant for all states. Probabilities for all other natural disturbances were based solely on expert opinion.

During the interviews, I also asked experts to estimate the proportion of land currently in each state for the initial conditions of VDDT models. I did not validate the proportion of land in each state with data due to the limited availability of data and time restrictions. After the interviews, I undertook follow-up discussions with all experts in an attempt to elicit estimates, correct models, and/or ensure model specification correctly reflected experts’ conceptual models.

41 Chapter Two

2.3.2 Disturbance and management scenarios

I modelled long-term changes in vegetation structure based on three management alternatives: a Natural Disturbance scenario, a Current Harvesting scenario, and an Ecological Thinning scenario. Differences between scenarios were based on which transition agents occurred in each scenario (Table 2.1). I modelled a constant management approach; areas were allocated to a specific scenario throughout the simulation and did not move between management scenarios.

The Natural Disturbance scenario included all natural disturbances specified by an expert, but did not include any timber harvesting, ecological thinning, or fuel reduction burns. This scenario models what may happen if the tenure of an area were changed to a national park or conservation reserve and left to develop under current climate conditions. I did not simulate potential effects of climate change in any of the scenarios.

Under the Current Harvesting scenario, I modelled the effects of natural disturbances and timber harvesting as specified by each expert. The rates of harvesting were calculated from the Box-Ironbark Timber Assessment (Department of Natural Resources and Environment 1998) and the Box-Ironbark Forests and Woodlands Investigation (Environment Conservation Council 2001). I also included fuel reduction burns in this scenario.

The Ecological Thinning scenario modelled the combined effects of natural disturbances, ecological thinning with poison, ecological thinning without poison, and fuel reduction burns. While ecological thinning is currently only a trial in Box-Ironbark woodlands, for comparison I specified that ecological thinning would occur at the same rate as timber harvesting in the Current Harvesting scenario.

2.3.3 Model construction and simulation

I constructed a STSM for each expert where the model structure was the same between experts but the annual probabilities for each transition agent and how transition agents affected state changes were based on that particular expert’s conceptual model. This was done to investigate the effects of between-expert uncertainty caused by differences in

42 Chapter Two how experts specified Box-Ironbark woodland dynamics. If experts specified transition agent estimates as annual return intervals during interviews, I converted them to annual probabilities for model construction.

I relied on VDDT’s Monte Carlo simulations to incorporate natural stochasticity. Each model produced 100 Monte Carlo simulations, which continued for 150 timesteps after preliminary simulations indicated that most model predictions stabilized within 100 years. VDDT models simulated the dynamics of 1000 independent cells, which were segregated equally into one of the three alternative management scenarios and run simultaneously. The cells were assigned a random age at the beginning of simulations to ensure that models reflected the current multi-age structure of the stands.

2.4 Results

2.4.1 Defining vegetation states

Eight experts completed the electronic survey to determine which vegetation components would be included in modelling. All experts indicated that the density of large overstory trees was the most important (5 experts) or second-most important (3 experts) habitat resource. Large overstory trees were defined as ≥60 cm DBH (6 experts) and were specified as needing to occur at densities of ≥10 stems/ha to indicate a healthy forest (5 experts). The density of smaller stems was another highly ranked habitat resource when averaged over experts. There was some disagreement in definitions between experts, but small stems were generally defined as 5-59 cm DBH with <100 stems/ha being a healthy forest. These two components with two levels were used to construct the four vegetation states: High density regrowth, Low density regrowth, High density mature, and Low density mature (Fig. 2.3). The presence of downed wood was also highly ranked, but to keep the model structure simple it was not included. The presence of large trees is generally considered as the most important indicator of a healthy forest, thus High Density mature and Low density mature are both important vegetation states, although lower densities of small trees make Low density mature the most desired vegetation state.

43 Chapter Two

To determine the proportion of each vegetation state currently present in the landscape, I averaged the estimates provided by experts during interviews. Experts agreed that current stands typically have high densities of small stems and lack large stems, particularly in relation to sites that resemble pre-European or older stands (Soderquist 1999; Venosta 2001), meaning the majority of current woodlands are High density regrowth. These initial conditions are referred to as Average initial conditions and contain 71.2% High density regrowth, 10.3% Low density regrowth, 12.5% High density mature, and 6% Low density mature.

2.4.2 Describing transition agents

During interviews, experts described how each transition agent in Table 2.1 was expected to affect vegetation states, predicted the return interval for each transition agent, and specified the probability or proportion of stands that would undergo a transition if the transition agent occurred. A schematic STSM with some of the more common transition agents can be found in Fig. 2.3, and details of all transition agents can be found in Appendix B. There was a great deal of disagreement between experts regarding how transition agents affected vegetation states and their annual probabilities, such that only a few parameters were the same between experts (Appendix B). Growth transitions were the most common between experts.

Of the eight experts interviewed, models could be constructed for five. The experts who could not be included were either uncomfortable providing the complete set of estimates required for model construction or they produced internally inconsistent models, which happened when experts specified that transition agents would result in conflicting biological processes.

Deterministic transitions for growth and coppice resprouting

Transitions resulting from growth and coppice resprouting were given a probability of 1.0, a proportion of 1.0, and occurred after a cell had stayed in a vegetation state in the absence of another transition for the number of years specified by the expert. There were two exceptions where experts specified that coppice occurred with a proportion of 0.5 or 0.7-0.8, rather than 1.0, making it a stochastic transition.

44 Chapter Two

Experts disagreed whether Box-Ironbark Eucalyptus species would self-thin substantially. Three of the five experts specified that all states would grow and self-thin to Low density mature in the absence of disturbances. The two other experts believed that Low density regrowth would grow and self-thin to Low density mature, whilst High density regrowth would mature and retain similar structure to be High density mature without substantial self-thinning.

The majority of Eucalyptus regeneration in these woodlands occurs as a result of coppice resprouting from the lignotuber (Kellas et al. 1998), which occurs intensively after most major disturbances. Natural regeneration from seed is rare in Box-Ironbark woodlands, and its causes are uncertain. Experts disagreed whether or not disturbances are required for regeneration from seed. Experts who believed that disturbances are required for regeneration from seed disagreed whether intense overstory wildfires or windthrow are needed. Intense fires provide heat to open woody fruits, an ash bed for germination, and a temporary reduction in light competition for seedlings as burnt stems resprout from epicormic buds, while windthrow provides disturbed soil for germination and reduced space and light competition for seedlings. There has been considerable discussion in the literature regarding both options (Kellas 1991; Orscheg 2006; Tolsma et al. 2007). One expert believed that the combination of an intense overstory wildfire after a windthrow event would be the only way to facilitate regeneration from seed, which is why it is rarely seen in these systems. Other experts believed that for most Box- Ironbark Eucalyptus species, seed fall is a continuous process (Tolsma et al. 2007), but seedling establishment does not occur frequently because mining has changed the topsoil and damaged the seed bank (Muir et al. 1995), or because grazing causes seedling mortality (Meers and Adams 2003).

Stochastic natural disturbances

Dodder laurel (Cassytha melantha), a native parasitic vine, was specified by one expert to potentially cause small tree mortality on a small scale, leading vegetation to change to Low density regrowth or Low density mature. Stands affected by dodder laurel were not expected to recover back to High density regrowth or High density mature, respectively, through a coppice response.

45 Chapter Two

The effects of drought on vegetation structure are highly uncertain. Some experts specified that drought would lead to mortality of trees in high density stands due to competition. This effect may be intensified for larger trees because of increased water demands, and one expert specified that drought would only affect large trees in the presence of parasitic mistletoe (Ameyma species). However, other experts identified that smaller trees are more susceptible to drought because of less developed root systems. Experts’ different opinions about the effects of drought highlight our current lack of understanding of how various species respond to changing precipitation patterns, and whether there are interacting effects of competition and tree size (but see Fensham and Holman 1999).

Insect attack can cause small tree mortality, but most experts specified that it would occur at such minor levels that it did not need to be considered in the models as a driver of state transitions. Experts who specified that insect attack would have a substantial effect also predicted that most stands would recover any lost small stems within 8 - 25 years of the attack.

The majority of experts predicted that intense overstory wildfires could cause small tree mortality, moving stands to either Low density regrowth or Low density mature. Some experts also predicted that intense overstory wildfires would cause sporadic mortality of large trees ≥60 cm DBH. Most experts also specified that stands would recover within 2-20 years because remaining trees would coppice. Experts’ estimates of the annual probability of an overstory fire (mean: 0.0117, range: 0.0063 to 0.0500) were similar to the annual probability calculated from GIS data (mean: 0.0069).

Windthrow or tornado events were considered by all experts as important processes for changing vegetation structure, although experts provided different estimates of the extent of damage that windthrow would cause annually. Experts specified that windthrow could cause the direct mortality of large trees through uprooting or indirectly through mechanical damage to stems and branches. There was disagreement about whether or not windthrow would lead to a coppice response; if windthrow was predicted to uproot trees then there was no coppice response, but if windthrow caused stem and branch damage then the remaining stem or stump would initiate a coppice response.

46 Chapter Two

Management actions

Annual probabilities for management actions were derived from existing literature. Management transitions were programmed as stochastic and the probabilities of transitions occurring were independent between years.

Some experts specified that the harvesting of smaller trees for firewood collection (approximately 15 cm DBH), compared to the harvesting of larger trees for sawlogs (approximately 30-50 cm DBH) would have differential effects on vegetation structure, while other experts specified a general response to timber harvesting. General timber harvesting had an annual probability of 0.037, while the probabilities for firewood and sawlog harvesting were 0.025 and 0.012, respectively. In terms of vegetation change, one expert predicted that firewood harvesting would immediately lead to a decreased density of small trees, but that stands would then experience an intense coppice response that would slow the development of large stems relative to unharvested stands. Another expert stated that firewood harvesting would increase growth rates of retained stems through reducing competition and, even with a coppice response, would help stands develop large trees faster than without harvesting. It was generally agreed that sawlog harvesting had the potential to decrease the rate at which large trees developed because they were removing medium sized stems that were most likely to develop into large trees. When specified as general timber harvesting, experts predicted that harvesting would change vegetation from High to Low density regrowth, but would then undergo a strong coppice response within 3-20 years to return stands to High density regrowth with ≥100 small stems/ha.

Ecological thinning is different from traditional timber harvesting because it specifies that stems desirable as habitat are retained and any nearby competitive stems are removed to promote growth of retained stems. It was specified as two separate transitions because at 2/3 of all trial sites, stump poison was applied to prevent a coppice response but due to root grafting, particularly in Red Box (Eucalyptus polyanthemos), stump poison was not applied at the other 1/3 of sites. To match the annual rate of 0.037 for timber harvesting, ecological thinning plus poison had a rate of 0.0278 and ecological thinning without poison had a rate of 0.0093. Four of five experts believed that ecological thinning would move stands from High density

47 Chapter Two regrowth below the threshold of 100 small stems/ha to Low density regrowth. With the application of stump poison, retained stems would grow without a coppice response from cut stems and develop into Low density mature. Without stump poison, most experts believed that stands would undergo a strong coppice response and, within several years, would return to High density regrowth or High density mature. One expert predicted that ecological thinning with stump poison would expedite the development of large trees, but that ecological thinning without stump poison would decrease overall stand growth rates.

Some experts predicted that fuel reduction burns could kill very large, old trees, whereas other experts believed that they would not affect large trees because of thick bark and the ability to undergo epicormic growth, but may kill some very small trees. In addition, some experts predicted that fuel reduction burns could cause some regeneration from seed. Finally, mining was not modelled as it currently occurs in very few Box-Ironbark parks.

2.4.3 Results from each management scenario

Natural Disturbance scenario

Due to the growth of stems, High density regrowth decreased to a mean of 52-62% of the land in this scenario in 150 years (Fig. 2.4a; Experts 1, 3, and 4). Most experts’ models predicted that Low density regrowth would increase only slightly to 13-18% because, while many natural disturbances moved stands into Low Density regrowth, this was balanced by a coppice response and growth that caused stands to change state to High density regrowth or Low density mature, respectively (Experts 2, 3, and 4). In addition, High density mature increased up to an average of 18-31% in 150 years (Experts 1-4). Nearly all models predicted an increase in Low density mature through time up to 8-40% (Fig. 2.4a; Experts 1, 2, 4, and 5), which was primarily a result of tree growth and stand self-thinning.

Despite general trends, there were differences between experts. Expert 1 predicted Low density regrowth would be absent within 60 years because this model specified that no natural disturbances would move stands below the 100 stem/ha threshold for small

48 Chapter Two stems (Fig. 2.4a). For Expert 2 there was a substantial decrease in High density regrowth and an accelerated accumulation of Low density mature compared to other experts. This occurred because of faster growth rates from High density regrowth to Low density mature relative to other experts. Furthermore, Expert 3 predicted that the amount of Low density mature would remain roughly constant through time. For Expert 5, there were more transition agents that caused stands to convert from High density regrowth to Low density regrowth without a corresponding coppice response that moved stands back into High density regrowth over time (i.e. dodder laurel and windstorms), which resulted in a large decrease in High density regrowth and a corresponding increase in Low density regrowth. Also, the model for Expert 5 predicted more often than other experts that natural disturbances would reduce the density of small stems below the 100 stem/ha threshold, which caused High density mature to almost disappear. Near the end of simulation for this expert, the percent of land in Low density mature began to increase because of the growth of stems (Fig. 2.4a).

Current Harvesting scenario

Under the Current Harvesting scenario, the amount of High density regrowth decreased to between 10-55% over 150 years for all experts, with decreases being of greater magnitude than in the Natural Disturbances scenario (Fig. 2.4b). The decreases occurred because timber harvesting moved stands from High density regrowth to Low density regrowth. Consequently, Low density regrowth increased relative to current conditions for all experts and increased more in this scenario than the Natural Disturbance scenario for Experts 1 and 2. Because timber harvesting diverted stands from High density regrowth to Low density regrowth, rather than allowing them to develop into High density mature as in the Natural Disturbance scenario, the proportion of stands in High density mature fell below current levels (Experts 1, 2, and 5). Low density mature increased over time, with increases being similar to the Natural Disturbance scenario (Fig. 2.4b; Experts 3, 4 and 5).

There were several exceptions to these patterns. Expert 1 predicted that timber harvesting would accelerate the development of Low density mature because, when harvested, stands moved from High to Low density regrowth, which had a much faster growth rate to Low density mature relative to other experts. The model for Expert 2

49 Chapter Two predicted slightly less Low density mature in the Current Harvesting scenario than the Natural Disturbance scenario (Fig. 2.4a and b; 40% vs. 31%). This occurred because timber harvesting changed stands from High density regrowth, which would otherwise develop quickly into Low density mature, into Low density regrowth, which was expected to develop very slowly into Low density mature. Because Expert 3 believed that timber harvesting increased the growth rate from High density regrowth to High density mature, instead of diverting stands to Low density regrowth, the proportion of stands in High density mature was much higher in this scenario than in the Natural Disturbances scenario (37% vs. 20%). This specification also resulted in no change in the proportion of Low density regrowth in the Current Harvesting scenario compared to the Natural Disturbance scenario. Furthermore, Expert 4 and 5 predicted similar outcomes from the Natural Disturbance and Current Harvesting scenarios. This likely occurred because the large number of, and high probabilities for, natural disturbances that Experts 4 and 5 specified would change stands overwhelmed the effects of timber harvesting (Fig. 2.4b).

Ecological Thinning scenario

The amount of land in High density regrowth decreased over time and was lower than both the Natural Disturbance and Current Harvesting scenarios. There was a corresponding increase in Low density regrowth over time that was higher than in the Natural Disturbance or Current Harvesting scenarios (Fig. 2.4c; Experts 2-5). This happened because ecological thinning changed High density regrowth to Low density regrowth. Most models indicated that ecological thinning would lead to a decrease in High density mature over time, which was more substantial than in the other scenarios (Experts 2, 3, and 5). This occurred because once stands moved from High density regrowth to Low density regrowth they were then confined in that state because of stump poison and could not develop into High density mature. The amount of land in Low density mature in this scenario was expected to increase with time, similar to the results for the Natural Disturbance scenario (Fig. 2.4c; Experts 1, 2, and 4).

Even though general trends under Ecological Thinning were apparent, there was variation between experts. The model for Expert 1 made substantially different predictions from the other experts because this model specified that ecological thinning

50 Chapter Two would not force stands below the small stem threshold, but would accelerate stem growth to High density mature. As a result, the model for Expert 1 predicted less High density regrowth than in the Natural Disturbance scenario and more land in High density mature over time and relative to the other scenarios (Fig. 2.4c). This model provided similar predictions for Low density regrowth as in the Natural Disturbance scenario, as ecological thinning was not predicted to move stands to Low density regrowth. Another exception was the model for Expert 3, which predicted that Low density mature would decrease over time. This occurred because once ecological thinning plus poison moved stands into Low density regrowth, there were no transitions to move out of that state except for growth to Low density mature, which occurred very slowly. In addition, there was very little change in High or Low density mature for Expert 4 in this scenario relative to the other scenarios. Again, the large number of and high probabilities for natural disturbances outweighed the effects of ecological thinning. Finally, the model for Expert 5 indicated that ecological thinning would lead to more Low density mature than timber harvesting or natural disturbances alone due to stands moving to that state from Low density regrowth (Fig. 2.4c).

2.5 Discussion

The majority of experts’ models predicted that Low density mature stands will increase to no more than 20% of the landscape in the next 150 years regardless of management. In addition, most models indicated that ecological thinning and timber harvesting do not expedite the development of Low density mature stands relative to natural disturbances alone. It is also likely that ecological thinning and timber harvesting will lead to a decrease in High density mature stands over the next 150 years. Over a much shorter management timeframe, such as 20-50 years, effects of timber harvesting and ecological thinning will be minimal if applied at the modelled rate. In addition, while general trends in vegetation change were apparent, model predictions varied widely because experts disagreed.

2.5.1 Management scenarios

In all scenarios, models generally predicted that High density regrowth will decrease and Low density regrowth will increase over time. This pattern was amplified in the

51 Chapter Two

Current Harvesting and Ecological Thinning scenarios and least pronounced under the Natural Disturbance scenario, which indicated that overstory removal of any sort will facilitate the movement of current stands with high densities of small stems to lower density stands. This change may be beneficial to understory flora because of reduced competition for light and resources (Homyack et al. 2004). However, while thinning may be beneficial for reducing competition, experts have also indicated that a drastic decrease in stem densities may lead to a stand collapse, where limited numbers of retained stems undergo high mortality rates due to increased susceptibility to windthrow (Lindenmayer and Franklin 1997), reduced rainfall (e.g. Butcher and Chandler 2007), or herbivory (Rogers and Leathwick 1997). When stand collapse occurs it can potentially have multiple negative impacts on the environmental services provided by forests (Oliveira et al. 2008). However, there are currently no data to indicate whether stand collapse will occur in Box-Ironbark woodlands and, if so, whether that threshold might be reached with either of the harvesting regimes under consideration.

Timber harvesting and ecological thinning accelerated movement from High density regrowth to Low density regrowth and, because High density regrowth would otherwise progress to High density mature, both harvesting methods will likely lead to a decrease in High density mature over time. This is problematic because the presence of large trees can be correlated with other important habitat features, such as canopy hollows for nesting (Alexander 1997; Wormington et al. 2003) and increased floral food resources (Wilson 2002), which indicates that High density mature provide higher quality habitat than High or Low density regrowth. Two notable exceptions include Ecological Thinning for Expert 1 and Current Harvesting for Expert 3, where ecological thinning and timber harvesting, respectively, were not expected to change stands to below the 100 small stem/ha threshold and, as a result, stands moved from High density regrowth to High density mature rather than to Low density regrowth.

In all scenarios, Low density mature increased over time as stands developed large trees and self-thinned. These increases, however, were minimal. This may be because Box- Ironbark woodlands are a slow-growing system (Kellas et al. 1998) and require more than the 150 years simulated in these models to produce greater increases in Low density mature. Interestingly, for most models neither timber harvesting nor ecological

52 Chapter Two thinning substantially altered the rate of Low density mature development relative to when there is no timber harvesting. Thus, for both High and Low density mature, it appears that neither management action will likely be beneficial for expediting the development of large trees with old-growth characters over the next 150 years. There were again two exceptions: the Current harvesting scenario for Expert 1 and the Ecological Thinning scenario for Expert 5, where relatively quick growth rates from Low density regrowth to Low density mature caused substantial increases in Low density mature.

