Modeling Predation Across Spatial Scales

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Delphastus catalinae and the silverleaf whitefly, Bemisia tabaci biotype B, on tomato: modeling predation across spatial scales

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Diego Fernando Rincon Rueda

Graduate Program in Entomology

The Ohio State University

2015

Dissertation Committee:

Luis A. Cañas, Advisor

Casey W. Hoy, Co-advisor

P. Larry Phelan

Laurence V. Madden

Robin A. J. Taylor

Diego F. Rincon

2015

Abstract

Ecological models are developed to gain understanding and generate predictions about ecosystems. Predator-prey models, for example, are used to integrate information on the biology and ecology of predators and prey with the aim of predicting the dynamics of the system over time. Model predictions are often used to guide the management of production systems, such as the application of biological control of pests in agriculture.

One important component of the theory on predator-prey interactions is the predator functional response, which is a mathematical description of predation rates based on the number of prey available. Functional responses are important components of predator- prey models because they are often used as a link between predator and prey population sub-models. Thus, the accuracy of predictions derived from predator-prey models depend upon the accuracy and precision of estimates derived from functional responses.

However, functional responses are often estimated through small-scale laboratory experiments, which are then used to model dynamics of larger natural systems, leading to biased estimations of the predation capabilities of natural enemies.

Understanding behavioral traits that determine the ability of predators to suppress pest populations at spatial scales larger than those evaluated in the laboratory may help in selecting the right species and release rates for biological control programs. My thesis is ii that predation rates within whole plants are driven by the interaction between prey distribution, individual predator patch-to-patch behavior and consumption rates within patch units. I propose that results derived from simple laboratory settings can be useful to predict predation rates within whole plants, if they are combined with spatially explicit descriptions of prey distribution and predator movement patterns. I assume that the leaflet is a spatial scale at which predators and prey behave as in laboratory settings, at least in experiments without replacement of consumed prey. My study extended from the leaflet to the plant scale, encompassing both the relatively homogeneous prey patch unit, leaflet, and the more structurally complex combination of leaflets, leaves, branches and main stem.

My study system consisted of the silverleaf whitefly (SWF), Bemisia tabaci biotype B and the coccinellid predator Delphastus catalinae inhabiting tomato plants in a greenhouse environment. The SWF is a significant worldwide pest of a variety of production systems including field and greenhouse tomato. Because of persistent application of chemical insecticides, the SWF has developed resistance to an array of chemical pesticides. For this reason, alternative control methods, such as the use the beneficial insects, have been encouraged but few have shown success. The predator D. catalinae is currently the only coccinellid predator that is commercialized in the USA for whitefly control. However, the use of D. catalinae as biological control agent has been limited because of its variable degree of success suppressing SWF populations. Despite the relatively high predation rates that have been reported for D. catalinae in laboratory

iii settings, its consumption capacity in more realistic scenarios, whole plants or fields, remains largely unknown.

To support my thesis, I reviewed the functional response theory emphasizing the most accepted methods to scale up laboratory results. I also explored what is known about the biology of the tomato-SWF-D. catalinae system (Chapter 1). I then evaluated key predator behavioral patterns by modeling the interaction between the spatial distribution of the SWF and the search behavior of D. catalinae. First, I developed an algorithm to generate within-plant spatial distributions of the SWF, based on aggregation patterns observed within and among tomato leaves (Chapter 2). Second, I described the spatial interaction between the SWF and D. catalinae at the within-plant scale and examined its effects on D. catalinae predation rates and functional response. I found that prey and predator prefer different regions within plants and that predation rates and the functional response at the scale of a leaflet are comparable to what have been observed in the laboratory. In contrast, I observed that predation rates are lower and that the functional response changes qualitatively when the scale of observation is increased from the leaflet to the plant (Chapter 3). To gain understanding of the processes that drive such a change in predation rates and functional response with scale transition, I developed an individual-based model that incorporates the observed behavioral patterns of D. catalinae individuals when preying on SWF nymphs within tomato plants (Chapter 4). I found that the number of leaflets visited per plant by predators and the degree of spatial alignment between predator and prey distributions impact predation rates significantly at the spatial scale of the whole plant. Also, I demonstrated that simple measures of prey distribution

iv and predator foraging patterns can be used to scale up functional responses estimated through small, homogeneous laboratory settings. Altogether, my research shows that non- random distributions and movement patterns of prey and predators can be predicted, at least within plant structures, and that simple measures of such patterns can be used to accurately model predation rates within plants using observations from laboratory settings. My thesis can be applied to overcome current limitations in the extrapolation of data collected in the laboratory to the field, which ultimately will help fine-tune release procedures of biological control programs.