Previous studies have shown thinning to be a successful method of promoting growth of small stems into larger size classes (e.g. Marquis and Ernst 1991; Sullivan et al. 2001; Homyack et al. 2004) and silvicultural thinning is used in forestry operations worldwide. The results for this case study may differ because of the slow growth of the overstory species of interest, such that the 150-year timeframe may have been insufficient to show conclusive changes. This is particularly noticeable in the results for Expert 5, where the proportion of Low density mature slowly begins to increase late in all management scenarios. In addition, Ross et al. (2008) found that stems responded to thinning with increased growth rates in white cypress pine (Callitris glaucophylla) savanna woodlands of south-eastern Australian, but that this effect was short-lived due to recruitment and gap-refilling. In Box-Ironbark woodlands, this effect may arise because a continued coppice response will quickly return thinned stands to densities similar to pre-thinning conditions (Kellas 1991), thus removing any competitive benefit of thinning stands. It is also possible that the increased growth rates seen in small stems after thinning diminishes with increased size such that, as trees get larger over time, stems’ growth rates decrease (Kellas et al. 1998).

2.5.2 Model uncertainty

Models are uncertain, which is important to account for because it can affect predictions (e.g. McCarthy and Burgman 1995; Morgan et al. 2001; Pellikka et al. 2005). I focused on uncertainty that arises due to differences between experts. Between- expert uncertainty arose because experts have different conceptual models of Box- Ironbark vegetation dynamics, which led them to differentially specify how transition agents will affect states and their probabilities of occurrence. As a result of this

53 Chapter Two uncertainty, the proportion of land in each vegetation state varied widely between experts, making decisions about management in Box-Ironbark forests and woodlands difficult.

The results from this case study highlight the importance of incorporating different opinions between experts. Most previous STSMs rely on the opinions of one or few experts or a forced consensus between experts (e.g. Forbis et al. 2006; Hemstrom et al. 2007; Vavra et al. 2007). Consulting multiple experts rather than just one expands the knowledge included in the assessment, akin to increasing the sample size (Clemen and Winkler 1999). Yet using forced consensus with multiple experts can have several disadvantages relating to the psychology of providing estimates in groups. For example, some experts dominate discussions in an effort to convince others, new ideas are often discouraged, people hold opinions that are not explained, the group may ignore pertinent information, and there is pressure to conform to the group majority (Clemen and Winkler 1999; Burgman 2005). Group interaction also tends to make participants overconfident (Cooke 1991).

By incorporating the opinions of multiple experts in this case study, it was possible to illustrate the large extent of disagreement and variation in expected outcomes. This was also seen when Morgan et al. (2001) elicited opinions from various experts regarding the effects of climate change on forest ecosystems. Their results depicted substantial differences between experts, which is not represented in consensus assessments. In addition, when Martin et al. (2005) included expert opinion in their assessment of the affects of grazing on birds, they found that disagreement between experts either reflected limited understanding about the ecology of a species or implied complex responses to disturbances, such that responses varied between locations. The disagreement between experts seen in these studies highlights the current state of understanding in their fields and allows researchers to focus on potential research needs (Morgan et al. 2001; Martin et al. 2005).

2.5.3 Limitations

There were several limitations to this study. First, I relied almost completely on expert opinion to parameterize models. It is desirable to validate expert-based models in order

54 Chapter Two to test model predictions and ensure model credibility (e.g. Kilgo et al. 2002; Walker et al. 2003; Shlisky et al. 2005). Validation can be done by using historical data, employing chronosequence methods, and by collecting ongoing monitoring data. Validation of parameter estimates for fire return intervals was possible using fire history data. Historical aerial photography might also be used to quantify some aspects of vegetation change, and would be a useful future extension to this research. Unfortunately in this case study, and others (e.g. Bellamy and Lowes 1999), validation of most parameter estimates and model outputs was not possible due to the lack of historical data and the time required to collect vegetation change data. This is a common problem in ecosystem modelling, where the only course of action is to carefully assess the model’s logic and assumptions (Starfield et al. 1994). For my models, I conferred with a member of the VDDT software development team, Leonardo Frid (ESSA Technologies; Vancouver, BC; 2007), and a long-standing software user, James Merzenich (USDA Forest Service; Portland, OR; 2007), to ensure that the models were constructed correctly and accurately reflected the conceptual models of experts. Further work might also involve debriefing experts on model predictions to determine if predictions matched experts’ expectations. Steps could then be taken to re-assess parameters, as necessary, to better reflect experts’ conceptual models. However, even though my models have not been validated and relied on expert opinion, they are useful initial steps in an adaptive management program that can be updated iteratively as monitoring data are collected regarding vegetation dynamics (Starfield and Bleloch 1991; Forbis et al. 2006).

As with any modelling framework, there are several limitations to STSMs. VDDT is a non-spatial model, meaning that it cannot model contagions, such as fire spread or insect infestations. Using a spatially-explicit single-tree model, Coates et al. (2003) found that the spatial distribution of the canopy had substantial impacts on stand growth and concluded that the spatial positioning of retained overstory stems is an important factor in harvesting planning. While spatially-explicit ecosystem models would provide useful information, such as patch sizes and the effects of contagions, they are complex in design and difficult to parameterize (Keane et al. 2002). With the limited data available for this research, I chose to use VDDT models, which are relatively easy to parameterize. Further research could attempt to convert these models

55 Chapter Two to TELSA, the spatially-explicit software equivalent to VDDT (ESSA Technologies Ltd. 2005).

Eliciting conceptual ecological models is a demanding and time-consuming activity, which can lead to interviewee fatigue and incorrect assessments (Burgman 2005). I constructed models with only four vegetation states in order to reduce interviewee confusion and fatigue. Several other studies using state-and-transition models have relied on only four to six vegetation states (e.g. Bestelmeyer et al. 2004; Shlisky et al. 2005; Bashari et al. 2008) and it is best to use more simple models when dealing with limited data (Barry and Elith 2006), such as in this case study. However, by categorizing vegetation into such broad categories, experts often found that estimating parameter values was quite difficult due to the large variation possible within any one state.

In addition, I did not incorporate the effects of climate change. It is possible that future transition probabilities may be changed as a result of climate change, but no data or past experience were available to inform how these changes might manifest. Previous studies have found that models incorporating the potential effects of climate change stabilize much slower than those without climate change predictions (see Rupp et al. 2000; Vavra et al. 2007). Projections of the effects of climate change in this region include a decrease in rainfall with increases in temperature and evaporation rates (CSIRO and Australian Bureau of Meteorology 2007) following declines in rainfall since the 1950’s (Cai and Cowan 2008). It is possible that decreased rain due to climate change may lead to reduced growth. If so, then self-thinning may also be delayed, leading to slower overall development of Low density mature woodlands. However, if mortality of small stems is increased as a result of reduced rainfall, self-thinning may proceed more quickly and/or the probability of stand collapse may increase.

Finally, management actions are typically considered deterministic transitions because the rate of harvesting is constant and occurs at regular intervals depending on the age or size of trees, while natural disturbances are stochastic and probabilistic. Stochastic transitions often occur regardless of the age or size of trees and, over time, lead to a lower proportion of the landscape being disturbed relative to deterministic transitions that occur with the same frequency (McCarthy and Burgman 1995). Using VDDT, it

56 Chapter Two is not possible to model management actions as deterministic. As a result, I underestimate the proportion of the landscape subject to timber harvesting and ecological thinning over time because there is a probability of stands escaping management in large successive intervals when programmed as stochastic disturbances. To compensate for underestimating the proportion of land affected by management actions, I completed a trial using an increased harvesting rate for both management actions. I found that this marginally increased the amount of land in the desired state for some experts, but did not lead to a general over- or underestimate of the benefits of either harvesting technique. Despite this, because both harvesting methods are specified similarly and I was concerned about the relative effects of these two types of timber harvesting, I can confidently conclude that the predictions for timber harvesting and ecological thinning are comparable and that neither harvesting technique is more beneficial for expediting the development of mature woodlands.

2.6 Conclusions

These preliminary results indicated that the timber harvesting methods currently being undertaken in Box-Ironbark woodlands will likely not expedite the development of large trees and may, in some cases, lead to a decrease in the proportion of stands with large trees. However, there was large variation in the model results because experts disagreed. Further analyses should be undertaken to quantitatively determine whether between-expert uncertainty or other sources of uncertainty, such as within-expert uncertainty, parameter uncertainty, natural stochasticity, or the effects of initial conditions, have greater effects on model predictions. This information can be used to identify current gaps in knowledge, such as the rates of rare stochastic disturbances and their impacts on overstory vegetation structure. These gaps can then be targeted for monitoring and data collection to help reduce uncertainty. Such analyses appear in Chapter Three. In addition, because experts provided different predictions and I have assumed that each expert is equally credible, a structured decision analysis that incorporates differences between experts could help determine which management action has the highest probability of expediting the development of large trees; these analyses will appear in Chapter Four.

57 Chapter Two

Table 2.1. All transition agents that cause transitions between vegetation states and the management scenario they are included in.

Transition Agent Natural Current Ecological Disturbance Harvesting Thinning Growth 9 9 9 Coppice response 9 9 9 Dodder laurel 9 9 9 Drought 9 9 9 Drought + mistletoe 9 9 9 Insect attack 9 9 9 Wildfire 9 9 9 Windthrow 9 9 9 Windthrow + wildfire 9 9 9 General timber harvesting 9 Firewood harvesting 9 Sawlog harvesting 9 Ecological thinning 9 Ecological thinning + poison 9 Fuel reduction burn 9 9

58 Chapter Two

Kilometers 0 40 80 160 240 320 -

Fig. 2.1. A map of Victoria, Australia, showing the historic extent of the Box-Ironbark region (in grey) and areas of remaining forests and woodlands (in green).

59 Chapter Two

For each For each For each Monte Carlo timestep cell simulation

Draw a random Randomize number from the list of a uniform possible distribution transition between 0.0 agents for and 1.0 that state

Is the random number less No Yes No transition than the STOCHASTIC sum of all TRANSITION transition agent probabilities?

Is the cell old enough Does the to undergo a Yes transition agent deterministic/ alter the age of growth the cell? transition?

Yes No No

DETERMINISTIC Increment Increment TRANSITION cell age by 1 cell age by 1 plus/minus age timestep specified by transition agent

Fig. 2.2. A schematic of the VDDT simulation algorithm.

60 Below threshold: Large trees (≥ 60cm DBH) Above threshold: <10 stems/ha ≥10 stems/ha Sawlog harvest

Above threshold: Growth ≥100 stems/ha High density regrowth High density mature Windthrow Coppice Ecothin, Firewood harvest, Small trees Wildfire, Windthrow (5-59cm DBH) Self-thin, Firewood harvest Coppice

Growth Below threshold: Low <100 stems/ha Windthrow density mature Low density regrowth Low density mature Sawlog

Wildfire Chapter Two harvest

Fig. 2.3. The four vegetation states used in state-and-transition simulation models according to the density and DBH of Eucalyptus species. Arrows describe some of the common transition agents and how they affect the vegetation states. Probabilities for these and all transition agents are found in Appendix 2.B. Photos care 61 of Terry Walshe. Chapter Two

(A) Natural Disturbance (B) Current Harvesting (C) Ecological Thinning

100 High density regrowth Low density regrowth 80 High density mature Low density mature 60 E1 40

0 100

60 E2 40

0 100

60 E3 40

0 100

80 Land in vegetation state (mean %)

60 E4 40

0 100

60 E5 40

0 0 30 60 90 120 150 0 30 60 90 120 150 0 30 60 90 120 150 Time (years) Fig. 2.4. e mean percent of land in each vegetation state over time according to experts (E1-E5) and arranged by management scenario, (A) Natural Disturbance, (B) Current Harvesting, and (C) Ecological inning. Analyses are for the Average initial condition and the median/mean parameter estimates for each expert. 62 CHAPTER THREE

Assessing sources of uncertainty in expert state-and-transition simulation models

64 Chapter Three

3.1 Abstract

State-and-transition simulation models (STSMs) are widely used to predict vegetation dynamics. However, uncertainty is often not acknowledged in model construction, even though it may affect model predictions. I constructed STSMs using expert opinion to predict the future vegetation structure of Box-Ironbark forests and woodlands in Victoria, Australia. I incorporated uncertainty due to differences between experts (between-expert uncertainty), imperfect knowledge of experts (within-expert uncertainty), natural stochasticity, and the effects of initial conditions. Between-expert uncertainty contributed most to variance in model predictions, suggesting that modelling strategies that rely on consensus between experts may lead to overconfident models and, when management decisions are based on models results, may have a greater likelihood of leading to the wrong decision.

65 Chapter Three

3.2 Introduction

State-and-transition models were originally described to assist rangeland managers organize information on vegetation dynamics and guide management (Westoby et al. 1989). Under this framework, vegetation composition and structure are used to define discrete states that are connected by transition agents, such as growth, natural disturbances, or management actions, which cause transitions between vegetation states (Westoby et al. 1989). This conceptual modelling system has since been incorporated into simulation models and applied to various vegetation dynamics applications. State- and-transition simulation models (STSMs) have been used to predict the effects of management on vegetation (e.g. Hemstrom et al. 2002; Forbis et al. 2006; Wales et al. 2007), to determine how changes in disturbance regimes have altered vegetation (e.g. Perry and Enright 2002a; Shlisky et al. 2005; Wondzell et al. 2007b), and to predict how animal behaviours and distributions are affected by vegetation (e.g. Moser et al. 2003; Letnic et al. 2004).

It is often difficult to parameterize STSMs due to a lack of data pertaining to ecosystem dynamics and species responses (Cooke 1991; Hemstrom et al. 2001). When empirical data are unavailable and decision support is needed immediately, expert opinion can provide information regarding ecosystem dynamics and species responses (e.g. Kuhnert et al. 2005; Martin et al. 2005; Forbis et al. 2006). While expert opinion is not a substitute for empirical research (Morgan et al. 2001), it can be integrated into STSMs to provide preliminary predictions of vegetation dynamics (Forbis et al. 2006).

It is important to incorporate uncertainty into forestry models as it is known to affect model predictions and management decisions (Ducey 2001; Kangas and Kangas 2004). There are many sources of uncertainty, such as natural variation and expert knowledge, among others (Regan et al. 2002). Natural variation arises because biological processes and disturbances occur randomly in space and time (‘stochastic ordering’, e.g. McCarthy and Burgman 1995) and their rates vary through time (‘stochastic frequency’, e.g. Raulier et al. 2003; Kangas and Kangas 2004). Stochastic ordering is commonly modelled in STSMs through Monte Carlo methods that randomize the order of disturbances over multiple simulations (e.g. Forbis et al. 2006; Wales et al.

66 Chapter Three

2007). There are several examples where stochastic frequency has been broadly taken into account in STSMs by programming variation in the rates of wildfire, drought, and insect attack (e.g. Merzenich et al. 2003; Forbis et al. 2006). However, stochastic frequency is often ignored and only single point estimates are used for the rates of disturbances (e.g. Hemstrom et al. 2002; Hemstrom et al. 2007; Wondzell et al. 2007b).

When relying on subjective judgement, between- and within-expert uncertainty are rarely incorporated into model construction. Between-expert uncertainty arises because of differences in opinions between experts, which can reflect spatial heterogeneity in processes (Martin et al. 2005), a dearth of knowledge about processes (Morgan et al. 2001), or cognitive biases (Burgman 2005). To my knowledge, there are no other published studies that rely on expert knowledge to parameterize STSMs and take between-expert uncertainty into account. Researchers may rely on the opinions of one or only a few experts with shared opinions. Relying on limited numbers of experts to inform management decisions can be a risky strategy considering they might provide incorrect estimates. As a result, it is usually recommended that multiple experts are consulted to incorporate more complete information, akin to increasing the sample size (Clemen and Winkler 1999). Often researchers rely on Delphi methods or other types of forced consensus (Burgman 2005) with groups of experts to provide one single model of how transition agents affect states and their probabilities (e.g. Forbis et al. 2006; Hemstrom et al. 2007; Vavra et al. 2007). However, forced consensus estimates are susceptible to psychological frailties. These include when participants dominate discussions and try to convince others of their opinions, when participants hold opinions that they will not explain, and when participants feel pressured to conform to the group majority (Clemen and Winkler 1999; Burgman 2005). In addition, forced consensus can lead to over-confidence (Cooke 1991). Between-expert uncertainty was incorporated into models in Chapter Two and was shown to cause large differences in STSM predictions between experts.

A second source of uncertainty that arises because of subjective knowledge is within- expert uncertainty, or how confident an expert is of their estimate. Within-expert uncertainty occurs when there is a true parameter value, but experts are unsure of its value. There are examples of studies that have assessed uncertainty in experts’ estimates

67 Chapter Three

(e.g. Walker et al. 2001), but I am unaware of any that incorporate within-expert variation of parameter estimates in STSMs. By presenting only the average expected value for parameters, models do not account for the imperfect knowledge of experts (Forbis et al. 2006). Finally, model uncertainty can arise through error in modelling initial conditions (Fan et al. 2000; Keane et al. 2002). Because model predictions may be affected by each of these sources of uncertainty, stochastic frequency, between-expert uncertainty, within-expert uncertainty, and the effects of initial conditions, it is important to understand their relative influences.

In this chapter, I quantified the magnitudes of several types of uncertainty and investigated their impacts on STSMs constructed to predict the dynamics of overstory Eucalyptus vegetation in the Box-Ironbark forests and woodlands of Victoria, Australia (Fig. 2.1). I quantified disagreement between experts, or between-expert uncertainty, by allowing experts to specify their own STSMs. Second, I quantified imperfect knowledge, or within-expert uncertainty, by asking experts for a range of parameter values, rather than single estimates. I also incorporated natural stochasticity through VDDT Monte Carlo simulations and investigated the effects of uncertainty arising from model initial conditions.

3.3 Methods

A detailed description of model parameterization and construction is included in Chapter Two. I used STSMs implemented in VDDT (ESSA Technologies Ltd. 2007) to predict the long-term vegetation dynamics of Box-Ironbark forests and woodlands in Victoria, Australia. All data were provided in the form of elicited expert opinion and were gathered through electronic surveys and interviews. Electronic surveys were used to determine the states in the VDDT models (Fig. 2.3) and interviews were conducted so that experts could specify the rates and effects of transition agents on vegetation states (Table 2.1; Appendix B). While there were three alternative management strategies being considered to restore this system (altered timber harvesting regulations, ecological thinning, and natural development without harvesting), for this chapter, land was allocated equally between strategies and results were combined for all management actions. Managers were most interested in the development of Low density mature

68 Chapter Three woodlands, which have ≥10 stems/ha of large Eucalyptus trees (≥60 cm diameter at breast height, or DBH) and <100 small stems/ha (5-59 cm DBH). As a result, analyses for this chapter will focus only on Low density mature woodlands. VDDT model predictions are in the form of proportion data, which were logit transformed to improve normality for data analysis. I completed data analyses on VDDT model predictions using the software R version 2.6.2 (R Development Core Team 2008)(Appendix C for code).

3.3.1 Modelling uncertainty

VDDT incorporates an element of natural variation, stochastic ordering, by performing multiple Monte Carlo simulations where the order of transitions is randomly selected for each cell over time. This is done according to the following procedure. First, the list of possible transition agents with their attendant probabilities is specified according to a cell’s current vegetation state. At one timestep and for a given Monte Carlo simulation, VDDT randomizes the list of possible transition agents and arrays their probabilities consecutively from 0, such that each probability covers a distinct interval. VDDT then sums the probabilities of the possible transition agents and draws a random number from a uniform distribution between 0 and 1. If that random number is greater than the summed probabilities of transition agents possible in that state, then no transition occurs. If the random number is less than the summed probabilities, then the transition agent whose distinct probability interval corresponds to the random number is selected and a transition occurs Cells’ ages are also adjusted if specified by the transition agent (ESSA Technologies Ltd. 2007). With this process, no more than one transition agent may be selected for a cell at one timestep. The probability distribution that describes the number of cells in a given state that undergo transitions at one timestep is the binomial distribution (Equation 3.1)(McCarthy 2007):

n! kxP )( == k − pp −kn ,)1( − knk )!(! (3.1) where n is the total number of cells, p is the probability of a transition, and k is the observed number of cells undergoing a transition. The mean, μ , of this distribution is

69 Chapter Three np and the variance, σ 2 , is np(1-p). Using this procedure, stochastic ordering in space and time in incorporated. However, this is only a portion of natural variation. In VDDT, it is possible to program broad inter-annual variation, for example by programming some proportion of years to have low, normal, or high probabilities of fire, but because uncertainty in parameter values cannot be modelled with VDDT Monte Carlo simulations, stochastic frequency in the rates of disturbances is not incorporated.