Dedicated to Said, Stella, Andrea, Ana Maria and Emilia

Acknowledgments

I would like to thank my advisors, Dr. Luis Canas and Dr. Casey Hoy, who knew how to make the perfect fit with my expectations and guided me patiently through this long process. I appreciate the tons of things they have taught me but, especially, that they always made me feel in family.

I thank my committee members, Dr. Larry Phelan, Dr. Larry Madden and Dr.

Robin Taylor who stimulated my critical thinking and with whom I enjoyed sharing my thoughts and receiving their always refreshing points of view. Especial thanks to Dr.

Larry Phelan for his enthusiasm teaching science and for being such a great instructor in the journal clubs.

I enjoyed the company of excellent lab mates during my first years. Claudia

Kuniyoshi and Karla Medina made my life easier during my years in Wooster (OH).

Especially, I would like to thank Alfredo Rios who became my friend and who taught me a great portion of what I have learnt.

I served as a Teaching Associate for several years, and was lucky enough to have worked with Dr. Carol Anelli with whom I learn to love education. Not only is she an outstanding instructor, but also a wonderful human being.

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I want to thank Aristóbulo López, my first research advisor at Corpoica, who introduced me to science and showed me the value of knowledge and creativity.

Funding for my dissertation project was received from Corpoica, Colciencias, The

Fulbright Program, the development fund contributed by K. W. Zellers and Sons Farms, and the Department of Entomology at The Ohio State University. Research support was also provided by state and federal funds appropriated to the Ohio Agricultural Research and Development Center, through SEEDS Research Enhancement Competitive Grants

Program (Grant OHOA1006).

The result of this work would not have been possible without the company and support of friends that knew how to keep me sane and healthy. I am thankful for the great memories I share with Santiago Sanchez, Constanza Echaiz, Wilmer Rodriguez, Ben

Phillips, Silvia Duarte and Agus Muñoz. I am fortunate to have met Carolina Camargo a long time ago because she has repeatedly made me understand how valuable a true friendship is.

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Vita

2005 ...... B.S. Biology, Pontificia Universidad

Javeriana (Colombia)

2008 to present ...... Research Scientist, Corpoica (Colombia)

2009 to 2012 ...... Fulbright fellow, Department of

Entomology, The Ohio State University.

2012 to present ...... Graduate Teaching Associate, Department

of Entomology, The Ohio State University.

Publications

Rincon, D. F., C. W. Hoy, and L. Canas. 2015. Generating within-plant spatial distributions of an insect herbivore based on aggregation patterns and per-node infestation probabilities. Environmental Entomology. DOI: 10.1093/ee/nvu022

Rincon, D. F.; C. Camargo, E. Valencia, and A. López-Avila. 2010. Localización de hospedero por larvas neonatas de Tecia solanivora (Lepidoptera: Gelechiidae). Corpoica

Ciencia y Tecnologia Agropecuaria 11: 5 -10. ix

Rincon, D. F., and J. Garcia. 2007. Frecuencia de cópula de la polilla guatemalteca de la papa Tecia solanivora (Lepidoptera: Gelechiidae). Revista Colombiana de Entomología

33: 133-140

Lopez-Avila, A., and D. F. Rincón. 2006. Diseño de un olfatómetro de flujo de aire para medir respuestas olfativas de insectos de tamaño mediano y pequeño. Revista Corpoica,

Ciencia y Tecnología Agropecuaria 7: 61 - 65.

Rincon, D. F., and A. Lopez-Avila. 2004. Dimorfismo Sexual en Pupas de Tecia solanivora (Povolny) (Lepidoptera: Gelechiidae). Revista Corpoica, Ciencia y

Tecnología Agropecuaria 5: 41 - 42.

Fields of Study

Major Field: Entomology

Table of Contents

Abstract ...... ii

Acknowledgments...... vii

Vita ...... ix

Publications ...... ix

List of Tables ...... xv

List of Figures ...... xvi

Chapter 1: Introduction ...... 1

Predator-prey models and biological control ...... 1

Functional response theory...... 3

Fit of data to functional response models ...... 6

Scale transition and functional responses ...... 9

Research foci ...... 13

Study system ...... 13

The tomato plant ...... 14

The silverleaf whitefly, Bemisia tabaci biotype B ...... 15

The predator Delphastus catalinae ...... 17

Research objectives ...... 20

Research hypotheses ...... 21

Chapter 2: Generating within-plant spatial distributions of an insect herbivore based on aggregation patterns and per-node infestation probabilities ...... 23

Abstract ...... 23

Introduction ...... 24

Materials and methods ...... 29

Plant-insect system ...... 29

Data collection ...... 31

Algorithm description ...... 32

Calibration of models ...... 35

Algorithm verification and validation ...... 39

Individual parameter value sensitivity analysis ...... 43

Results ...... 45

Algorithm verification and validation ...... 45

Model elasticity and individual parameter value sensitivity analysis ...... 49

Discussion ...... 53

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Assumptions ...... 53

Ecological Interpretation of Parameters ...... 58

Application of the algorithm...... 60

Chapter 3: Intra-plant spatial interaction between Delphastus catalinae (Coleoptera:

Coccinellidae) and Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) and its effect on predation rates...... 62

Abstract ...... 62

Introduction ...... 63

Materials and Methods ...... 69

Data analysis ...... 71

Results ...... 75

Discussion ...... 81

Chapter 4: Scaling up functional responses from Petri dish to plant scale based on predator patch residence times and patch visitation patterns ...... 93

Abstract ...... 93

Introduction ...... 94

Study system ...... 97

Materials and methods ...... 99

Model description ...... 100

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Model calibration ...... 102

Model validation ...... 108

Sensitivity analysis ...... 110

Results ...... 113

Model validation ...... 113

Sensitivity analysis ...... 116

Discussion ...... 117

Chapter 5: Conclusions and future directions ...... 131

Distribution of silverleaf whitefly nymphs within tomato plants ...... 132

Differences in leaf nodal preference between the silverleaf whitefly and the predator

Delphastus catalinae ...... 133

Relationship between predator foraging behavior and type of functional response ... 133

Prediction of predation rates within plant structures ...... 135

Future directions ...... 136

References ...... 138

Appendix A: R code of the algorithm to generate silverleaf whitefly distributions within tomato plants ...... 170

Appendix B: R code of a spatially explicit individual-based model to scale-up a functional response estimated in the laboratory for the system silverleaf whitefly-

Delphastus catalinae inhabiting tomato plants...... 175 xiv

List of Tables

Table 1. Algorithm parameters and associated models, estimates and standard errors (SE).

See text for details...... 45

Table 2. Comparison between logistic models to describe the number of Delphastus catalinae individuals found per tomato leaflet...... 79

Table 3. Regression coefficients of the polynomial logistic models fitted to describe the distribution of silverleaf whitefly nymphs, Delphastus catalinae, and silverleaf whitefly nymph mortality across leaf nodal positions of tomato plants. Estimates followed by an asterisk are significantly different from the rest in the same row...... 80

Table 4. Parameter estimates of functional response models of the predator Delphastus catalinae at two spatial scales. Parameters were estimated by Maximum likelihood...... 80

Table 5. Description of parameters used in the model...... 115

Table 6. Whole-plant functional response parameter estimates obtained from observed and simulated values. Numbers in parenthesis are Standard Errors of the Estimates. ... 116

Table 7. Comparisons of functional response models after changing parameter values using Likelihood Ratio Test. When significant differences between type II and III models were no detected (α = 0.1), the type II model was selected because of its reduced number of parameters...... 119 xv

List of Figures

Figure 1. Schematic representation of a tomato plant (a) and leaf (b), showing how leaf nodal positions and leaflets were numbered to record the location of the silverleaf whitefly nymphs...... 30

Figure 2. Flow diagram of the algorithm used to distribute silverleaf whitefly nymphs and eggs on a tomato plant, showing the process for generating counts in leaves (a) and leaflets (b). Functions f, F, g, and G represent equations 6, 7, 8, and 9, respectively (see

Table 1)...... 36

Figure 3. Comparison between generated (dark bars) and observed (light bars) data, by leaf nodal position of tomato plants and associated regression analyses. (a) Verification analysis of Coefficients of variation (CV’s) about the mean proportion of silverleaf whitefly nymphs. (b) Validation analysis of the mean proportion of silverleaf whitefly nymphs. (c) Validation analysis of the mean proportion of silverleaf whitefly eggs.