Each expert used the same four-state model structure (Fig. 2.3) and transition agents (Table 2.1) to parameterize a single STSM that depicted their understanding of the effects of growth, natural disturbances, and management actions on vegetation structure. To incorporate between-expert uncertainty, each expert provided their own transition probabilities and specified how each transition agent would affect vegetation change. Many of the same transition agents were used in multiple experts’ models, although some transition agents were not included if the expert did not believe the transition had an effect on the vegetation states (Appendix B).

To account for within-expert uncertainty, experts provided all estimates in the form of a three-point estimate where the low and high estimates represent the “plausible bounds” and the middle estimate represents the mean (Morgan and Henrion 1990; Stainforth et al. 2005). People are often most comfortable providing a best guess and extreme values (Burgman 2005), which can be converted to probability distributions to incorporate uncertainty in parameter estimates (Regan et al. 2002; Dorazio and Johnson 2003). I assumed the estimates would follow a beta distribution (Vose 1996; Burgman 2005) and calculated the variance of the distributions from the normal bounds of the middle and lower probability estimates (Equation 3.2):

2 ⎛ μ − l ⎞ σ 2 = ⎜ ⎟ , ⎝ z ⎠ (3.2) where σ 2 is the variance, μ is the mean or middle estimate from the three-point estimate, l is the lower estimate from the three-point estimate, and z is the z-value derived from a normal distribution. Because the interview format sought only best

70 Chapter Three estimates and plausible bounds with the three-point estimates, the degree of confidence intended by the bounds was unspecified. As a result, it was necessary to assume something about that interval. When calculating the variance, three sets of beta distributions were generated for each transition agent; I assumed that the intervals provided were likely to be 90% confidence intervals, though I tested the sensitivity of model predictions to this assumption by running simulations in which the bounds also represented 80 and 95% confidence intervals. The variances for the three confidence intervals relied on z-values derived from a normal distribution: 1.282 (80%), 1.645 (90%), and 1.960 (95%).

Shape parameters (α and β ) for the beta distributions were calculated using the mean, μ , and the variance imputed for a given confidence interval, σ 2 , for each transition agent and for each expert (Equation 3.3)(McCarthy 2007),

⎡ (1− μμ ) ⎤ = μα −1 ⎢ 2 ⎥ ⎣ σ ⎦

⎡ ()1− μ ⎤ 1−= μμβ − .1 ()⎢ 2 ⎥ ⎣ σ ⎦ (3.3)

Once beta distributions were generated for each transition agent in an expert’s model, a single probability estimate was drawn at random from the beta distribution. These single estimates were drawn from the beta distributions of each transition agent specified by an expert and compiled to provide a complete set of transition probabilities for each expert’s model. This process was repeated to make 25 replicate models for each expert. I completed one trial where 100 replicates were constructed for each expert using the 90% confidence interval. I visually inspected probability density functions showing the proportion of cells in Low density mature for 100 replicates versus 25 replicates and found that the distributions were virtually identical. In addition, because the time required to construct and simulate the additional replicates was extremely onerous, only 25 replicates were constructed for each expert.

Generating beta distributions from experts’ three-point estimates was meant to incorporate within-expert uncertainty by showing how confident experts were in their estimates of parameter values. However, during interviews I asked experts to estimate

71 Chapter Three how variable each parameter’s true value could be – i.e. to estimate stochastic frequency. As a result, these two sources of uncertainty are conflated in the elicited parameters. However, for simplicity, I have referred to these two sources of uncertainty only as within-expert uncertainty for the rest of this paper.

Initial conditions were generated by asking experts to predict the current percent of land in each of the four vegetation states. I averaged these values to generate an Average initial condition (Table 3.1). However, during interviews it became apparent that there were two divergent concepts of how the landscape looked before European settlement, which may have affected Box-Ironbark development. The second initial condition approximated historical accounts of Box-Ironbark woodlands as an open and grassy with a park-like appearance (Howitt 1855; Newman 1961; Benson and Howell 2002). This state is referred to as the Open initial condition; it contains predominantly Low density mature and, to a slightly lesser degree, Low density regrowth (Table 3.1). The third initial condition followed references that refer to Box-Ironbark country as very densely wooded (Mitchell 1839; MacKay 1891; Palmer 1955; Perry 1978; Clacy 2003). I refer to this third initial condition as Dense; it contains mostly High density mature with a slightly smaller proportion of High density regrowth (Table 3.1). I modelled three ensembles of models with these initial conditions to investigate how initial conditions may affect or limit this system’s ability to respond to transition agents.

3.3.2 VDDT Model simulation

Each model produced 100 Monte Carlo simulations, which continued for 150 timesteps after preliminary simulations showed most model predictions stabilized by roughly 100 years. VDDT models simulated the dynamics of 1000 independent cells, which were assigned a random age at the beginning of simulations to ensure that models reflected the current multi-age structure of the stands. The three ensembles of models based on the initial conditions specified the proportion of cells in each state at the beginning of the simulation.

For each of the five experts there were 225 models. There were 25 replicate models constructed using the values drawn from beta distributions for transition agent

72 Chapter Three estimates; these 25 replicate models were generated for each of the three possible confidence intervals (80%, 90%, or 95%) and each of these was repeated for the three sets of initial conditions.

3.3.3 Data analysis

Variance parameter estimates

Variance Components Analyses (VCAs) are commonly used to determine group effects by measuring the proportion of the total variance in the population that can be explained by group variance (Quinn and Keough 2002). I used hierarchical mixed- effects linear models to determine whether variation in the percent of land in the desired vegetation state, Low density mature, was primarily due to differences between experts’ models, within each expert’s set of 25 replicate models, or between the 100 VDDT Monte Carlo simulations (Faraway 2006; Gelman and Hill 2007; Baayen 2008). Maximum likelihood estimators of variance and covariance parameters can be negatively biased if fixed effects are specified and the degrees of freedom are not reduced appropriately (Robinson 2008). As a result, models were fit with Restricted Maximum Likelihood (ReML), which is not biased by fixed effects and more accurately estimates parameters (Robinson 2008). Modelling was completed using the nlme (Pinheiro et al. 2008) and lme4 (Bates 2008) packages in R.

I first completed VCAs for each of the three confidence interval levels at every ten timesteps. I used varying-intercept hierarchical mixed-effects models to determine how variance in the response variable, the percent of land in Low density mature, changed throughout model simulations (Equation 3.4).

ijk = α + + bby 00 ()+ ε ijkjkj 2 0 j Nb ( ,0~ σ j ) , for j = K 25,1, 2 0 ()jk Nb ( ,0~ σ k ) , for k = K 5,1, 2 ijk N( ,0~ σε ) , for i = K ,300,1, (3.4)

where yijk is the percent of land in Low density mature, α is a fixed effect for the initial conditions that includes a binary dummy variable to accommodate the three

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initial conditions, b0 j is the random group level intercept for experts’ replicate models

2 with variance σ j , b0 ()jk is the random group level intercept for experts with variance

2 σ k , and ε ijk represents the residuals (VDDT Monte Carlo simulations) with variance σ 2 .

VCA parameter estimation was completed using a Bayesian framework to produce posterior density distributions for the variance estimates (between-expert, b0 ()jk ; within- expert, b0 j ; and VDDT Monte Carlo/residual,ε ijk ) based on VDDT model predictions from the final timestep in the simulation (150 years). Bayesian modelling was done with the coda package in R (Plummer et al. 2008) using Markov Chain Monte Carlo (MCMC) estimation. MCMC simulations ran for 110,000 iterations using a Gibbs sampler. The prior on the fixed effects parameter is taken to be locally uniform, while the prior on the variance-covariance matrices of the random effects is taken to be the locally non-informative prior described in Box and Tiao (1973); the priors are sampled from a Wishart distribution, depending on the current values of the random effects (Bates 2008). Convergence was assessed using the potential scale reduction factor (Rhat statistic) based on three chains and a 10,000 iteration burn-in (Gelman et al. 2000). Posterior density distributions from one of the three chains were used to determine the uncertainty in variance parameter estimates; analyses were repeated for each of the three confidence interval levels separately.

Effects of initial conditions

I then investigated the specific effects of Initial Conditions on model predictions. This was done by comparing the mean percent of land in Low density mature over time for each of the experts’ models based on the three initial conditions. Model predictions based on the Open and Dense initial conditions were plotted relative to the predictions of models based on the Average initial condition. This was done to compare how model predictions varied for the historic conditions, Open and Dense, relative to predictions using Average.

I also completed a second set of VCAs for each of the three initial conditions separately to determine the variance in the percent of land in Low density mature at timestep 150

74 Chapter Three and using the 90% confidence interval. These analyses used a varying-intercept hierarchical mixed-effects model fit by ReML (Faraway 2006; Gelman and Hill 2007; Baayen 2008; Robinson 2008) because initial conditions were analyzed separately, there were no fixed effects (Equation 3.5).

ijk = + bby 00 ()+ ε ijkjkj 2 0 j Nb ( ,0~ σ j ) , for j = K 25,1, 2 0 ()jk Nb ( ,0~ σ k ) , for k = K 5,1, 2 ijk N( ,0~ σε ) , for i = K ,100,1, (3.5)

where yijk is the percent of land in Low density mature, b0 j is the random group level

2 intercept for experts’ replicate models with variance σ j , b0 ()jk is the random group

2 level intercept for experts with variance σ k , and ε ijk represents the residuals (VDDT Monte Carlo simulations) with variance σ 2 .

Sensitivity analyses

I was interested in how the percent of cells in Low density mature at 150 years varied as a result of variation in the 25 transition agent estimates drawn from beta distributions. I conducted sensitive analyses using linear regressions to describe how variation in the percents of cells in Low density mature varied according to variation in each transition agent specified by an expert. Slopes of the regression lines were used to determine which transition agents had the greatest effect on each expert’s predictions (Equation 3.6).

i = α + βxy + ε ii , (3.6)

where yi is the percent of cells in Low density mature of 150 years,α is the intercept of the linear regression, β is the slope of the linear regression, xi is either the probability that a probabilistic transition agent will occur or the minimum age that growth with occur, andε i is the error term. I did not include timber harvesting or ecological thinning in the sensitivity analyses because management actions were modelled with a single probability rather than a range of probability estimates.

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3.4 Results

3.4.1 State-and-transition simulation model predictions

There was a great deal of disagreement between experts regarding annual probabilities of transition agents and how they affected vegetation states, such that only a few parameters and beta distributions were similar between experts (Appendix B; for example, Fig. 3.1 and 3.2). Growth transitions were the most common between experts, with overlap of experts’ beta distributions (Fig. 3.1a, 3.2a, and 3.2b). There was some overlap between experts who specified that intense overstory wildfire would change stands from High to Low density regrowth (Fig. 3.2c). Two experts specified that windthrow could change stands to Low density regrowth, but they disagreed substantially about the annual probability of the transition agent, which lead to quite divergent beta distributions (Fig. 3.2d). The confidence intervals used to specify the variance in experts’ transition agent estimates altered transition agents’ beta distributions so that 95% confidence intervals selected parameter estimates nearer to the mean more often, while the 80% confidence interval was more uncertain (Fig. 3.1b-f). Nonetheless, the changes in the distributions between confidence intervals were small relative to the interval widths of the distributions and the differences between experts’ distributions (Fig. 3.1 b-f).

The flexibility in parameterizing VDDT models led to large differences in the percent of land in Low density mature within and between experts (Fig. 3.3). Differences between experts arose mostly because of the high predictions of Expert 2 and, to a lesser degree, Expert 1. Most other experts provided similar average predictions of 7.5-11.9% of land in Low density mature over time. Within-expert variation increased as model predictions diverged and became more difficult to predict over time. VDDT Monte Carlo variation also differed between experts; the standard deviations for Experts 3 and 4 were much smaller than Experts 1 and 5. Overall, differences between experts appeared to be larger than variation within experts or in VDDT Monte Carlo simulations (Fig. 3.3).

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3.4.2 Variance parameter estimates

VCAs that did not include the effects of initial conditions illustrated that the magnitude of variance in the percent of land in Low Density mature was similar for each of the three confidence interval levels and increased over time (Fig. 3.4a-c). Between 80 and 100 years, total variance was maximized and then declined. This occurred as model predictions for each expert stabilized (Fig. 3.3). The majority of variance in model predictions arose from differences between experts, which contributed a maximum of 0.9 of variance. Residual variance due to VDDT Monte Carlo simulations caused less of the variation in model predictions than between-expert variance, with a maximum variance of 0.15 at timestep 150 (Fig. 3.4a). Within-expert variance was least influential, with a maximum value of 0.02 at timestep 110 (Fig. 3.4c). Also, within- expert variance did not appear until timestep 40 (Fig. 3.4a) or 70 (Fig. 3.4b and c) as VDDT model predictions diverged. By the final timestep, the variances for each of the three confidence intervals were practically indistinguishable between experts, within experts, or between VDDT Monte Carlo simulations (Fig. 3.4a-c).

MCMC estimation to determine the posterior density distributions for these VCA parameters showed that, in addition to being the largest source of variation, between- expert variance was the most uncertain (Fig. 3.5a and b). The 95% Bayesian credible intervals for between-expert variance ranged from 0.261 to 6.013 (Fig. 3.5b), while the Bayesian posterior density distribution for within-expert variance was considerably more certain with a 95% credible interval of 0.010 to 0.018 (Fig. 3.5a and c). The variance in the VDDT Monte Carlo simulations was best defined (Fig. 3.5d). The posterior density distribution for between-expert variance arises from the estimates of the intercepts for each expert (Fig. 3.5b inset). These intercepts appear to cluster for three experts (Experts 3-5), from which the intercepts for Experts 1 and 2 are offset. Estimates for all variance components parameters were little changed whether experts’ transition agent estimates were considered to represent 80%, 90%, or 95% confidence intervals (Table 3.2).

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3.4.3 Effects of initial conditions

In some experts’ models based on the Open or Dense initial conditions, the percent of land in Low density mature has converged, or is moving to convergence, with the predictions for the Average initial condition (Fig. 3.6). This is particularly true for Dense, which began simulations with a similar percent of land in Low density mature as Average. If simulations continued for a longer period, it is possible that most models would converge with the predictions for Average. The only exception was for Expert 1, where the percent of land in Low density mature was much higher in the Open initial condition than other experts (Fig. 3.6a). Convergence between initial conditions implies that the models are somewhat insensitive to changes in initial conditions, and stable states over the long term are dictated mostly by the transitions specified by each expert. However, over a shorter timeframe, sensitivity is increased and initial conditions have substantial impacts on model predictions. Note that the higher initial proportion for the Open-Average initial condition was balanced with much lower predictions for High density regrowth (Appendix D).

VCAs to determine the effect of each initial condition on the variance in model predictions indicated that variance was lowest in simulations using the Average initial condition, while the Dense initial condition led to much higher estimates of variance (Fig. 3.7). As in the previous VCA, the majority of variation arose due to differences between experts, whereas differences within experts and between VDDT Monte Carlo simulations were minimal. The variance estimates were based on percent of land in Low density mature for each expert at 150 years (Fig. 3.8). The results clumped for all experts, except Expert 2, under the Average initial condition and were widely spaced for predictions using the Open initial condition. The Dense initial condition resulted in a heterogeneous clumping pattern with Expert 2 much higher than other experts (Fig 3.8).

3.4.4 Sensitivity analyses

Sensitivity analyses illustrated how variability in transition agent parameter estimates affected the variation in the percent of cells in Low density mature after 150 years.

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While there are more complex analytical methods available to perform sensitivity analyses on simulation models, such as boosted regression trees, these linear regressions were able to provide some insight into how variation in parameter estimates impacted model predictions. The three transition agents with the steepest slopes for each expert were compared (Table 3.3). Drought, intense overstory wildfire, and growth to Low density mature were the most influential transition agents for most experts. However, the effect of the transition agents on vegetation states – i.e. the “From” and “To”- differed substantially across all experts, making it difficult to compare sensitivity of transition agent parameter estimates between experts. The majority of R2 values, the steepness of regression line slopes, and the standard errors of the slope estimates all increase as within-expert uncertainty increases, such that Experts 1, 2, and 5 have high values for the sensitivity analysis and large within-expert bounds (Table 3.3; Fig.3.3).

3.5 Discussion

This work has shown that differences between experts were by far the greatest source of variation in model predictions and, consequently, were the most important source of uncertainty to be characterized. Within-expert uncertainty, Monte Carlo variation, and the long-term effects of initial conditions were relatively unimportant.

3.5.1 Incorporating uncertainty

Most variance arose from differences between experts, which occurred because experts provided quite different estimates of how transitions would affect vegetation states and their probabilities of doing so. In addition, the estimate for between-expert variance was the least certain because it was based on five experts’ estimates that were heterogeneous and thus were difficult to generalize. As a result, not only was most of the variance in model predictions due to differences between experts, but it was also not possible to precisely define the between-expert estimate of variance.

These results showed how variable predictions were across experts and highlighted the importance of allowing experts to provide unique opinions. This is in stark contrast to the majority of previous STSMs, which have undergone consensus to provide one

79 Chapter Three model structure or single parameter estimates (e.g. Hemstrom et al. 2002; McIntosh et al. 2003; Forbis et al. 2006; Hemstrom et al. 2007; Wales et al. 2007). Forced consensus can often be a better forecaster of parameter estimates than a single expert’s estimate (Cooke 1991). However, because the experts in this case study provided such different estimates and there is no a priori reason to choose one expert over another, using a single model would have omitted crucial information in this study. This research gives an estimate of how poorly the outcome could be by basing management decisions on the advice of one expert in this highly uncertain system. A similar result was seen by Morgan et. al. (2001), where experts provided a diversity of opinions on climate change, which is not observed in consensus summaries.

Furthermore, because variance was due mostly to differences between experts, I recognize the importance of selecting experts. By interviewing experts from a variety of backgrounds, including ecology, forestry, botany, and zoology, I hoped to encapsulate the majority of knowledge relating to Box-Ironbark dynamics. This has analogues to stratified sampling, where different environmental gradients are deliberately sampled in order to include environmental diversity. Likewise, the diversity of expert opinion should be incorporated in process models. This is particularly true in this case study, where the models for experts of similar domains clustered (e.g. Fig. 3.5b).

In addition, I only managed to build models for five experts after interviewing eight and, because increasing the number of experts can be thought of as akin to increasing sample size (Clemen and Winkler 1999), further interviews may be beneficial in reducing uncertainty if additional experts agree and patterns begin to emerge. However, if further experts disagree, definite conclusions would not be available until monitoring data is collected to update model predictions.

VDDT Monte Carlo variance arose because of stochastic ordering of disturbances. Within-expert variance arose because of the uncertainty around point estimates for transition agent probabilities, which reflected stochasticity in their rates and experts’ uncertainty in parameter estimates. Stochasticity in the rates of disturbances was unintentionally conflated with within-expert uncertainty because of the way transition agent estimates were elicited, and were grouped together as a result. Experts displayed different levels of within-expert and Monte Carlo variance, such that some model

80 Chapter Three predictions were highly uncertain and others were very confident. Larger within-expert uncertainty arose because experts provided wide bounds on their transition agent estimates and larger Monte Carlo variance arose because experts specified a large number of transition agents between states and, consequently, the increased movement between states was more difficult to predict. More confident predictions resulted from small bounds on estimates or limited numbers of transition agents.

Even with the conflation of stochastic rates and within-expert uncertainty, the results of this study indicated that these sources of uncertainty were unimportant compared to between-expert variance. However, because experts can be relatively poor estimators of their own uncertainty (Cooke 1991), it is possible that this result was driven by overconfidence, a common bias in subjective assessments (Morgan and Henrion 1990; Ayyub 2001; Soll and Klayman 2004). I treated experts’ upper and lower estimates as 80%, 90%, or 95% confidence intervals. However, experts’ reasonable bounds are often much narrower bounds than absolute bounds (Stainforth et al. 2005). For example, Teigen and Jørgensen (2005) found that when asked for 90% confidence estimates, people often provided less than 50% confidence intervals. Yaniv and Foster (1995) also found that when asked for 98% confidence intervals, people often provided roughly 60% intervals. This occurs because there is a cognitive trade-off between providing wide, less informative intervals and providing precise, informative estimates that may not include the true estimate (Yaniv and Foster 1995). I did not explicitly ask experts to provide upper and lower estimates that reflected a given confidence interval; instead I asked for the “most reasonable” upper and lower limits. The small within-expert variance may have resulted from this linguistic uncertainty (Burgman 2005). However, the within-expert variances from the 80%, 90%, and 95% confidence intervals were minute relative to between-expert variance and, even with large increases, the within- expert uncertainty would likely still be much less important.

The initial conditions affected model predictions substantially over the short-term, but began converging over time for most experts. Those models that have different predictions between initial conditions, however, may converge over a longer time period. As a result, it appears that this system is not completely limited by initial conditions and that, over time, the dynamics and structure will be dictated by natural disturbances and management. However, if a range of initial conditions currently exist

81 Chapter Three in the landscape, these subunits may react differently to management actions on a short management timeframe and may require different management strategies. I also found that variance was lowest when using the Average initial condition. This was not surprising because the Average initial condition was the mean of percent estimates provided by experts and thus more closely matched the conceptual models of experts. The Open initial condition matched most experts’ conceptual models of the likely future stable state of Box-Ironbark woodlands and forests; whereas the Dense initial condition was expected by fewer experts. As a result, most experts’ models behaved more erratically when beginning under the Dense initial condition, leading to more heterogeneous predictions between experts and an increase in the variance estimate. Interestingly, this may mean that models are less reliable when initiated with conditions similar to where restoration efforts are being focused; high density stands.

Sensitivity analyses investigated which transition agents most strongly influenced the development of Low density mature. For many of the experts’ models, these transition agents were drought, intense overstory fire, and growth to Low density mature. The most influential transition agents are ideal candidates for building simple models. It would also be beneficial to compare which transition agents were most influential across all experts. However, because experts specified their own models and transition agents were predicted to affect vegetation differently between models, it was not possible to compare most parameters between experts. As a result, it was only possible to determine the most influential parameters for each expert’s model. However, transition agents that were influential in some experts’ models but were uncertain between experts are very useful parameters for monitoring because, by collecting data, mangers can learn about the effects of these processes, update models, and reduce uncertainty in the effects of transition agents over time. In addition, sensitivity analyses showed that the regression relationships are stronger when experts’ within-expert variance was high. This occurred because parameters with large ranges are more likely to show an effect.

3.5.2 Limitations

I incorporated uncertainty due to stochastic ordering through Monte Carlo simulations, partial expert knowledge and stochastic rates of disturbances through three-point

82 Chapter Three parameter estimates, and different knowledge between experts through constructing unique STSMs. However, there are other sources of uncertainty that may affect model predictions. These include underspecificity and model uncertainty (Regan et al. 2002).

Underspecificity is a type of linguistic uncertainty that occurs when there is unwanted generality (Regan et al. 2002). Underspecificity occurred in this case study because the vegetation states I defined were very broad; within one vegetation state, small trees could have any DBH between 5-59 cm and densities of small stems could be anything greater than 100. This broad classification was necessary to minimize the number of vegetation states and the possible confusion in specifying model estimates for many vegetation states. However, some experts found that these broad categories made it difficult to estimate the rates and effects of disturbances. Underspecificity also arises due to the ambiguity in asking experts for “reasonable” bounds, as discussed above.

I did not investigate structural uncertainty that arises because I chose one model structure and ignored other equally plausible models (Wintle et al. 2003). Structural uncertainty could have been incorporated by constructing several models and asking experts to parameterize each one, or by allowing each expert to construct their own model using unique vegetation states. Using multiple model structures would have been onerous for experts, considering that interviews for one model structure took between 2 to 4 hours per expert. Using unique model structures for each expert would also have been detrimental as I would not be able to directly compare results of experts’ models. As such, I chose to use one simple model structure for all analyses.

3.6 Conclusions

In addition to incorporating various sources of uncertainty into models, I have also been able to classify and quantify these different sources of uncertainty. I discovered that between-expert uncertainty in this case study was quantitatively high, and appeared to be more important than other sources of uncertainty. As a result, it is now possible to try and reduce this uncertainty. Specifically, in this case study, experts agreed that drought, wildfire, and growth have strong influences on the development of Low density mature, but strongly disagreed about how these agents affect vegetation. There

83 Chapter Three are several options to address this uncertainty. First, increasing the number of experts sampled might discern if there is agreement and clumping between experts. In addition, current predictions made by these experts could be weighted, if there were some reason to select one expert over another, and combined with adaptive management to determine which expert is best over time. Adaptive management is a formal decision- making framework that can be used to make decisions under uncertainty (Holling 1978). It is an iterative process that integrates monitoring directly with current models, and works to decrease uncertainty in biological processes through time (Nichols et al. 1995; Nichols and Williams 2006). While it is beyond the scope of this work to collect long-term monitoring data, I explore the application of adaptive management as a decision-making tool in Box-Ironbark forests and woodlands in Chapter 4. Specifically, I investigate how applying various decision criteria to model predictions affects management decisions.

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Table 3.1. The percent of cells in each vegetation state at the beginning of simulations for each of the three initial conditions. Average represents the current estimated proportion of vegetation in each state, while Open and Dense represent two possible historic conditions.

Initial High density Low density High density Low density Condition regrowth regrowth Mature mature Average 71.2 10.3 12.5 6 Open 5 30 15 50 Dense 30 5 50 15

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Table 3.2. The mean variance estimates and 95% Bayesian Credible Intervals (in square brackets) for the VCA parameters, between-expert, within-expert, and Monte Carlo/residual, derived from posterior probability density distributions. Values are provided for each transition agent confidence interval, 80%, 90%, and 95%.

Parameter Confidence interval Variance estimate 80% 0.957 [0.263, 6.070] Between-expert 90% 0.955 [0.261, 6.013] 95% 0.983 [0.269, 6.234] 80% 0.023 [0.018, 0.030] Within-expert 90% 0.013 [0.010, 0.018] 95% 0.019 [0.014, 0.024] 80% 0.140 [0.138, 0.142] Monte Carlo / Residual 90% 0.117 [0.115, 0.119] 95% 0.132 [0.130, 0.134]

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Table 3.3. The three transition agents for each expert with the steepest slope for the regression line between transition agent parameter values and the percent of cells in Low density mature at 150 years. Results are sorted in descending order by absolute value of the slope.

Expert Transition agent From To Slope Standard error R-squared 5 Drought + Mistletoe LDR LDR -1.497 0.785 0.137 5 Drought + Mistletoe HDR HDR -1.468 0.787 0.131 5 Growth LDR LDM -1.438 0.789 0.126 1 Drought HDM HDR -1.187 0.533 0.178 1 Growth HDM LDM -1.136 0.537 0.163 1 Coppice LDR HDR -1.135 0.537 0.163 2 Growth HDR LDM -1.095 0.408 0.238 2 Overstory wildfire HDM LDM 1.058 0.413 0.222 2 Growth HDR HDM -0.991 0.420 0.195 4 Drought HDM LDM 0.440 0.251 0.118 4 Coppice LDR HDR 0.422 0.253 0.108 4 Drought LDM LDM -0.387 0.255 0.091 3 Overstory wildfire LDR LDR -0.371 0.241 0.094 3 Overstory wildfire HDM LDR -0.258 0.247 0.045 3 Windstorm HDM LDR 0.257 0.247 0.045

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Expert 1 0.025 Expert 2 0.020 Expert 3 Expert 4 (A) 0.015 Expert 5 Density 0.010

0.005

0.000

0.020

0.015 (B) 0.010 Density

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0.030

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0.020 (C) 0.015 Density 0.010

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0.000

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(D) 0.010 Density

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(E) 0.006

Density 0.004

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0.000 0.020

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(F) 0.010 Density 0.005

0.000 0 100 200 300 400 500 Years until transition occurs Fig. 3.1. Probability density functions for parameter values drawn randomly from beta distributions for transition agent: ‘Growth from High density regrowth to High density mature.’ Graphs are for (A) all experts, 90% conﬁdence interval, (B) Expert 1, (C) Expert 2, (D) Expert 3, (E) Expert 4, and (F) Expert 5. Lines represent distributions drawn from 88 experts’ 80% (dotted lines), 90% (solid lines), or 95% (dashed lines) conﬁdence intervals. Chapter ree

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Fig. 3.2. Probability density functions for parameter values drawn randomly from beta distributions for transition agent: (A): ‘Self-thin from High density mature to Low density mature,’ (B): ‘Growth from Low density regrowth to Low density mature,’ (C): ‘Wildﬁre,’ and (D): ‘Windthrow.’ Distributions were drawn from experts’ 90% conﬁdence intervals. Experts not shown did not specify these transitions. 89 90

Land in Low density mature (mean % ± 2 x st.dev.) 35 40 10 15 20 25 30 0 5 all management scenarios. variation (innersolid lines)andMonte Carlosimulations(outerdotted lines).Results are fortheAverage and initial condition, 90%conﬁdence interval, Fig. 3.3. emeanpercent oflandin Low density mature over Error timeforeachexpert. barsrepresent twostandard deviations forwithin-expert 0 ● ● Expert 5 Expert 4 Expert 3 Expert 2 Expert 1 ● 15 ● 30 45 ● 60 Time (years) ● 75 ● 90 105 ● 120 ● 135 ●

150

a h C ree r e t p years. Results areforthe(A)80%,(B)90%, and(C)95%conﬁdence intervals. expert uncertainty,andVDDT MonteCarlosimulations/residualsevery10years for150 Fig. 3.4.Barplotsshowing the totalvarianceduetobetween-expertuncertainty,within Variance 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (A) (C) (B) Between−expert Within−expert Monte Carlo/Residuals 10 20 30 40 50 60 Time (years) 70 80 90 100 110 120 130 C h 140 a - p t e r 150 ree 91

Density 0.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 0.4 0.5 found in (B). in found variance. Estimatesfor the variance ( intercepts foreachexpert variance, variance,(B), between-expert and(D)Monte (C)within-expert Carlo/residual Fig. 3.5. Bayesian posteriorprobability density functionfor(A)allvariance parameters, 10 20 30 40 50 60 10 20 30 40 50 0 0 (D) (C) (B) (A) 0.12 0.01 0.008 0.114 0.010 0.12 1.00 Variance (naturallogscale) ● 0.116 0.012 1.00 7.50 0.015 7.50 0.118 0.018 60.00 ± standard deviation) are are deviation) standard ± Monte Carlo/residualvariance Within-expert variance Between-expert variance 60.00 0.022 0.120 C ● h 400.00 a Expert 5 Expert 4 Expert 3 Expert 2 Expert 1 400.00 p t e r ree

Chapter ree

(A) Open-Average (B) Dense−Average

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Differences in amount of land in 0 30 60 90 120 150 0 30 60 90 120 150 Time (years)

Fig. 3.6. e diﬀerences in amounts of land in Low density mature between initial conditions over time: (A) Open and (B) Dense are compared to Average. Analyses are for the 90% conﬁdence interval and the Natural Disturbance scenario only.

93 Chapter ree

Between−expert Within−expert

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Fig. 3.7. Total variance due to between-expert uncertainty, within-expert uncertainty, and VDDT Monte Carlo / residuals for each initial condition. Results are for the 90% conﬁdence interval at year 150.

94 (A) (B) (C) ● Expert 1 Expert 2 60 Expert 3 Expert 4 2 x st.dev.)

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Fig 3.8. e mean percent of land in Low density mature at 150 years for each expert and the (A) Average initial condition, (B) Open initial condition, h a p

and (C) Dense initial condition. Errorbars represent two standard deviations for within-expert variance (inner solid lines) and Monte Carlo simulations t e r (outer dotted lines). Results are for all management scenarios and the 90% conﬁdence interval. ree 95

96 CHAPTER FOUR

Incorporating uncertainty into decisions about forest restoration; towards adaptive management

98 Chapter Four

4.1 Abstract

Forest management and restoration are inherently uncertain. Formal decision analysis frameworks can assist in making responsible management decisions and reduce uncertainty in management actions over time. Here I explore the first three steps of a particular framework, adaptive management, with particular focus on the ability to explicitly incorporate uncertainty and risk into decisions about forest restoration. I rely on the case study of restoring Box-Ironbark forests and woodlands in Victoria, Australia, where managers are aiming to expedite the development of mature woodlands through several novel management options: ecological thinning, current harvesting regulations, and natural development without harvesting. Expert models were used to predict the potential effects of management actions on vegetation structure but due to large differences between experts’ models, the potential outcomes of management were highly uncertain. I applied several explicit management objectives and decision criteria, with varying levels of complexity and associated risk, to uncertain model predictions to assist in decision-making. Results indicated that the optimal management action depends on the objective chosen, which highlights the importance of selecting clear objectives, acknowledging the level of risk managers are willing to accept, and utilizing a decision analysis framework that can facilitate learning about the effects of uncertain management actions. In this case study, the most risky and uncertain management action is ecological thinning, while natural development without harvesting is the most risk-averse strategy.

99 Chapter Four

4.2 Introduction

Uncertainty is inherent in forest management and restoration. This is because forest management often involves broad areas, long time horizons, and multiple stakeholders (Tester et al. 1997; Rieman et al. 2001; Kangas and Kangas 2004). In addition, the complete effects of management actions on ecosystems are typically unknown (e.g. Williams et al. 1996; Prato 2000). When management actions are relatively novel, there will likely be limited monitoring data and few appropriate field sites available for chronosequence studies to inform management actions. Also, due to the slow growth of the dominant species, it is typically not feasible to collect new monitoring data on the long-term effects of forest management before decision support is required. Finally, in addition to management actions, stochastic processes, such as wildfire and insect outbreaks, alter vegetation but occur randomly in space and time and have uncertain rates of occurrence (McCarthy and Burgman 1995).

To assist in making responsible management decisions in the presence of uncertainty, it is wise to make use of a formal decision analysis framework (Clemen 1996). Adaptive management (AM) is a structured, iterative process that aims to decrease uncertainty in management actions through time by integrating monitoring data and current quantitative models (Holling 1978; Nichols et al. 1995). To do this, AM requires five elements: explicit management objectives; possible management actions; models that predict the effects of management actions; measures of confidence in the models; and a monitoring program to validate and update model predictions over time (Nichols and Williams 2006; Duncan and Wintle 2008). In this chapter, I investigate formal decision analysis using the first three elements of AM - objectives, actions, and models - in a case study of the restoration of Victoria’s Box-Ironbark forests and woodlands.

Due to a history of intense utilization dating from the 1830s (Forests Commission of Victoria 1928; Environment Conservation Council 2001), Victoria’s Box-Ironbark forests and woodlands are highly degraded ecosystems lacking mature stands with large, hollow-bearing trees (Kellas 1991; Soderquist 1999). Managers are now interested in expediting the development of large, hollow-bearing trees because they provide primary nesting and foraging sites for many fauna species (Bennett et al. 1994; Alexander 1997;

100 Chapter Four van der Ree et al. 2006; Selwood et al. 2009), which are declining due to a lack of appropriate habitat (Calder and Calder 1994; Alexander 1997; Soderquist 1999; Environment Conservation Council 2001). Managers of the Box-Ironbark forests and woodlands are relying on an AM framework in the implementation of novel restoration strategies. However, while the general aim of restoration in Box-Ironbark forests and woodlands is to “accelerate the development of older-growth characteristics” (Parks Victoria 2007) through management actions, this is a vague objective.

The first step in AM is to establish unambiguous and explicit objectives (Shea et al. 1998; Possingham et al. 2001; Nichols and Williams 2006). I investigated the impacts of different management objectives on decision-making by applying several explicit decision criteria to predictions of the expected outcomes of alternative management actions. Decision criteria included several deterministic criteria for decision-making under risk, a probabilistic objective function, stochastic dominance, and risk-spreading. Criteria were selected because they are of varying complexity, there are different levels of risk associated with the criteria, and each technique has its own set of strengths and weaknesses. Deterministic criteria for decision-making under uncertainty have been developed extensively in decision theory, game theory, and statistics to select an action that maximizes a positive expected outcome or minimizes a negative expected outcome (Neter et al. 1973). Probabilistic objective functions are used to evaluate alternative management actions by taking into account both benefits and costs of each action (Dorazio and Johnson 2003). They can take different functional forms to represent decision-makers’ attitudes towards risk, and are typically used to maximize the probability of achieving a desired outcome (Frank 1994). Stochastic dominance takes entire distributions into account and incorporates risk-aversion to compare uncertain outcomes of management actions (Levy 1998; Graves and Ringuest 2009). Finally, risk- spreading is used to maximize the positive expected outcomes of a set of actions while minimizing the risk, or variance, of that set of actions by allocating resources to various actions or portfolios (Markowitz 1952; Markowitz 1991).

The second step of AM involves identifying management actions to achieve objectives (Nichols and Williams 2006). Managers are considering three alternative management actions to assist in expediting large tree development in Box-Ironbark forests and woodlands: ecological thinning, current timber harvesting regulations, and growth

101 Chapter Four without timber harvesting (Department of Natural Resources and Environment 1998; Sutton 2000; Parks Victoria 2004). Ecological thinning is currently being trialled as an intensive harvesting method that removes all stems within a certain distance around candidate trees to reduce competition and allow retained trees to grow faster than those without thinning. Current harvesting aims to retain increased numbers of medium and large trees in harvesting coupes to allow development of hollows, and growth without timber harvesting relies on stem growth and natural disturbances to develop large, hollow-bearing trees.

The third step of AM involves constructing quantitative models that predict the effects of management actions. I constructed state-and-transition simulation models (STSMs) that describe Box-Ironbark forest and woodland dynamics by cataloging vegetation into alternative states according to vegetation structure and by specifying possible transition agents between states (Westoby et al. 1989). These transition agents include processes such as fire, windthrow, timber harvesting, and vegetation growth. Since their inception as conceptual models, state-and-transition models have been integrated into computerized simulation models and used extensively to predict the effects of natural disturbances and management actions on forest and woodland vegetation (e.g. Forbis et al. 2006; Barbour et al. 2007b; Wales et al. 2007). The additional requirements for STSMs over conceptual state-and-transition models are estimates of transition probabilities.

Due to the novelty of management actions being used in this system, there are limited field sites currently available to perform chronosequence studies and a dearth of monitoring data to inform predictive models. Similar to many other studies that are limited in empirical data (e.g. Morgan et al. 2001; Martin et al. 2005; Forbis et al. 2006), I relied on expert opinion to construct and parameterize STSMs. Almost all previous STSMs that use expert knowledge have relied on consensus estimates to construct a single model (e.g. McIntosh et al. 2003; Bestelmeyer et al. 2004; Hemstrom et al. 2007; Wales et al. 2007). By constructing a single model that reflects the average views of several experts, managers ignore the uncertainty that arises because of differences between experts, which can be quite substantial (Chapter Three)(Morgan et al. 2001; Pellikka et al. 2005; Duncan and Wintle 2008). I constructed STSMs for each of several experts, which resulted in numerous different predictions for a single

102 Chapter Four management action (Chapter Two). Consequently, deciding between management actions is difficult because there is large uncertainty in how management actions will affect vegetation. To assist with management decisions under uncertainty in this case study, I relied on an AM framework by applying various decision criteria to predictions from STSMs that depict the effects of management and natural disturbances on Box- Ironbark vegetation structure.

4.3 Methods

A detailed description of model parameterization and construction is included in Chapter Two. I used STSMs implemented in VDDT (ESSA Technologies Ltd. 2007) to predict the long-term vegetation dynamics of Box-Ironbark forests and woodlands in Victoria, Australia. All data were provided in the form of elicited expert opinion and were gathered through electronic surveys and interviews. Electronic surveys were used to determine the states in the VDDT models (Fig. 2.3) and interviews were conducted so that experts could specify the rates and effects of transition agents on vegetation states (Table 2.1; Appendix B). I completed data analyses on VDDT model predictions using the software R version 2.6.2 (R Development Core Team 2008)(Appendix E for code).