Dashed and continuous lines in regression plots represent the concordance and the best-fit lines, respectively...... 48

Figure 4. Validation analysis of the mean frequency of silverleaf whitefly densities per leaflet within tomato leaves, showing the comparison between generated (dark bars) and observed (light bars) data...... 49 xvi

Figure 5. Sensitivity analysis of parameters with elasticity coefficients higher than 0.3 after changing their values by +20% (∆), +50% (□), -20% (x) and -50% (◊) with respect to the original estimates (bold line, ○). (a) Effect of varying pv and θv on the generated mean proportion of eggs per leaf nodal position. (b) Effect of varying av, bv and pv on the generated Coefficients of variation. (c) Effect of varying bl on the generated mean frequency distribution of nymph numbers among leaflets...... 52

Figure 6. Response of Delphastus catalinae adults to different silverleaf whitefly nymph densities on tomato plants. a) Proportion of leaves visited by predator (○, solid line) in comparison with the proportion of leaves infested with silverleaf whitefly nymphs (x, dashed line) as a function of silverleaf whitefly density. b) Proportion of predators observed in the tomato plant as a function of silverleaf whitefly density...... 76

Figure 7. Intra-plant distribution of silverleaf whitefly nymphs (a), the predator

Delphastus catalinae (b) and silverleaf whitefly mortality (c)...... 79

Figure 8. Functional response of Delphastus catalinae preying on silverleaf whitefly nymphs at the intra-leaf (a) and the intra-plant (b) spatial scales...... 81

Figure 9. Flowchart describing the routine and decisions followed by a modeled predator in a four-day predator-prey interaction period. See text for details...... 105

Figure 10. Distribution of the proportions of silverleaf whitefly nymphs (light bars) and the predator Delphastus catalinae visits (dark bars) across leaf nodal positions within tomato plants. For illustrative purposes, I present a 12-leaf plant, being “1” the lowest nodal position. a) Perfect alignment between proportions of silverleaf whitefly nymphs

xvii and predator visits. b) 40% reduction in parameter p. c) 60% reduction in parameter p.

See Table 1 for details...... 112

Figure 11. Residence time of the predator Delphastus catalinae as a function of density of silverleaf whitefly nymphs. a) Number of leaflets visited by predators per tomato plant as a function of silverleaf whitefly nymph density per plant. b) Time spent (in hours) of predators within tomato leaflets as a function of silverleaf whitefly nymph density per leaflet...... 114

Figure 12. Validation analysis of an individual-based model developed to scale up laboratory functional responses to the scale of plant. a) Comparison between observed

(○) and simulated (∆) predation rates, showing the observed (dashed curve) and simulated (solid curve) functional response models at the scale of plant. b) Relationship between observed and simulated predation rates, showing the best-fitting line (solid) and the perfect concordance line (dashed)...... 116

(dotted line). b) Effect in functional response of changing parameter VT (max. residence time in leaflets of predators) by +50% (dashed line), -50% (dash-dot line) and when the time spent by predators within leaflets by predators, g(z), is set constant (solid line), compared with the output from unchanged parameters (dotted line). c) Effect in

xviii functional response of changing parameter p by -40% (dash-dot line) and -60% (dotted line), compared with the output obtained when p = pv and θ = θv (dotted line)...... 120

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Chapter 1: Introduction

Predator-prey models and biological control

Generations of ecologists have worked to apply ecological theory to the selection of appropriate natural enemies for biological control in pest management programs.

Information about the natural history, phenology and behavior of both predators and pests allows the formulation of hypotheses and predictions on the outcome of the interaction between biological control agents and pests in the field. To this end, mathematical models are a useful tool to reduce the complexity of dynamic, real-world phenomena by including only the most relevant ecosystem variables and assessing their interactions under a wide range of hypothetical scenarios.

Nonetheless, some argue that ecological modeling has had little success in guiding biological control programs (Murdoch et al. 1985, Liebhold 1994). For example,

Barlow (1999) reviewed 50 models of biological control systems and found that none successfully predicted the long-term outcome of classical biological control programs.

Yet, the author found moderate success of models in predicting the outcome, optimum release strategies and selection of appropriate natural enemies for short-term augmentative and inoculative biological control programs.

Biological control models focus on the interactions and factors affecting predator- prey dynamics to help make decisions about agent releases that either induce stability under low pest densities for long time periods (i.e. on perennial crops), or prevent a pest population from exceeding an economic injury level over short cropping seasons (i.e. on annual crops). In both cases, two main components are needed to describe predator-prey dynamics: insect development and the rate at which an individual predator consumes prey

(consumption rate). Development is usually modeled for each species as a function of temperature, whereas the consumption component of the model is used to link the predator and prey sub-models. Several methods have been used successfully to model insect development using data collected in laboratory experiments (e.g. Goudriaan 1973,

Manetsch 1976, Richter 2008). This success is not surprising given the biochemical nature of development, which leads to a relatively deterministic outcome (Wagner et al.

1984). However, the consumption component usually fails to predict the predator’s performance in the field from data collected in laboratory experiments, mainly because consumption is mostly a behavioral, stochastic phenomenon affected by a number of aspects that are often left out of models (Abrams 1982). The development of appropriate models to successfully implement biological control programs requires an understanding of factors affecting variation in prey consumption by predators.