4.3.1 Modelling the effects of management

I constructed a model in VDDT for each expert using the same vegetation states and transition agents (Chapter Two; Fig 2.3; Table 2.1; Appendix B). Experts’ models varied from each other because the effects of transition agents on vegetation states and the probabilities of transitions were specified by each expert, and were therefore unique. The treatment of uncertainty in this chapter was limited to between-expert uncertainty because that was demonstrably the largest source (up to 90%) of uncertainty in model predictions (Chapter Three). As a result, other sources of uncertainty, including the effects of initial conditions and within-uncertainty, were not included. Only the Average initial condition and the mean/middle estimates from experts’ three-point estimates were used for parameterization of these models.

103 Chapter Four

To investigate the effects of management actions on vegetation states, I varied the proportion of area in the landscape allocated to each of the three management scenarios – Ecological Thinning, Current Harvesting, and Natural Disturbances. The proportion of land in each management scenario was adjusted by 0.1, between 0.0 and 1.0, such that if 0.5 of land was in the Ecological Thinning scenario and 0.3 of land was in the Current Harvesting scenario, then 0.2 of land would be in the Natural Disturbance scenario. This resulted in a total of 66 replicate VDDT models for each expert and was done to show how the entire landscape might be affected depending on the distribution of land to different management actions. VDDT models simulated the dynamics of 1000 independent cells, which were assigned a random age to ensure the model simulation reflected the current multi-age structure of the stands. For each model, Monte Carlo methods, which are used to model natural stochasticity (Vose 1996), completed 100 simulations over 150 timesteps. In VDDT, Monte Carlo methods randomize the order that transitions occur by sampling from a uniform distribution (ESSA Technologies Ltd. 2007). In total, VDDT models included predictions for five experts each with 100 Monte Carlo simulations, which resulted in 500 simulations for each of the 66 replicate models.

The 66 replicate VDDT models predicted the percent of land in each vegetation state over time. Managers are interested in the development of vegetation states with large Eucalyptus trees (≥60 cm diameter at breast height) due to their role as habitat for many fauna species (Alexander 1997; Soderquist 1999; van der Ree et al. 2006). As a result, I applied the decision criteria to the combined amount of High and Low density mature, both of which contain large trees. Low density mature could not be considered alone due to the slow development of this vegetation type (Chapter Two).

4.3.2 Decision criteria

To assess which management scenario would be optimal for facilitating the development of woodlands in High and Low density mature, I applied a selection of decision criteria to the VDDT model results at 50 years. Forest dynamics models tend to focus on slightly shorter timeframes (e.g. 20 years: Raulier et al. 2003; Knowe et al. 2005). However, Box-Ironbark forests and woodlands are quite slow-growing (Department of Natural Resources and Environment 1998) and 50 years allowed

104 Chapter Four sufficient time for the models to simulate vegetation growth and distinguish between management regimes.

Deterministic criteria for decision-making under risk

I adapted the first set of deterministic criteria for decision-making under risk from Neter et al. (1973) and Morgan and Henrion (1990) and applied them to the 500 results pooled for each replicate. The first criterion I investigated was the more risk- prone Maximax criterion, where the management action that results in the highest possible percent of land in High and Low density mature is optimal, but ignores the possibility of negative consequences. I also investigated a variant of the Maximax criterion that I called the Average Maximax criterion. For this criterion only, I calculated the average percent of land in High and Low density mature for each of the 66 replicate VDDT models, instead of the pooled 500 results, and applied the Maximax criterion to these data. Third, I considered a permutation of the Minimax Regret criterion, where any of the 66 replicates that led to a negative outcome was removed and the remaining replicates were assessed to determine which management scenario maximized the percent of land in High and Low density mature. In this study, a negative outcome occurred when a model predicted the percent of High and Low density mature would fall below the arbitrary regret threshold of 8%. I also investigated the outcome if the regret threshold were set at 5% or 10%. After removing any replicate that predicted the amount of High and Low density mature might fall below the threshold, I then selected the management action that maximized the expected percent of land in High and Low density mature from remaining replicates. Finally, I examined which management scenario would be optimal according to the Maximin criterion, which is a more risk-averse criterion and maximizes the lowest possible estimates of land in High and Low density mature.

Probabilistic objective function

The VDDT predictions were quite heterogeneous between experts and I wished to represent this variation by converting predictions to continuous probability density functions (pdfs). This was done by combining all of the experts’ predictions of the percent of land in High and Low density mature at 50 years for a given replicate, and

105 Chapter Four generating Bayesian posterior pdfs from these data. I converted VDDT predictions to Bayesian posterior pdfs for only three of the 66 replicates. I chose the three replicates where 100% of the landscape was placed in one of the management scenarios because models showed the greatest differentiation when the entire landscape was dedicated to only one management scenario. Converting experts’ model predictions to Bayesian posterior pdfs first involved logit transforming VDDT model results, which are in the form of proportion data, to improve normality (McCarthy 2007). Logit transformed model predictions were pooled and sampled in order to generate normal pdfs. These normal pdfs were used to predict new hypothetical normal posterior pdfs; the Bayesian posterior pdfs. I selected vague priors for this analysis; they were drawn from a normal distribution with mean = 0 and variance = 100 (McCarthy 2007). Analyses were undertaken with WinBUGS version 3.0.3 (Lunn et al. 2000) and the coda package (Plummer et al. 2008) in R; see Appendix E for code.

For the probabilistic objective function, I converted the Bayesian posterior pdfs for the three management scenarios to cumulative density functions (cdfs) to indicate the probability of having a given percent, or less, of land in one of the desired vegetation states, High and Low density mature. I applied an objective function to the cdfs to decide between management actions; the function I chose minimized the probability of falling below the current level of land in the desired vegetation states in 50 years. This objective takes risk into account by acknowledging the probability that management may decrease the amount of High and Low density mature on the landscape.

Stochastic dominance

Another common method to decide between management actions in the presence of uncertainty is stochastic dominance (Levy 1992; e.g. Goldstein et al. 2006; Knoke et al. 2008; Graves and Ringuest 2009). Stochastic dominance takes whole cdfs into account, instead of moments such as the mean or variance (Graves and Ringuest 2009). First- order stochastic dominance is satisfied assuming that the utility function increases with x – i.e. the value of the management action increases as the proportion of land in High and Low density mature increases – and if

B (xF ) ≥ F A (x ), (4.1)

106 Chapter Four

where FA is the cumulative density function for management scenario, A, and FB is the cumulative density function for management scenario, B (Levy 1992; Graves and Ringuest 2009). When the cdfs intersect for two management scenarios, first-order stochastic dominance cannot be used to determine dominance, and second-order stochastic dominance is used instead. Second-order stochastic dominance again assumes that the value of the utility increases with x, but also assumes risk-aversion is present, such that utility is a decreasing function of the outcome and the action with the lowest risk is preferable (Knoke et al. 2008; Graves and Ringuest 2009). Under second-order stochastic dominance, management action A dominates B if

z []B () - FxF A () ≥ 0,dxx ∫ ∞− (4.2) holds for all z over x, where z represents all possible values of x, given the strict inequality holds for at least one point (Levy 1998; Graves and Ringuest 2009), which means the area under the cdf of A must be less than or equal to the area below the cdf for action B for every value of z (Knoke et al. 2008). I applied first-order stochastic dominance rules to the three cdfs generated from the Bayesian posterior pdfs and second-order stochastic dominance to the integrated areas under the cdf curves for the management actions that did not display first-order stochastic dominance.

Risk spreading

Risk-spreading is related to portfolio theory for financial allocation (Markowitz 1991), and is used to maximize the expected outcome of management while incorporating the potential risk, or variance, of management actions. This is done by allocating a different proportion of total land to each of the three management scenarios in order to attain a certain percent of land in the desired vegetation states, High and Low density mature, in 50 years. The percent of land that managers wish to have in High and Low density mature is called the critical level, Lcrit.

To determine the optimal allocation of land, I first calculated the means, μi , and

2 variances, σ i , from the posterior probability density functions for the three replicates

107 Chapter Four where 100% of land is allocated to one of the three management scenarios (Table 4.1). The values of i range from A to C, where A is Natural Disturbance, B is Current Harvesting, and C is Ecological Thinning. Next, I calculated the expected mean percent of land in High and Low density mature, Μ , and expected variance, Σ 2 (Equation 4.3).

μ +=Μ μ + xxx μCCBBAA 2 222222 xxx σσσ CCBBAA +++=Σ X2

X = xrx σσ + xrx σσ + xrx σσ CBCBCACABABA , (4.3)

where xA, xB, and xC are the proportions of land allocated to the three management

2 2 2 scenarios, A, B, and C, such that xA + xB + xC = 1.0; μ A , μ B , μC , σ A , σ B , andσ C are as defined above; Χ is the covariance matrix; r is the correlation coefficient; and σ A ,

σ B , and σ C are the standard deviations when all land is placed under one management scenario, A, B, or C (Table 4.1).

I then calculated a beta distribution, with shape parameters α and β , to describe the distribution of land in High and Low density mature, dependent on allocation (Equation 4.4)(McCarthy 2007),

⎡ (1 Μ−Μ ) ⎤ α Μ= −1 ⎢ 2 ⎥ ⎣ Σ ⎦

⎡ ()1 Μ− ⎤ β 1 ΜΜ−= − .1 ()⎢ 2 ⎥ ⎣ Σ ⎦ (4.4)

Next, I set Lcrit, the percent of land in High and Low density mature below which it is undesirable to fall. I performed a non-linear optimization in Microsoft Excel (Microsoft

Corporation 2002) to choose the allocations, xi, that minimized the probability of falling below Lcrit. I was most interested in the optimal allocation of total land when the critical level was set at 18.5%, which is the current estimated percent of land in High and Low density mature. However, I also investigated how allocation would change depending on the critical level.

108 Chapter Four

The correlation coefficient, r, is a term of the covariance and indicates whether the variance in one management action is affected by the variance in another. For example, development of mature Box-Ironbark forests and woodlands might be strongly affected by rainfall, where this dry ecosystem displays markedly increased growth rates with high rain years. It is then reasonable to expect that if there is high rainfall, then the growth rates and development of mature forests would increase for all land regardless of management action. In this example, the correlation coefficient would be high such that when one management scenario has a higher probability of success then all strategies have a corresponding increase in success. I first assumed that the correlation coefficient, r, was 0. I also investigated how the allocation would change if values for r were increased to 0.3 and 0.7.

4.4 Results

4.4.1 Vegetation dynamics according to management scenarios

Low density mature increased slowly from the initial value of 6% and thus could not be used to discriminate between the optimality of management scenarios, especially early during simulations (Fig. 4.1a and b). It was not until near the end of model simulations that most experts’ models distinguished between the optimality of management scenarios (Fig. 4.1c). The proportion of High density mature, however, increased quickly from the initial value of 12.5% and differentiated between management scenarios early in the simulation (Fig. 4.2). To provide better insight into effects of management at a shorter and more reasonable management timeframe, the predictions for High and Low density mature were combined for all further analyses.

The combined percent of High and Low density mature began to differentiate between management scenarios as early as timestep 10 for some experts (Fig. 4.3a; Experts 1 & 3). Throughout simulations, patterns developed and were consistent for most experts. At 50 years, the results for Expert 1 supported Ecological Thinning, the results for Expert 3 promoted a landscape primarily under Current Harvesting, and all other experts’ results indicated that no timber harvesting (Natural Disturbance) was most beneficial for the development of forests with large trees (Fig. 4.3b). Interestingly, at

109 Chapter Four timestep 150, the model for Expert 5 changed to promote Ecological Thinning because of the delayed growth response of trees promoted by thinning (Fig. 4.3c).

In addition to predicting different effects of management actions, experts’ models also predicted a very different percent of the landscape in the desired vegetation states (Fig. 4.3 insets and 4.4). In fact, due to differences between experts, the lowest estimate for Expert 1 was greater than the highest estimate for Expert 5.

4.4.2 Deciding between management scenarios

Deterministic criteria for decision-making under risk

I applied each of the deterministic criteria for decision-making under risk to the predicted percent of land in High and Low density mature in 50 years. Data were pooled together for 5 experts and 100 Monte Carlo simulations for three of the criteria and were averaged for Maximax Average. Criteria were considered in order from the most risky to the most risk-averse.

The Maximax criterion selects the management scenario that maximizes the percent of land in High and Low density mature. I applied this criterion to the pooled VDDT model results and found that Ecological Thinning was expected to be the most beneficial management action, with 62.2% of land in High and Low density mature (Fig. 4.5a). This result occurred because Expert 1 predicted the highest overall percent of land in High and Low density mature (Fig. 4.4) and, thus, the Maximax result was based on the model for Expert 1 (Fig. 4.3b).

A second analysis using the Maximax criterion was conducted using VDDT model results averaged over experts and Monte Carlo simulations; the Maximax Average criterion. This criterion indicated that Natural Disturbance was the most beneficial management strategy with 30.3% of vegetation in High and Low density mature (Fig. 4.5b), which was similar to the results of most experts (Fig. 4.3b; Experts 2, 4, & 5).

I also investigated the Minimax Regret criterion, where all replicates that predicted the amount of High and Low density mature woodlands will fall below 8% were removed

110 Chapter Four

(Fig. 4.4) and only remaining replicates were considered when selecting the management scenario that maximizes the percent of High and Low density mature. Putting the majority of land in Natural Disturbance with some land in Ecological Thinning was preferred (Fig. 4.5c). This occurs because, with an 8% regret threshold, replicates with the majority of land in Ecological Thinning fall below the threshold for Expert 5, thus removing them from consideration (Fig. 4.4). The remaining replicates are maximized, which in this case represents the replicates from Expert 1 (Fig. 4.4), not including the high predictions when most land is allocated to Ecological Thinning, and reflects the predictions of this expert (Fig. 4.3b; Expert 1). If the regret threshold were set at 5%, then only replicates allocated with predominantly Ecological Thinning would be removed (Fig. 4.4), meaning that replicates with Natural Disturbances and Current Harvesting would be considered (Fig. 4.5). However, if the threshold were set at 10%, then most replicates for Expert 5 would fall below this threshold (Fig. 4.4) and only three replicates with predominantly Natural Disturbance would be considered (Fig. 4.5).

The final criterion I investigated for decision-making under risk was the Maximin criterion, which maximizes the lowest predictions. Under this criterion, Natural Disturbance was slightly better than the other scenarios, but a large proportion of the landscape could be under Current Harvesting and still be beneficial to vegetation (Fig. 4.5d). This criterion selected the expert with the lowest possible predictions, Expert 5 (Fig. 4.4), and reflected the model predictions for that expert (Fig. 4.3b).

Probabilistic objective function

The criteria for decision-making under risk selected the predictions of single experts because there were large differences in predictions between experts. To incorporate the heterogeneity between experts into a single distribution, I conducted additional analyses using Bayesian posterior probability density functions (pdfs) for three of the 66 replicate models where all of the landscape was in one of the three management scenarios (Fig. 4.3b: vertices of the graphs). I selected only these three replicates because they displayed the greatest differentiation in vegetation. The Bayesian posterior pdfs for Natural Disturbance and Current Harvesting were very similar with means of 30.2% and 26.2%, respectively, of land in High and Low density mature in 50 years (Fig. 4.6a and

111 Chapter Four b; Table 4.1). While the distribution of Natural Disturbance predicted slightly more land in High and Low density mature, the distribution for Current Harvesting was more certain, with narrower 95% credible intervals and a smaller standard deviation. The Bayesian posterior pdf for Ecological Thinning was highly uncertain with tails greater than those of Natural Disturbance and Current Harvesting (Fig. 4.6c; Table 1). There was a total of 18.5% of land in High and Low density mature at the beginning of the simulation and models predicted that all management scenarios could lead to a decrease from current levels.

The criteria for decision-making under risk select management actions that minimize or maximize a given outcome, but do not incorporate the probability of achieving a management goal. I was interested in minimizing the probability that there will be less land in High and Low density mature in 50 years than there is currently, 18.5%. I applied this probabilistic objective to cumulative density functions (cdfs) derived from the Bayesian posterior pdfs and found that Natural Disturbance minimized the probability of falling below this undesired threshold (P=0.19; Fig. 4.7). The estimated probabilities that Current Harvesting and Ecological Thinning will result in this undesired outcome were higher: P=0.28 and 0.40, respectively (Fig. 4.7). A more risky objective function could be constructed where managers minimized the probability that there will be less than 50% of land in High and Low density mature in 50 years. This would result in Ecological Thinning being the desired management action (Fig. 4.7), even though the likelihood of achieving this threshold is quite low (Fig. 4.6).

Stochastic dominance

Using stochastic dominance rules, Natural Disturbance dominated Current Harvesting according to first-order stochastic dominance because the cdf of Natural Disturbance is entirely below/to the right of the cdf of Current Harvesting (Fig. 4.7)(Graves and Ringuest 2009). The cdf for Ecological Thinning, however, intersected with the cdf for Natural Disturbance and, thus first-order stochastic dominance could not determine which of these two scenarios was preferred (Fig. 4.7). As a result, I applied second-order stochastic dominance rules to the predictions for Natural Disturbance and Ecological Thinning. Natural Disturbance dominated Ecological Thinning according to second- order stochastic dominance because the curve of the area under the cdf for Natural

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Disturbance was below/to the right of the curve of the area under the cdf for Ecological Thinning (Fig. 4.8).

This result holds true with the assumption that these decisions are based on a linear utility function, which indicates that any increase is desired over a decrease in High and Low density mature. If managers were risk-averse and weighted a small loss of mature woodlands as being more important than a substantial gain, using a monotonic increasing utility function, then the differences between Natural Disturbance and Ecological Thinning would be amplified at the lower section of the distribution (Fig. 4.7 and 4.8), which would reinforce the decision that Natural Disturbance is preferred.

Risk Spreading

According to risk-spreading analyses, to minimize the probability of falling below the critical level of 18.5% of land in High and Low density mature in 50 years, 0.46 of land should be allocated to Natural Disturbances, 0.38 to Current Harvesting, and 0.16 to Ecological Thinning (Fig. 4.9a). If, however, managers selected a more risky critical level and aimed for higher percentages of land in High and Low density mature, then an increasing proportion of land should be allocated to Natural Disturbance, up to a threshold of approximately 35% where the optimal management strategy alters and all land should be allocated to Ecological Thinning (Fig. 4.9a). This threshold was reached as the probability of Natural Disturbance falling below the critical level exceeded the probability of Ecological Thinning falling below that same critical level (Fig. 4.7). If the critical level was very high and managers aimed to have over 75% of land in High and Low density mature in 50 years, these analyses indicated that the allocation of land to each management scenario could be equal and was relatively inconsequential (Fig. 4.9a). This was because the probability of achieving the risky objective of 75% in High and Low density mature was very small for all scenarios (Fig. 4.6) and, as a result, land could be allocated to any scenario without substantially altering the minute probability of achieving this critical level in 50 years.

Higher correlation coefficients, r, indicated that an increased probability of achieving a desired outcome with one management scenario will also result in an increased probability of achieving that outcome with another scenario. When r was increased,

113 Chapter Four there was a shift to allocate more land to Natural Disturbance for a given critical level and the threshold at which managers should switch to Ecological Thinning was increased (Fig. 4.9b and c). This occurred because Natural Disturbance was a low risk/low variance strategy and Ecological Thinning was a high risk/high variance strategy (Fig. 4.6), but Natural Disturbance had a lower probability of achieving high critical levels than Ecological Thinning (Fig. 4.7). As a result, when r was high, all strategies had a higher probability of success and, thus, there was less benefit to allocating land to a risky action, Ecological Thinning.

4.5 Discussion

In this chapter, I investigated the implications of applying multiple decision criteria to uncertain forestry management actions. This research highlights the importance of applying explicit objectives when making management decisions for uncertain systems because the optimal action depends on the management objective chosen. In addition, the decision criteria provided objective and explicit methods to make management decisions, and would oblige managers to describe their willingness to accept risks.

I found that Low density mature woodlands will likely develop slowly on the landscape and, as a result, any management action that leads to a decrease in this vegetation will be highly detrimental. Ecological Thinning was the most risky management action because its effects were highly uncertain and could lead to large increases or losses in desired vegetation. If managers preferred more conservative decision-making, natural disturbances without harvesting was the preferred management option.

4.5.1 Criteria for decisions under uncertainty and risk

The criteria investigated in this case study provide explicit methods to make management decisions. They vary in their analytical complexity and have their own set of positive and negative attributes. In addition, each method requires that managers acknowledge their attitudes towards risk.

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When applied to pooled VDDT model results, deterministic decision criteria chose between experts because of large differences between experts’ models. Experts often disagree about the effects of management (Chapter Two)(Alho and Kangas 1997; Duncan and Wintle 2008). However, considering that three of the five experts agreed that timber harvesting may be detrimental to the development of large trees, perhaps this pattern would be reinforced if I increased my sample size by canvassing additional experts (Clemen and Winkler 1999). Selecting between experts is akin to consulting or relying on the opinions of only one expert, which does occur in forest planning (i.e. Kangas et al. 2000). However, the method I utilized is preferable because the choice is explicit and based on objective decision criteria rather than the subjective choice of an individual.

When there are multiple plausible models, it may be more appropriate to integrate over models than to select a single model because distributions generated from integrating over models can assist in characterizing uncertainty (Dorazio and Johnson 2003). I selected a normal distribution to characterize the Bayesian posterior pdfs based on expert model predictions. Due to large differences between experts, it may also have been appropriate to generate a multi-modal distribution that more closely reflected individual experts’ results. However, I assumed that results for the five experts I interviewed reflected a wider population of opinions and, if other experts were sampled, these additional results would reinforce the normal distribution I generated, whereas a multi-model distribution would be less easily generalized to further results. I found that the large differences between experts led to very uncertain Bayesian posterior pdfs. If additional models were constructed using monitoring data or more experts, the predictions from new models could be used to update these preliminary models in an adaptive management framework (Ellison 1996; Dorazio and Johnson 2003; Prato 2008). This would provide narrower pdfs and allow more confident predictions about the effects of these alternative management scenarios on vegetation dynamics.

I applied a probabilistic objective function to cdfs derived from Bayesian posterior pdfs in order to minimize the probability of decreasing mature woodlands from the current level. By incorporating the risk that any of the management actions could lead to a decrease in mature woodlands and providing a probability of achieving the management goal, this method could be considered more applicable than deterministic criteria for

115 Chapter Four decision-making under risk. With this method, managers can explicitly incorporate their willingness to take risks by altering the management objective and determining the corresponding change in the probability of success.

Stochastic dominance is a useful method for decision analysis because it takes entire probability distributions into account (Levy 1998). However, using variance as a measure of risk ignores skewness and the fact that negative deviations are often much less desired than positive deviations (Graves and Ringuest 2009). And, while stochastic dominance is often the choice for deciding between actions because it can incorporate a variety of utility functions (Levy 1992), it often requires too many pairwise comparisons to be useful (Graves and Ringuest 2009). In this case study with only three management alternatives, I first considered a linear utility function to select between management scenarios. However, by considering different attitudes to risk and using monotonic increasing utility function to weight a loss of High and Low density mature as being more important than a substantial increase, it was possible to see how preferences about management strategies would change.

Finally, risk-spreading aims to minimize the probability of falling below a desired critical level and takes variance into account via the portfolio effect, where a proportion of land is allocated to different management scenarios (Markowitz 1952; Markowitz 1991). Risk-spreading is an advantageous method because it maximizes the probability of attaining a desired objective, while facilitating learning about the effects of multiple management actions through monitoring areas under different management actions. While it I did not consider constraints to allocation in this case study, if there are economic or political constraints to allocation, risk-spreading can be optimized to consider these factors. Similar to objective functions and stochastic dominance, risk- spreading can also incorporate attitudes towards risk through setting different critical levels. While it is meant to be a relatively risk-averse strategy, managers can alter the critical level to higher and riskier values to investigate the potential effects on management decisions.

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4.5.2 Managing Box-Ironbark forests and woodlands

Due to the slow growth of Box-Ironbark eucalypts (Department of Natural Resources and Environment 1998) and their ability to withstand self-thinning, Low density mature woodlands will develop slowly on the landscape regardless of management actions. This highlights the importance of maintaining stands with large trees and preventing actions that may lead to the loss of current large trees (Wales et al. 2007). High density mature woodlands, however, may develop sooner because stands with small stems will grow into larger trees before undergoing self-thinning.

I found that more risky decision criteria, such as the Maximax criterion, a probabilistic objective function that attempts to achieve at least 50% of the landscape in High and Low density mature, and risk-spreading with a high critical level, all recommended Ecological Thinning. This occurred because the effects of Ecological Thinning were very uncertain and may substantially expedite development of mature woodlands, albeit with a low probability. On the contrary, more risk-averse decision criteria, such as the Maximax Average criterion, the Minimax Regret criterion, the Maximin criterion, a probabilistic objective function based on minimizing the loss of mature woodlands from the current level, stochastic dominance, and risk-spreading with lower critical levels all recommended Natural Disturbance. This occurred because experts agreed more about the effects of natural disturbances on vegetation dynamics than the effects of ecological thinning, which led to more confident predictions about Natural Disturbance. In addition, several criteria and most experts indicated that current harvesting regulations would likely be superior for developing mature woodlands than ecological thinning, but would not be preferred over Natural Disturbance.

4.5.3 Limitations

There are several areas in which this research could be extended. First, I focused on one timestep, 50 years and only three of 66 replicates where all land is under one management scenario for the Bayesian posterior pdfs, the objective function, and stochastic dominance. Future research could extend these decision analyses throughout all timesteps of model simulation and to the other 63 replicate models to discern if this affects management decisions.

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I speculated which management scenario is optimal based on a single management objective – to expedite the development of High and Low density mature woodlands. Realistically, there may be multiple objectives needing simultaneous consideration. While it is beyond the scope of this paper, it would be possible to undertake multi- criteria decision analyses using our model results. The need for multi-criteria decision analyses is increasing in forest management as managers begin to balance timber harvesting objectives with conservation goals (e.g. Alho and Kangas 1997; Kangas et al. 2000; Barbour et al. 2007a).

Finally, I acknowledge that my models were based on expert opinion rather than empirical data. This does not mean they are not useful; in fact, they are a necessary first step in undertaking an adaptive management plan because they reflect the current available knowledge regarding this system (Kangas et al. 2000; Duncan and Wintle 2008). Martin et al. (2005) illustrated the benefit of using expert knowledge by showing that, when there were limited data, using expert opinion leads to more confident and cost- and time-effective predictions than if expert opinion is ignored. The model predictions generated in this case study can be updated with monitoring data over time in an iterative fashion to modify management decisions and actions (Ellison 1996; Dorazio and Johnson 2003). This could be done by generating distributions for VDDT parameter estimates provided by experts to be used as Bayesian prior probabilities. Managers could then update these prior probability distributions with monitoring data over time to generate posterior distributions for VDDT parameters that more closely reflect current knowledge. This would allow immediate decisions to be made using informed predictions of the effects of management, while accommodating new data and becoming more precise through time.

4.5.4 Conclusions

By applying multiple management objectives to model predictions, I have shown that there is no one optimal solution. This highlights the importance of making management decisions using a framework that is clear and can be interrogated, and emphasizes the necessity of specifying explicit management objectives. The approaches presented here also allow managers to evaluate very uncertain management actions and

118 Chapter Four make informed decisions about restoration activities while expressing their own willingness to take risks. With our current state of knowledge, ecological thinning is the most risky action to employ if managers would like to expedite the development of mature Box-Ironbark forests and woodlands. However, by optimizing the allocation of land to different management scenarios such that most land is in Natural Disturbance and some land is allocated to harvesting, data can be collected to illustrate the effects of management actions on vegetation through time. With this information, model predictions can be updated in an adaptive management framework to better reflect newly acquired knowledge through time in an effort to reduce uncertainty and optimize management strategies.

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Table 4.1. The mean percent of land in High and Low density mature, with upper and lower 95% Bayesian credible intervals (in square brackets), and standard deviations derived from Bayesian posterior pdfs when all of the landscape is allocated to one management scenario, (A) Natural Disturbance, (B) Current Harvesting, or (C) Ecological Thinning. Results are at 50 years.

Standard Management scenario Mean Percent Deviation (A) Natural Disturbance 30.2 [8.2, 55.1] 12.7 (B) Current Harvesting 26.2 [7.0, 48.6] 11.3 (C) Ecological Thinning 27.1 [1.3, 63.6] 18.2

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(A) 10 years (B) 50 years (C) 150 years

1.0 ● 1.0 ● 1.0 ● ● 8.2% ● 13.3% ● 13.9% ● ● ● ● ● 8.4% ● 13.8% ● ● ● 21.2% ● ● ● 0.8 ● ● ● ● 8.6% 0.8 ● 14.8% 0.8 ● ● ● ● 36.0% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 10.8% ● 23.6% ● 29.7% ● ● ● ● ● ● ● 12.6% ● 25.2% ● 33.3% ● ● ● ● ● ● 0.8 ● ● ● ● 13.8% 0.8 ● 26.1% 0.8 ● 39.7% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 5.9% ● 5.7% ● 4.3% ● ● ● 6.3% ● ● ● 6.8% ● ● ● 7% 0.8 ● ● ● ● 6.8% 0.8 ● ● ● ● 9.15% 0.8 ● ● ● ● 12.8% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 9.2% ● 8.2% ● 6.9% ● ● ● 9.5% ● ● ● 9.1% ● ● ● 7.7% 0.8 ● ● ● ● 10% 0.8 ● ● ● ● 9.75% 0.8 ● ● ● ● 8.2% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ●

Proportion of cells under Current Harvesting ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 5.4% ● 3.9% ● 10.1% ● ● ● ● ● 6.07% ● ● ● 4.4% ● 13.2% ● ● ● 0.8 ● ● ● ● 7% 0.8 ● ● ● ● 5.05% 0.8 ● 20.3% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of cells under Ecological Thinning Fig. 4.1. e median percent of land in Low density mature over time. Each dot represents one of the 66 replicate models where a proportion of cells was allocated to the three management scenarios, as indicated by graph axes. Darker/larger dots indicate more land in mature woodlands. Each graph is independent; the darkest/largest dots in each graph have a diﬀerent percent of land in mature woodlands (see inset on each graph). When the proportion of cells under Current Harvesting and Ecological inning are both 0.0 (bottom left corner), then all cells were in Natural Disturbance. Graphs are arranged in columns by year: (A) 10, (B) 50, and (C) 150, and in rows by experts (1−5). 121 Chapter Four

(A) 10 years (B) 50 years (C) 150 years 1.0 ● 1.0 ● 1.0 ● ● 13.3% ● 4.8% ● 2.5% ● ● ● ● ● ● ● 24.5% ● 31.9% ● 31.9% ● ● ● ● ● ● ● ● ● 0.8 ● 47.4% 0.8 ● 63% 0.8 ● 61.5% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 5.05% ● 1.3% ● 1.2% ● ● ● ● ● ● ● 8% ● 6.75% ● 10.2% ● ● ● ● ● ● ● ● ● 0.8 ● 13.6% 0.8 ● 18.6% 0.8 ● 30.2% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 12.8% ● 10.3% ● 5.1% ● ● ● 16.8% ● ● ● 25.8% ● ● ● 22.1% 0.8 ● ● ● ● 24.1% 0.8 ● ● ● ● 51.6% 0.8 ● ● ● ● 43.4% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 9.5% ● 11.9% ● 12.6% ● ● ● ● ● ● ● 10.6% ● 13.5% ● 15.2% ● ● ● ● ● ● ● ● ● 0.8 ● 11.2% 0.8 ● 15.2% 0.8 ● 18.4% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ●

1.0 ● 1.0 ● 1.0 ● ● 3.7% ● 0% ● 0% ● ● ● ● ● ● 7.8% ● ● 3.8% ● 0.55% ● ● ● ● ● ● 0.8 ● ● ● 12.1% 0.8 ● ● 8.5% 0.8 ● 1.5% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of cells under Ecological Thinning Fig. 4.2. e median percent of land in High density mature over time. Each dot represents one of the 66 replicate models where a proportion of cells was allocated to the three management scenarios, as indicated by graph axes. Darker/larger dots indicate more land in mature woodlands. Each graph is independent; the darkest/largest dots in each graph have a diﬀerent percent of land in mature woodlands (see inset on each graph). When the proportion of cells under Current Harvesting and Ecological inning are both 0.0 (bottom left corner), then all cells were in Natural Disturbance. Graphs are arranged in 122 columns by year: (A) 10, (B) 50, and (C) 150, and in rows by experts (1−5). Chapter Four

(A) 10 years (B) 50 years (C) 150 years 1.0 ● 1.0 ● 1.0 ● ● 23.8% ● 25.6% ● 41.5% ● ● ● ● ● ● ● 28.6% ● 43.6% ● 49% ● ● ● ● ● ● ● ● ● 0.8 ● 37.8% 0.8 ● 58.4% 0.8 ● 59.9% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 23.2% ● 31.2% ● 35.5% ● ● ● ● ● ● ● 23.8% ● 35.2% ● 46.5% ● ● ● ● ● ● ● ● 0.8 ● ● 24.4% 0.8 ● 44.1% 0.8 ● 70.5% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 19.5% ● 18.8% ● 11.4% ● ● ● 21.1% ● ● ● 27.9% ● ● ● 27.4% 0.8 ● ● ● ● 24.1% 0.8 ● ● ● ● 41.3% 0.8 ● ● ● ● 49.7% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

1.0 ● 1.0 ● 1.0 ● ● 20.1% ● 22.5% ● 22.8% ● ● ● ● ● ● ● 20.5% ● 23.5% ● 24.2% ● ● ● ● ● 0.8 ● ● ● ● 20.8% 0.8 ● ● 24.9% 0.8 ● 26.3% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ●

1.0 ● 1.0 ● 1.0 ● ● 13.7% ● 5.1% ● 10.5% ● ● ● ● ● 16.5% ● ● ● 9.57% ● 13.7% ● ● ● 0.8 ● ● ● ● 17.8% 0.8 ● ● ● ● 12.9% 0.8 ● 20.3% ● ● ● ● ● ● ● ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● 0.4 ● ● ● ● ● ● ● E5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of cells under Ecological Thinning Fig. 4.3. e median percent of land in High and Low density mature over time. Each dot represents one of the 66 replicate models where a proportion of cells was allocated to the three management scenarios, as indicated by graph axes. Darker/larger dots indicate more land in mature woodlands. Each graph is independent; the darkest/largest dots in each graph have a diﬀerent percent of land in mature woodlands (see inset on each graph). When the proportion of cells under Current Harvesting and Ecological inning are both 0.0 (bottom left corner), then all cells were in Natural Disturbance. Graphs are arranged in columns by year: (A) 10, (B) 50, and (C) 150, and in rows by experts (1−5). 123 Chapter Four

3 Expert

Natural Disturbance 5 Current Harvesting Ecological Thinning

0 10 20 30 40 50 60 Percent of land in High and Low density mature

Fig. 4.4. Box and whisker plots showing the percent of land in High and Low density mature for the three replicates where the entire landscape is allocated to only one of the management scenarios. Results are at 50 years. e vertical lines represent diﬀerent regret thresholds, 5% (dot-dashed), 8% (solid), and 10% (dashed).

124 mature line) or5%(dot-dashedline),ratherthan8%(results presented) inHigh andLow density left ofthelineswouldbeconsidered iftheregret threshold were changedto10%(dashed Average, (C)Minimax Regret, and(D)Maximin. In (C),onlythereplicates below/to the by deterministiccriteriafordecision-makingunderrisk:(A)Maximax, (B)Maximax 0.0 (bottomleftcorner),thenallcellswereinNaturalDisturbance. When theproportionofcellsunderCurrentHarvesting andEcological inningareboth graph haveadiﬀerentpercentoflandinmaturewoodlands (seeinsetoneachgraph). land inmaturewoodlands.Eachgraphisindependent; thedarkest/largestdotsineach three managementscenarios,asindicatedbygraphaxes.Darker/largerdotsindicatemore represents oneofthe66replicatemodelswhereaproportioncellswasallocatedto Fig. 4.5. epercent oflandinHigh andLow densitymature in50years. Proportion of cells under Current Harvesting 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ● ● 0.0 0.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● . ● ● 0.1 0.1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (B) MinimaxRegretcriterion ● ● ● ● 0.2 0.2 ● ● ● ● ● ● ● ● ● ● (A) Maximaxcriterion ● ● ● 0.3 0.3 ● Proportion of cells under EcologicalThinning Proportion ofcellsunder ● ● ● ● ● ● ● ● 0.4 0.4 ● ● ● ● ● ● ● ● ● 0.5 0.5 ● ● ● ● ● ● 0.6 0.6 ● ● ● ● ● 0.7 0.7 ● ● ● ● ● ● ● ● ● 0.8 0.8 53.6% 43.1% 39.1% 62.2% 47.2% 39.1% ● ● 0.9 0.9 ● 1.0 1.0 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.1 ● ● ● ● ● (B) MaximaxAveragecriterion ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 0.2 ● ● ● ● (B) Maximincriterion ● ● ● ● ● ● ● ● ● ● ● 0.3 0.3 ● ● ● ● Graphs are arranged ● ● ● ● ● ● ● 0.4 0.4 ● ● ● ● ● ● ● ● Each dot ● ● ● 0.5 0.5 ● ● ● ● ● ● ● ● ● ● ● 0.6 0.6 ● ● ● ● ● ● ● ● ● 0.7 0.7 ● ● ● ● ● ● ● ● ● ● C ● ● ● h 0.8 0.8 ● a ● ● 10.3% 7.5% 3.8% 30.3% 28% 26.3% ● ● ● p t e 0.9 0.9 ● ● r ● ● Four 125 1.0 1.0 ● ●

Chapter Four 4 (A) Natural Disturbance 3 2 1 0 4 (B) Current Harvesting 3 2 Density 1 0 4 (C) Ecological Thinning 3 2 1 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Proportion of land High and Low density mature Fig. 4.6. Bayesian posterior probability density functions showing the predicted percent of land in High and Low density mature in 50 years when the entire landscape is allocated to only one action, (A) Natural Disturbance, (B), Current Harvesting, or (C) Ecological inning. Solid grey lines represent Bayesian 95% credible intervals. 126 Chapter Four

1.0

0.8

0.6

0.4

0.2 Natural Disturbance Current Harvesting Ecological Thinning

Probability of developing the given percent vegetation or less 0.0

0 20 40 60 80 100 Percent of land in High Density or Mature woodland

Fig. 4.7. Cumulative density functions illustrating the application of an objective function and ﬁrst order stochastic dominance. Cdfs indicate the probability of achieving a given percent of land, or less, in High and Low density mature in 50 years when the entire landscape is allocated to only on management scenario. e vertical dotted line at 18.5% represents the current estimated amount of land in High and Low density mature.

127 Chapter Four

0.7

0.6

0.5

0.4

0.3

0.2 Area under the cumulative density function 0.1 Natural Disturbance Ecological Thinning 0.0

0 20 40 60 80 100 Percent of landscape in High Density or Mature woodland

Fig. 4.8. Areas under the curves of the cumulative density functions for Natural Distur- bance and Ecological inning to illustrate second order stochastic dominance. Results are at 50 years. e vertical dotted line at 18.5% represents the current estimated amount of land in High and Low density mature.

128 the regression coeﬃcient,r, is:(A)0.0,(B)0.3,and (C) 0.7. estimated amount oflandinHigh andLow densitymature. Results depictallocations when density mature in50years. dottedlineat18.5%represents evertical thecurrent critical level represents thepercent ofland that managerswishtohave inHigh and Low should beallocatedtoeach management scenariodependingonthecriticallevel (L Fig. 4.9. Anapplicationofrisk-spreading oftotallandthat indicatingtheproportion Proportion of land allocated to each of the management scenarios 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.7 0 0 0 (A) (C) (B) r r r =0.3 =0 =0.7 10 10 10 20 20 20 30 30 30 Critical level(L 40 40 40 50 50 50 crit 60 60 60 ) 70 70 70 Ecological Thinning Current Harvesting Natural Disturbance 80 80 80 crit ). e C h 90 90 90 a p t e r Four 129 100 100 100

130 CHAPTER FIVE

Synthesis and conclusions

131

132 Chapter Five

5.1 Introduction

The Box-Ironbark forests and woodlands of Victoria, Australia, have been heavily degraded, but are now receiving the attention of managers who wish to restore the vegetation to conditions that more resemble old-growth forests and woodlands. This thesis investigated decision-making for management of these ecosystems using simulation models parameterized using expert opinion. Chapters Two to Four contain analyses that address three broad questions of interest:

1. Can simulation models be constructed using expert opinion to predict the effects of management actions on vegetation dynamics if historical ecological data are lacking?

2. How can uncertainty be incorporated into models and what sources of uncertainty most strongly impact model predictions?

3. How can decisions most effectively be made when model predictions are uncertain?

In the final chapter, I summarize the major findings of this research and discuss the implications for management and restoration activities.

5.2 A summary of findings

In Chapter Two, I constructed and parameterized state-and-transition simulation models (STSMs) using information elicited from experts in Box-Ironbark ecology and management. Models predicted whether management will assist in the restoration of Box-Ironbark forests and woodlands by expediting the development of mature stands with large trees. The three candidate management actions for restoration included modified timber harvesting regulations, establishing no-cutting National Parks, or a new ecological thinning technique. Models predicted that mature woodlands with large trees will likely not develop sooner with either ecological thinning or modified harvesting, and either management action may actually lead to a loss of desired

133 Chapter Five vegetation. In this first set of models, I also investigated the importance of incorporating uncertainty caused by differences between experts into models. I found that variation in model predictions caused by between-expert uncertainty was substantial. However, further analyses were needed to quantitatively determine whether between-expert uncertainty or other sources of uncertainty had greater effects on model predictions.

In Chapter Three, I used Variance Components Analyses and Markov Chain Monte Carlo sampling to determine which of four sources of uncertainty - between-expert variation, within-expert variation, natural stochasticity, or the effects of initial conditions - most affected model predictions. Between-expert uncertainty contributed substantially more to variance in model predictions than all other sources of uncertainty, suggesting that modelling strategies that rely on consensus between experts may lead to overconfident models. If management decisions are based on consensus model results, there may be a greater likelihood of making the wrong decision by ignoring other plausible outcomes of management. Furthermore, in this case study, experts agreed that several transition agents - drought, wildfire, and growth - have strong influences on the development of low density mature woodlands, but strongly disagreed about the nature of the effects on vegetation. Because transition agents were predicted to affect vegetation differently between experts, it was not possible to compare most parameters between experts. However, transition agents that were influential in some experts’ models, but were uncertain between experts, can be very useful parameters for monitoring because mangers can apply adaptive management to learn about the effects of transition agents, update models, and reduce uncertainty in model predictions over time.

In Chapter Four, I investigated the utility of a structured decision-making framework by applying different management objectives to STSMs that predict the effects of the three management strategies. The goal was to assist in decision-making by exploring the outcomes when criteria with varying levels of complexity and inherent risk were employed. I found that the optimal management action depends on the objective chosen. This finding is important because it highlights the importance of selecting clear objectives and acknowledging the level of risk managers are willing to accept. The methods presented also facilitate decisions regarding very uncertain management

134 Chapter Five actions. Models indicated that natural development without harvesting is the most risk- averse strategy to employ if managers would like to expedite the development of mature Box-Ironbark forests and woodlands. Ecological thinning is the most risky action to employ because it has a small likelihood of greatly expediting the development of mature stands. By employing risk-spreading from portfolio theory, it is possible to maximize the possibility of achieving a desired outcome while allowing managers to learn about the effects of different management actions. In Box-Ironbark forests and woodlands, risk-spreading analyses indicated that the optimal solution is to allocate the majority of forests to no-cutting National Parks and to allocate a small amount land to the harvesting techniques so that monitoring data can be collected to determine the effects of different management actions on vegetation. These data could be used to update model predictions in an adaptive management framework to reduce uncertainty and optimize management strategies through time.

5.3 Implications

The results of this thesis have several implications for forest simulation modelling and management. First, expert opinion is often implicitly incorporated into experimental design (McCarthy and Masters 2005) or model validation (Kuhnert et al. 2005). This research illustrated the validity of constructing simulation models using expert opinion when empirical data are lacking because doing so leads to predictions about the effects of management that would not otherwise be available. In addition, assembling expert knowledge into a quantitative and explicit modelling framework allows for the interrogation of conceptual models for logic and consistency. Importantly, sampling experts is akin to stratified sampling of heterogeneous environments. As such, it is necessary to sample experts from multiple domains to collect a more complete set of information. For this case study, I found that the conceptual models of experts with many years of experience in the ecology and dynamics of Box-Ironbark forests and woodlands displayed a diversity of knowledge not previously represented.

It is very common to collect the opinions of experts using consensus methods that ignore differences in knowledge between experts; this happens both generally in environmental management (Morgan et al. 2001; Regan et al. 2006) and specifically in

135 Chapter Five studies using STSMs (e.g. Forbis et al. 2006; Hemstrom et al. 2007; Vavra et al. 2007). The research presented here illustrated that ignoring differences between experts can be highly detrimental to modelling and management because ignoring this uncertainty can substantially underestimate variation in the possible effects of management actions, potentially leading to poor management decisions.

There is growing awareness that risk and uncertainty must be acknowledged in environmental management and decision-making (e.g. Burgman 2005; Hatton et al. 2006). However, important sources of uncertainty are still ignored in some modelling frameworks, such as STSMs, and there is often inadequate acknowledgement of the uncertainty and risks involved in making decisions based on model predictions. This thesis illustrated that making decisions with uncertain model predictions is possible and, when uncertainty is accounted, it is possible to discern the risk of negative or unexpected outcomes. These results also illustrated the importance of explicit and measurable management objectives because different objectives will lead to different management strategies. This is particularly important considering many existing management objectives are vague, ambiguous, and qualitative.

5.4 Future research

There are several areas in which this research could be expanded. Because models were constructed using expert opinion, it would be ideal to validate or update model predictions (e.g. Kilgo et al. 2002; Walker et al. 2003; Shlisky et al. 2005). Typically validation is done using historical data or by employing chronosequence methods. Unfortunately, these methods were not available for this case study. One possible validation method would be using aerial photography to determine vegetation transition rates, or by relying on empirical data of the effects of timber harvesting and ecological thinning from similar ecosystems. In the long-term, collecting monitoring data would be most beneficial for testing model predictions.

This work has also identified that between-expert variation leads to the majority of uncertainty in model predictions and that, while drought, wildfire, and growth to mature woodlands are the most influential transition agents for the development of

136 Chapter Five mature woodlands, there was large disagreement between experts regarding the impacts of those transition agents of vegetation. There are several ways to reduce uncertainty in the effects of management. First, because models could be constructed for only five experts, future research could focus on eliciting the knowledge of additional experts to determine if general patterns emerge. In addition, monitoring could focus on the most influential transition agents in areas where different management strategies are being employed and, over time, models could be updated using an adaptive management framework.

In addition, it may be beneficial to incorporate the effects of climate change on vegetation dynamics to provide managers with an estimate of the impact that reduced rainfall and increased average temperatures may have on these ecosystems (CSIRO and Australian Bureau of Meteorology 2007; Cai and Cowan 2008). Models could also be implemented into a spatial modelling framework to better explain the effects of contagions, such as wildfire and insect attack.

5.5 Conclusions

I have constructed coherent and explicit models that predict the response trajectory of Box-Ironbark forests and woodlands. The results presented here indicated that neither ecological thinning nor altered timber harvesting regulations are likely to expedite the development of mature Box-Ironbark forests and woodlands relative to growth and natural processes; though there is ample opportunity for learning because there was large uncertainty in the dynamics of this system and the effects of these novel management strategies. Since natural development was the least risky management strategy and ecological thinning had a small and high-risk chance of expediting mature stands, it would be optimal to prohibit harvesting in the majority of forests and woodlands, but also allocate a small proportion of land to ecological thinning and timber harvesting. Monitoring data could then be collected to determine the effects of these uncertain harvesting techniques over time.

The results of this thesis also illustrated that expert-based simulation models provide vital assistance to forest management and restoration. Simulation models must

137 Chapter Five incorporate the uncertainty that arises due to differences between experts because the variation in model predictions caused by this uncertainty can be large. In addition, making decisions based on uncertain model predictions is difficult, but not impossible when an explicit management framework is adopted and management objectives are clear and measurable. Finally, while this research was a first step in accumulating the available knowledge regarding Box-Ironbark forest and woodland dynamics, it provided the methodology to make optimal management decisions in the face of uncertainty, which is widely applicable in forest management and ecological restoration.

138

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158

APPENDICES

APPENDIX A. INTERVIEW PROCEDURE.

Activities - Review introductory information. - Discuss how to undertake formal elicitations and how to use the VDDT software. - Discuss the Group Model and estimate parameter values, which will involve: 1. Assigning deterministic transitions due to tree growth and estimating their rates. 2. Assigning probabilistic transitions due to disturbances and management and estimating their return frequencies or spatial extent.

Tricks to eliciting and using subjective data Problems with Subjective Data (Cooke 1991, pg. 27), (Ayyub 2001, Ch. 3): 1. Data spread – when extremely large confidence bounds are provided. This is especially common when dealing with very small numbers: Experts A, E, and G (Fig. 1.b) 2. Data dependence – leads to bias when a subject is consistently negative or positive, which leads to very low or high values. Commonly becomes apparent in risk-averse versus risk-taking personalities: Expert D overestimates (Fig. 1.a & b) 3. Poor Calibration – when estimates are highly inaccurate: Expert D (Fig. 1.a & b) and Expert E (Fig. 1.b); contrast to Experts B and C (Fig. 1.a & b). 4. Overconfidence – when subjects provide narrower intervals than reality; results from not thinking of how extreme situations might arise. This is the most pervasive bias in subjective probabilities (Morgan and Henrion 1990): Expert D (Fig. 1.a & b)

Fig. 1. Problems with subjective data, taken from Walker (2003). A: mean residential ambient benzene concentration. B: mean personal air benzene concentration. Estimates are median (+), interquartile range (box plots), and 90% credible intervals (tails). Straight line indicates mean measured value.

159 Appendix A

Heuristics used in Expert Opinion (Cooke 1991, pg 55, 64-79): 1. Availability – estimates tend to be based on the most memorable events. This may lead experts to rely on work they are familiar with or their own work when making assessments, while ignoring other sources of information. This will lead to overestimates if recall is enhanced through recent, dramatic or highly plausible scenarios, but will lead to underestimates if recall is difficult due to no recent experience or scenario is abstract (Morgan and Henrion 1990). - Example: Fig. 2 of Lichtenstein research taken from Cooke (1991).

Fig. 2. An example of Availability taken from Cooke (1991).

2. Anchoring – when a subject first makes an initial estimate of the central tendency and then adjusts this value for other estimates. Often individuals do not extrapolate to other situations successfully. - Example: Given five seconds to estimate the values of: 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1, or 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 two groups of high school students estimated an average 2250 for the first set and 512 for the second. The actual answer is 40,320 for both but the first few digits of the set bias the estimates.

3. Representativeness – Various examples exist; the most common are (1) when we assume commonality between objects of similar appearance and (2) the base-rate fallacy (discussed separately below), and (3) when sample size is ignored. - Example of when sample size is ignored: There are two populations of squirrel gliders. In the larger populations, about 45 babies are born each

160 Appendix A

year. In the smaller population, about 15 babies are born. On average, we would expect that 50% of births are female. Each population is monitored and after 10 years, the researchers report the years when more than 60% of births were female. Which population would be reported to have more such years? In a similar survey, 21 people said the larger population, 21 people said the smaller, and 53 thought either was likely. In reality, the smaller population is much more likely to have years with more than60% of births being female due to the random effects of having a smaller population size (i.e. 9 births versus 27).

4. Base-rate fallacy – a specific example of representativeness. Occurs when a subject relies on recent information and ignores the historical data for an event. - Example: There are two EVC types being surveyed with aerial photography to determine location on the landscape. Type A, a Coast Banksia Woodland, occurs on 10% of the landscape and is of conservation importance, while Type B, Dune Scrub, occurs on 90% of the landscape. When classifying the landscape, the aerial survey method is 80% accurate – meaning that out of 10 sites, the survey will accurately classify the vegetation 8 times and will misclassify the vegetation 2 times. The fallacy: that equal numbers of sites will be misclassified due to the 20% error rate. In reality: for 100 sites on the landscape 10 will be Type A, but the test will show 8 Type A samples and 2 Type B samples. Vice versa, for the same 100 sites 90 of them will actually be Type B, but the test will show 72 as Type B (80% of 90 sites correctly classified) and 18 as Type A (20% misclassified). This means that the aerial survey will show 26 samples as being Type A and 74 samples as being Type B.

Type A (Actual – Type B (Actual – Total (Sampled) 10%) 90%) Type A (Sampled) 8 18 26 Type B (Sampled) 2 72 74 Total (Actual) 10 90

5. Control – Situations in which subjects believe they have control over the outcome, when they do not, can lead to distorted assessments. - Example: Two groups of people are given lottery tickets priced at $1 each. The winning ticket will earn a prize of $50. The first group are given lottery tickets with random numbers, while the second group are able to select their own tickets. When asked how much the resale price of their ticket would be, the first (random) group chose an average of $1.96, but the second (control) group chose an average of $8.67 even though the probability of winning the lottery was equal for both groups.

Advice for Experts (Morgan and Henrion 1990 pg. 155, taken from Wallsten and Whitfield 1986): 1. Sequential effects – the order things are considered should not affect judgment. Perhaps keep a list of all facts being considered and then re-consider them in different sequences.

161 Appendix A

2. Memory effects – define various classes of information that seem relevant and then search for several examples of each. Especially try to think of conflicting information. 3. Estimate the reliability of information – think about sample sizes, power of statistical tests, and extrapolating to new systems. Note that study results may be probabilistic, subject to random error, or result from imprecise measurements. 4. Keep in mind that the importance of an event should not affect its probability. 5. In making probability assessments, think in the measure that is most comfortable, such as probability, odds, or return frequencies.

Examples – Run through trial elicitation techniques with participants and provide feedback (1) Estimate distance - A. Melbourne to Canberra – 669 km - B. Melbourne to Kakadu National Park – 3,666 km - C. Melbourne to Vancouver, Canada – 13,199 km (from Google Map)

(2) Estimate the trees/ha using five pictures provided. - A. Photo 51 – 263 stems - B. Photo 46 – 525 stems - C. Photo 39 – 766 stems - D. Photo 44 – 986 stems - E. Photo 49 – 1325 stems (Extrapolated from circular quadrats 1/15 to 1/8 of ha in size)

(3) Show pictures of 4 different ecosystems. Estimate fire frequency. - A. Boreal: mean 350, min 165, max 605 (Carcaillet et al. 2007). - B. Grassland: range 1-3 years (Lunt and Morgan 2002). - C. Rainforest: median 2380, min 450, max 8760(Lertzman et al. 2002; Gavin et al. 2003). - D. Mallee: mean 10-20 years (Bradstock and Cohn 2002); 5-10 years (Noble 1989).

Using the VDDT software Basic terminology: - VDDT: Vegetation Dynamics Development Tool - State or Class: a classification of vegetation based on dominant species and structure; measured in relation to stand features, i.e. density of trees or volume of downed wood. For example, cover type: Box-Ironbark or grassland. Structure: stand-initiation forest (Fig. 3). - Transition: causes changes between states. For example, growth or fire. - Deterministic transition: attributed to growth; time of transition varies; occurs with probability of 1 after avoiding probabilistic transitions for the age range specified. - Probabilistic transition: attributed to disturbances or management; time of transition is one time step; probability varies. - Cell or Pixel: the basic unit used in VDDT to describe a given stand or area of forest; will use 1000 cells in the simulation; represents 1 ha in this model. - Time step: equal to one year in this model.

162 Appendix A

Fig. 3. Explanation of a state/class in the VDDT software.

Review example of group model in VDDT - Model algorithm from User’s Guide – events are considered independent; the probability that a cell goes through a disturbance in one time step does not affect its probability of going through another disturbance - i.e. one cell can be burned in consecutive time steps in the model, even though in reality the probability would decrease due to low fuel loads; the total probability cannot sum to greater than 1; the model is non-spatial. - States/classes: means of distinguishing states; cover names and structure text files. - Deterministic transitions: growth, occurs with P=1 after cell avoids all probabilistic transitions during its specified age range. - Probabilistic transitions: identify causes, probabilities, ages, relative age, time since disturbance option; disturbance text files. - Initial conditions: time steps, cell number, initial ages, and multiple simulations - Run simulation. - Results: graphs and numerical outputs.

Group Model Establishing Model Parameters (States) To establish discrete vegetation states, experts filled out surveys ranking the ability of vegetation attributes, such as large trees or coarse woody debris, to indicate healthy Box- Ironbark woodlands. The attributes considered most important (the majority of votes) were selected for use in the group model. Once selected, I used the levels and thresholds for the attributes identified by experts to distinguish between vegetation states (Fig. 4)

Fig. 4. Group model with four states.

163 Appendix A

Establishing Model Values (Deterministic Transitions) To describe slow transitions that result from growth, consider the growth of trees and stands without management or other disturbances. Identify the gradual changes between states and estimate the timeframes for these deterministic transitions. Use the “most reasonable” slowest, fastest, and best estimates for growth rates.

Establishing Model Values (Probabilistic Transitions) To describe faster, probabilistic transitions, consider all natural disturbances and management actions that have occurred in Box-Ironbark woodlands in the last 200 years and may occur over the next 1000 that will cause vegetation to change between states. Please make a list of these transitions and describe how these transitions will each affect state changes. Then provide the “most reasonable” slowest, fastest, and best estimates for the return frequency of each transition.

Post interview Models can be reviewed and adjusted after elicitations are complete.

164 EXPERT. A a a b c d e Expert From Transition Agent To Management Scenario Probability MinAge MaxAge RelativeAge PPENDIX 1 HDR Ecothin HDR Ecothin 0.0093 0 999 -1 -150 1 HDR Ecothin+poison HDR Ecothin 0.0278 0 999 1 150 1 HDR Prescribed underburn HDR Harv, Ecothin 0.0010 0.0020 0.0100 0 999 -1 -2 -3 1 HDR Wildﬁre HDR NatDist, Harv, Ecothin 0.0063 0.0100 0.0167 0 999 -1 -30 1 HDR Windstorm HDR NatDist, Harv, Ecothin 0.0020 0.0040 0.0100 0 999 -1 -2 -3

1 HDR Timber harvesting LDR Harv 0.0370 0 999 0 B. A 1 HDR Growth HDM NatDist, Harv, Ecothin 1.0000 30 65 150 999 0 1 LDR Coppice HDR NatDist, Harv, Ecothin 0.7000 0.7500 0.8000 5 10 20 0 1 LDR Prescribed underburn HDR Harv, Ecothin 0.0010 0.0020 0.0100 0 999 -1 -2 -3 LL TRANSITION AGENTS SPECIFIED BYEACH 1 LDR Wildfire HDR NatDist, Harv, Ecothin 0.0063 0.0100 0.0167 0 999 0 1 LDR Windstorm LDR NatDist, Harv, Ecothin 0.0020 0.0040 0.0100 0 999 -2 -4 -8 1 LDR Growth LDM NatDist, Harv, Ecothin 1.0000 30 65 150 999 0 1 HDM Drought HDR NatDist, Harv, Ecothin 0.0083 0.0143 0.0250 0 999 0 1 HDM Prescribed underburn HDR Harv, Ecothin 0.0010 0.0020 0.0100 0 999 -1 -2 -3 1 HDM Wildfire HDR NatDist, Harv, Ecothin 0.0063 0.0100 0.0167 0 50 100 -1 -30 1 HDM Growth LDM NatDist, Harv, Ecothin 1.0000 50 100 150 999 0 1 LDM Prescribed underburn HDR Harv, Ecothin 0.0003 0.0007 0.0033 0 999 -1 -2 -3 1 LDM Wildfire HDR NatDist, Harv, Ecothin 0.0063 0.0100 0.0167 0 50 100 -1 -30 1 LDM Prescribed underburn LDR Harv, Ecothin 0.0003 0.0007 0.0033 0 999 -1 -2 -3 1 LDM Prescribed underburn HDM Harv, Ecothin 0.0003 0.0007 0.0033 0 999 -1 -2 -3 1 LDM Growth LDM NatDist, Harv, Ecothin 1.0000 999 999 0

a Vegetation state abbreviation: HDR - High density regrowth; LDR - Low density regrowth; HDM - High density mature; LDM - Low density mature. b e combined probability of the transition agent occurring and the transition between states occurring. For this column and all columns to the right: a single value is an absolute estimate; two values are the bounds of a uniform distribution; three values are the upper and lower bounds for a beta distribution. For this chapter, the middle estimate was used in modelling and, in the case of estimates from a uniform distribution, the mean value was used.* c MinAge is the number of timesteps a cell must remain in the From state before a transition agent can cause a transition. d MaxAge is the numbers of timesteps after which a cell in the From state is no longer susceptible to the transition agent. Blank rows indicate that the value is taken to be the same as the MinAge value. e e transition agent advances or delays growth transition by altering the number of timesteps a cell has been in the From state. 165 166

Expert From a Transition Agent Toa Management Scenario Probability b MinAge c MaxAge d RelativeAge e 2 HDR Drought LDR NatDist, Harv, Ecothin 0.0000 0.0001 0.0002 0 999 0 2 HDR Ecothin LDR Ecothin 0.0093 0 999 0 2 HDR Ecothin+poison LDR Ecothin 0.0278 0 999 11 2 HDR Timber harvesting LDR Harv 0.0370 0 999 0 2 HDR Prescribed underburn LDR Harv, Ecothin 0.0001 0.0002 0.0004 0 999 0 2 HDR Wildfire LDR NatDist, Harv, Ecothin 0.0067 0.0100 0.0133 0 999 0 2 HDR Growth HDM NatDist, Harv, Ecothin 0.7500 30 60 100 0 2 HDR Growth LDM NatDist, Harv, Ecothin 1.0000 150 200 300 999 11 2 LDR Coppice HDR NatDist, Harv, Ecothin 0.5000 3 10 0 2 LDR Drought LDR NatDist, Harv, Ecothin 0.0000 0.0001 0.0002 0 999 -999 2 LDR Prescribed underburn LDR Harv, Ecothin 0.0001 0.0002 0.0004 0 999 -999 2 LDR Wildfire LDR NatDist, Harv, Ecothin 0.0067 0.0100 0.0133 0 999 -999 2 LDR Growth LDM NatDist, Harv, Ecothin 1.0000 200 250 300 999 11 2 HDM Drought LDM NatDist, Harv, Ecothin 0.0000 0.0001 0.0002 0 999 0 2 HDM Ecothin LDM Ecothin 0.0093 0 999 0 2 HDM Ecothin+poison LDM Ecothin 0.0278 0 999 11 2 HDM Growth LDM NatDist, Harv, Ecothin 1.0000 100 150 250 999 11 2 HDM Timber harvesting LDM Harv 0.0370 0 999 0 2 HDM Prescribed underburn LDM Harv, Ecothin 0.0001 0.0002 0.0004 0 999 0 2 HDM Wildfire LDM NatDist, Harv, Ecothin 0.0067 0.0100 0.0133 0 999 0 2 LDM Coppice HDM NatDist, Harv, Ecothin 0.5000 3 10 0 2 LDM Drought LDM NatDist, Harv, Ecothin 0.0000 0.0001 0.0002 0 999 -999 2 LDM Growth LDM NatDist, Harv, Ecothin 1.0000 999 999 11 2 LDM Prescribed underburn LDM Harv, Ecothin 0.0001 0.0002 0.0004 0 999 -999 2 LDM Wildfire LDM NatDist, Harv, Ecothin 0.0067 0.0100 0.0133 0 999 -999

a Vegetation state abbreviation: HDR - High density regrowth; LDR - Low density regrowth; HDM - High density mature; LDM - Low density mature. b e combined probability of the transition agent occurring and the transition between states occurring. For this column and all columns to the right: a single value is an absolute estimate; two values are the bounds of a uniform distribution; three values are the upper and lower bounds for a beta distribution. For this chapter, the middle estimate was used in modelling and, in the case of estimates from a uniform distribution, the mean value was used.* c MinAge is the number of timesteps a cell must remain in the From state before a transition agent can cause a transition. Appendix B d MaxAge is the numbers of timesteps after which a cell in the From state is no longer susceptible to the transition agent. Blank rows indicate that the value is taken to be the same as the MinAge value. e e transition agent advances or delays growth transition by altering the number of timesteps a cell has been in the From state.

Expert From a Transition Agent Toa Management Scenario Probability b MinAge c MaxAge d RelativeAge e 3 HDR Firewood harvesting HDR Harv 0.0250 0 999 50 60 100 3 HDR Sawlog harvesting HDR Harv 0.0120 0 999 -5 -10 -15 3 HDR Ecothin LDR Ecothin 0.0093 0 999 0 3 HDR Ecothin+poison LDR Ecothin 0.0278 0 999 50 60 100 3 HDR Wildfire LDR NatDist, Harv, Ecothin 0.0040 0.0067 0.0100 0 999 0 3 HDR Windstorm LDR NatDist, Harv, Ecothin 0.0000 0.0001 0.0001 0 999 0 3 HDR Growth HDM NatDist, Harv, Ecothin 1.0000 200 250 300 999 0 3 LDR Coppice HDR NatDist, Harv, Ecothin 0.8500 0.9000 0.9500 5 10 15 0 3 LDR Firewood harvesting LDR Harv 0.0250 0 999 50 80 100 3 LDR Wildfire LDR NatDist, Harv, Ecothin 0.0040 0.0067 0.0100 0 999 -999 3 LDR Windstorm LDR NatDist, Harv, Ecothin 0.0000 0.0001 0.0001 0 999 -999 3 LDR Growth LDM NatDist, Harv, Ecothin 1.0000 200 250 300 999 0 3 HDM Wildfire LDR NatDist, Harv, Ecothin 0.0040 0.0067 0.0100 0 999 0 3 HDM Windstorm LDR NatDist, Harv, Ecothin 0.0000 0.0001 0.0001 0 999 0 3 HDM Growth HDM NatDist, Harv, Ecothin 1.0000 999 999 0 3 HDM Sawlog harvesting HDM Harv 0.0120 0 999 0 3 LDM Wildfire LDR NatDist, Harv, Ecothin 0.0040 0.0067 0.0100 0 999 0 3 LDM Windstorm LDR NatDist, Harv, Ecothin 0.0000 0.0001 0.0001 0 999 0 3 LDM Growth LDM NatDist, Harv, Ecothin 1.0000 999 999 0

e e transition agent advances or delays growth transition by altering the number of timesteps a cell has been in the From state. Appendix B 167 168

Expert From a Transition Agent Toa Management Scenario Probability b MinAge c MaxAge d RelativeAge e 4 HDR Drought LDR NatDist, Harv, Ecothin 0.0050 0.0140 0.0450 0 999 6 4 HDR Ecothin LDR Ecothin 0.0093 0 999 0 4 HDR Ecothin+poison LDR Ecothin 0.0278 0 999 26 4 HDR Firewood harvesting LDR Harv 0.0250 0 999 0 4 HDR Insects LDR NatDist, Harv, Ecothin 0.0025 0.0004 0.0001 0 999 6 4 HDR Wildfire LDR NatDist, Harv, Ecothin 0.0100 0.0200 0.0500 0 999 0 4 HDR Growth LDM NatDist, Harv, Ecothin 1.0000 150 300 400 999 26 4 LDR Coppice HDR NatDist, Harv, Ecothin 1.0000 2 5 0 4 LDR Recovery HDR NatDist, Harv, Ecothin 0.2000 0.6000 0.8000 8 25 0 4 LDR Drought LDR NatDist, Harv, Ecothin 0.0050 0.0140 0.0450 0 999 -999 4 LDR Insects LDR NatDist, Harv, Ecothin 0.0025 0.0004 0.0001 0 999 -999 4 LDR Wildfire LDR NatDist, Harv, Ecothin 0.0100 0.0200 0.0500 0 999 -999 4 LDR Growth LDM NatDist, Harv, Ecothin 1.0000 100 200 350 999 26 4 HDM Sawlog harvesting HDM Harv 0.0120 0 999 -1 -200 4 HDM Drought LDM NatDist, Harv, Ecothin 0.0050 0.0140 0.0450 0 999 6 4 HDM Firewood harvesting LDM Harv 0.0250 0 999 0 4 HDM Growth LDM NatDist, Harv, Ecothin 1.0000 80 150 200 999 26 4 HDM Insects LDM NatDist, Harv, Ecothin 0.0025 0.0004 0.0001 0 999 6 4 HDM Wildfire LDM NatDist, Harv, Ecothin 0.0100 0.0200 0.0500 0 999 0 4 LDM Coppice HDM NatDist, Harv, Ecothin 1.0000 2 5 0 4 LDM Recovery HDM NatDist, Harv, Ecothin 0.0500 0.3000 0.5000 8 25 0 4 LDM Drought LDM NatDist, Harv, Ecothin 0.0050 0.0140 0.0450 0 999 -999 4 LDM Growth LDM NatDist, Harv, Ecothin 1.0000 999 999 0 4 LDM Insects LDM NatDist, Harv, Ecothin 0.0025 0.0004 0.0001 0 999 -999 4 LDM Sawlog harvesting LDM Harv 0.0120 0 999 0 4 LDM Wildfire LDM NatDist, Harv, Ecothin 0.0100 0.0200 0.0500 0 999 -999

modelling and, in the case of estimates from a uniform distribution, the mean value was used.* Appendix B c MinAge is the number of timesteps a cell must remain in the From state before a transition agent can cause a transition. d MaxAge is the numbers of timesteps after which a cell in the From state is no longer susceptible to the transition agent. Blank rows indicate that the value is taken to be the same as the MinAge value. e e transition agent advances or delays growth transition by altering the number of timesteps a cell has been in the From state.

Expert From a Transition Agent Toa Management Scenario Probability b MinAge c MaxAge d RelativeAge e 5 HDR Drought+Mistletoe HDR NatDist, Harv, Ecothin 0.0033 0.0060 0.0100 0 999 -999 5 HDR DodderLaurel LDR NatDist, Harv, Ecothin 0.0010 0.0020 0.0035 0 999 36 5 HDR Ecothin LDR Ecothin 0.0093 0 999 0 5 HDR Ecothin+poison LDR Ecothin 0.0278 0 999 36 5 HDR Timber harvesting LDR Harv 0.0370 0 999 0 5 HDR Wind+Fire LDR NatDist, Harv, Ecothin 0.0044 0.0067 0.0083 0 999 6 5 HDR Windstorm LDR NatDist, Harv, Ecothin 0.0067 0.0100 0.0125 0 999 36 5 HDR Growth HDM NatDist, Harv, Ecothin 1.0000 150 200 250 999 0 5 LDR Coppice HDR NatDist, Harv, Ecothin 1.0000 3 5 0 5 LDR Recovery HDR NatDist, Harv, Ecothin 1.0000 20 30 35 0 5 LDR DodderLaurel LDR NatDist, Harv, Ecothin 0.0010 0.0020 0.0035 0 999 0 5 LDR Drought+Mistletoe LDR NatDist, Harv, Ecothin 0.0033 0.0060 0.0100 0 999 0 5 LDR Wind+Fire LDR NatDist, Harv, Ecothin 0.0044 0.0067 0.0083 0 999 -999 5 LDR Windstorm LDR NatDist, Harv, Ecothin 0.0067 0.0100 0.0125 0 999 0 5 LDR Growth LDM NatDist, Harv, Ecothin 1.0000 100 150 200 999 6 5 HDM Drought+Mistletoe HDR NatDist, Harv, Ecothin 0.0033 0.0060 0.0100 0 999 0 5 HDM Ecothin LDR Ecothin 0.0093 0 999 0 5 HDM Ecothin+poison LDR Ecothin 0.0278 0 999 36 5 HDM Wind+Fire LDR NatDist, Harv, Ecothin 0.0044 0.0067 0.0083 0 999 6 5 HDM Windstorm LDR NatDist, Harv, Ecothin 0.0067 0.0100 0.0125 0 999 36 5 HDM Growth HDM NatDist, Harv, Ecothin 1.0000 999 999 0 5 HDM DodderLaurel LDM NatDist, Harv, Ecothin 0.0010 0.0020 0.0035 0 999 6 5 HDM Timber harvesting LDM Harv 0.0370 0 999 0 5 LDM Drought+Mistletoe LDR NatDist, Harv, Ecothin 0.0033 0.0060 0.0100 0 999 36 5 LDM Wind+Fire LDR NatDist, Harv, Ecothin 0.0044 0.0067 0.0083 0 999 6 5 LDM Windstorm LDR NatDist, Harv, Ecothin 0.0067 0.0100 0.0125 0 999 36 5 LDM Coppice HDM NatDist, Harv, Ecothin 1.0000 3 5 0 5 LDM DodderLaurel LDM NatDist, Harv, Ecothin 0.0010 0.0020 0.0035 0 999 0 5 LDM Growth LDM NatDist, Harv, Ecothin 1.0000 999 999 6

modelling and, in the case of estimates from a uniform distribution, the mean value was used.* Appendix B c MinAge is the number of timesteps a cell must remain in the From state before a transition agent can cause a transition. d MaxAge is the numbers of timesteps after which a cell in the From state is no longer susceptible to the transition agent. Blank rows indicate that the value is taken to be the same as the MinAge value. 169 e e transition agent advances or delays growth transition by altering the number of timesteps a cell has been in the From state.

APPENDIX C. THE R CODE USED FOR DATA ANALYSES IN CHAPTER THREE.

The following script generates beta distributions from the three-point transition agent estimates. It also samples single values from the distributions generated for Probability, MinAge, MaxAge, and Relative Age (Appendix B) to be used in each of the 25 replicate models.

Ntables<-5 # experts its<-25 # replicate models for(i in 1:Ntables) { table1<-read.table(paste(“probestimates”,i,”.txt”,sep=“”),sep=“\t”, header=T) # read in data for(k in 1:its) { minage<-table1[,”MinAge”] maxage1<-table1[,”MaxAge”] maxage2<-maxage1 probs<-table1[,”Probability”] probs.var<-((probs-table1[,”LowProb”])/z)^2 # imputing variance from the normal bounds of the probabilities where z is the z-value for a given confidence interval, 80%, 90%, or 95%. a<-probs(probs(1-probs)/probs.var-1) b<-(1-probs)(probs(1-probs)/probs.var-1) # beta(a, b) relativeage<-table1[,”RelativeAge”] uniforms<-runif(length(relativeage),table1[,”HighRelAge”], table1[,”LowRelAge”]) normals<-rnorm(length(relativeage),relativeage,abs(relativeage– table1[,”LowRelAge”])/z) projectname<-table1[,”Project_Name”] projectname2<-paste(projectname[],rep(k,length(minage)),sep = “”) for(j in 1:length(minage)) { minage[j]<-ifelse(!is.na(table1[j,”LowMinAge”]), rnorm(1,minage[j], (table1[j,“MinAge”]–table1[j,”LowMinAge”])/z),minage[j]) maxage1[j]<-ifelse(!is.na(table1[j,”LowMaxAge”]),runif(1,table1 [j,“LowMaxAge”],table1[j,”HighMaxAge”]),maxage1[j]) maxage2[j]<-ifelse(is.na(table1[j,”MaxAge”]) & is.na(table1 [j,”LowMaxAge”]),minage[j],maxage1[j]) # determining the ages; this statement defaults to minage if maxage cols all blank (NA) probs[j]<-ifelse(!is.na(table1[j,”LowProb”]), rbeta(1,a[j],b[j]), probs[j]) # determining the probabilities relativeage[j] <-ifelse(is.na(table1[j,”LowRelAge”]),relativeage[j], normals[j]) relativeage[j] <-ifelse(is.na(relativeage[j]),uniforms[j], relativeage[j]) relativeage[j]<-ifelse(is.nan(relativeage[j]),runif(1,table1[j, ”LowRelAge”],table1[j,”HighRelAge”]), relativeage[j]) # determining the relative ages } table.new<-data.frame(projectname2, table1[,2:7],minage,maxage2, table1[,14:15],probs,table1[,19],relativeage,table1[,23:24]) names(table.new)[c(1,8,9,12,13,14)]<-c(“Project_Name”,“MinAge”, “MaxAge”,“Probability”,“Proportion”,“RelativeAge”)

170 Appendix C write.table(table.new,paste(“probestout90”,i,”it”,k,”.csv”,sep=“”), sep=“,”,row.names = FALSE) } # end iterations loop (k) } # end tables loop (i) R code

Variance components analyses (VCAs) were completed using a hierarchical mixed- effects model: model.CIx.Timey<-lmer(Area~IC+(1|Expert/Project),data= Table.CIx.Timey) R code where x is the confidence interval (80%, 90%, or 95%), and y is the timestep (every 10 years from 10 to 150 years), Area is the percent of land in Low density mature, IC is the fixed effects for the three initial conditions, Expert is the random group level intercept for experts, and Project is the random group level intercept for experts’ replicate models nested within Expert.

VCA parameter estimation was completed using Markov Chain Monte Carlo (MCMC) estimation at the final timestep, 150 years: mcmc.CIx<-mcmcsamp(model.CIx.Time150, n=110000, verbose=F) mcmc.CIx<-as.matrix(mcmc.CIx[10000:nrow(mcmc.CIx),]) R code where x is the confidence interval (80%, 90%, or 95%) and model.CIx.Time150 is the hierarchical mixed-effects model calculated from VCAs.

VCAs were completed to determine the effects of the initial conditions by estimating their variance separately, as opposed to as fixed effects. Analyses were completed using the 90% confidence interval and timestep 150: model.ICz.CI90.Time150<-lme(fixed=Area~1,random=~1|Expert/Project, data=Table.ICz.CI90.Time150, control=lmeControl(opt="optim")) R code where z is the initial condition (Average, Open, or Dense).

171 APPENDIX D. THE AMOUNTS OF LAND IN EACH VEGETATION STATE BETWEEN INITIAL CONDITIONS OVER TIME. STATES ARE HDR: HIGH DENSITY REGROWTH, LDR: LOW DENSITY REGROWTH, HDM: HIGH DENSITY MATURE. (A) Open−Average (B) Dense−Average 60 ● Expert 1 Expert 2 Expert 3 40 Expert 4 Expert 5 20

0 ● ● ● ● ● ● ● ● ● ● ● HDR ● ● −20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −40 ● ● ● −60 ● 60

20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● LDR

−20

−40

−60

40 ●

20 ● ● ● ● 0 ● ● ● ● HDM ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −20 ● ● ● ● ● ●

−40

−60

Differences in amount of land vegetation state (mean %) 60 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 40

20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 LDM

−20

−40

−60

0 30 60 90 120 150 0 30 60 90 120 150 172 Time (years)

APPENDIX E. THE WINBUGS AND R CODE USED FOR DATA ANALYSES IN CHAPTER FOUR.

Experts’ predictions of the percent of land in High and Low density mature at 50 years for the three replicates where all land was allocated to one management scenario were used to generate Bayesian posterior probability density functions in WinBUGS: model { # Likelihood: for (i in 1:500) { logit.Area[i] <- log((Area[i]/100) / (1-(Area[i]/100))) # logit transformed percent data logit.Area[i] ~ dnorm(logit.mean,tau) k[i]<-Expert[i] # dummy variable for Expert } # Priors: logit.mean ~ dnorm(0, 1.0E-6) tau ~ dgamma(0.01, 0.01) # Predictions and transformations: mean <-exp(logit.mean) / (1+exp(logit.mean)) new.logit.data ~ dnorm(logit.mean,tau) new.data <-exp(new.logit.data) / (1+exp(new.logit.data)) } # Initial values needed for initializing MCMC chains: list ( logit.mean = 0, new.logit.data = 0, tau = .01 ) Data Expert[] Area[] ...... WinBUGS code

The areas under the curves were generated in R using the posterior probability density functions generated in WinBUGS: for(i in 1:100) { iterno[i]<-integrate(ecdf(z.coda), 0, i/100) } R code where z is the management strategy (Natural Disturbance or Ecological Thinning) and z.coda is the output from WinBUGS.

173

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s: Czembor, Christina Anne

Title: Incorporating uncertainty into expert models for management of box-ironbark forests and woodlands in Victoria, Australia

Date: 2009

Citation: Czembor, C. A. (2009). Incorporating uncertainty into expert models for management of box- ironbark forests and woodlands in Victoria, Australia. Masters Research thesis , School of Botany, The University of Melbourne.

Persistent Link: http://hdl.handle.net/11343/35232

File Description: Incorporating uncertainty into expert models for management of box-ironbark forests and woodlands in Victoria, Australia

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