Variation in consumption rate has been associated with a number of factors, including the change in a predator’s behavioral and physiological patterns in response to host plants (Messina and Hanks 1998, Timms et al. 2008), prey type (Aljetlawi et al.

2004, Sarmento et al. 2007), interaction with other predators (Skalski and Gilliam 2001), spatial heterogeneity (Nachman 2006a, Englund and Leonardsson 2008) and several abiotic variables (Legaspi et al. 1996a, Logan and Wolesensky 2007). However, most theoretical constructs for predator-prey interactions describe the number of prey eaten by predators as a function of the number of prey available in a given area. Such a description is known as the functional response (Holling 1961).

Functional response theory

The general procedure to collect data to fit functional responses consists essentially of a bioassay in which individual predators, after a starvation period, are exposed to a range of fixed numbers of prey over a given time period. Usually, this experiment is performed under laboratory conditions, using prey numbers as treatments and small arenas (usually Petri dishes) with a substrate of leaf discs as experimental units.

The last step is the search for a mathematical function that best describes the number of prey eaten against several levels of prey offered, using nonlinear regression analysis

(Fenlon and Faddy 2006).

There are three basic types of functional responses. The first one (type I) is the simplest, and assumes a linear relationship between consumption rate and prey density.

Because type I functional response does not account for prey handling and searching times, individual predators increase their prey consumption as the prey population increases, without reaching an asymptote. Type I functional responses are found in

predator-prey systems in which a constant proportion of prey consumed across levels of prey density can be assumed, typical of filter feeders, or passive (i.e. sit-and-wait) predators.

The type II functional response model is a saturating function where, as prey density increases, consumption rate eventually reaches a theoretical maximum determined by the handling time. The type II functional response model includes two parameters: handling time and attack rate. Handling time includes time spent pursuing, subduing, and consuming each prey item plus the time spent preparing to search for the next prey item. As prey density increases, searching time approaches zero, but handling time remains constant and therefore it prevents the increase of consumption rates above a maximum. Attack rate, in turn, is a constant describing the number of prey attacked while the predator is searching for prey (not handling) per unit time. This type of response was modeled by Holling (1959) after an elegant experiment, in which he used blindfolded human subjects who were instructed to find and pickup small discs of sandpaper on a flat surface. The number of prey eaten, or discs collected, increased as the time spent searching, or not handling, increased. There were no satiation levels, no foraging strategies and the encounter probability was constant. Predators may exhibit the same behavior in response to different levels of prey density.

The type III functional response is similar to type II but resembles a sigmoid curve, where the consumption rate of predators increases at a high rate at low prey densities, but increases at a low rate at higher densities of prey. This functional response

typically emerges as a result of changes in the foraging behavior of predators, which make capture more efficient at low prey density levels. The only difference with respect to type II is that type III functional responses make attack rate a function of prey density, a relationship that can take a number of forms depending on the number of parameters included in the equation (Juliano 2001).

Classical functional response theory considers three general changes of predator behavior that can trigger a sigmoid (type III) functional response (Neal 2004). The most common situations that are known to cause the emergence of type III functional responses are: 1) predators switching prey from a less abundant prey type to a more abundant one (Cock 1978); 2) presence of refuges for prey (Berryman et al. 2006); and 3) predators may become better at capturing and subduing prey (i.e. learning), so experienced predators would be more efficient and successful in the next encounter

(Holling 1961). More recently, the emergence of type III functional responses has been associated with predators foraging in patchy habitats as a result of the establishment of co-aggregations between prey and predator populations (Nachman 2006a, Morozov 2010,

Cordoleani et al. 2013).

One important common assumption of the classical functional response models described above, oftentimes ignored, is that they describe predation at constant prey densities. However, in most laboratory settings, prey are either not replaced as they are consumed, or they are replenished at some time interval. Therefore, prey density typically declines as the experiment proceeds, implying that the classical models described by

Holling do not accurately describe functional responses. The Rogers random predation equation (Rogers 1972, Juliano 2001) describes the number of prey eaten by predators where prey population becomes depleted, a more appropriate model to use under the aforementioned experimental conditions. However, the Rogers equation does not have a closed-form solution for the number or the proportion of prey eaten, so it has to be written in terms of the Lambert W function to allow parameter estimation through

Maximum likelihood (Bolker 2008).

Fit of data to functional response models

Theory behind functional responses is relatively well supported by a wide body of empirical studies that document their soundness as models to describe consumption rates

(Juliano 2001, Neal 2004). However, the literature often shows that the results given by functional response experiments in the laboratory do not reflect the performance of predators in the field.

In a comparative study, Munyaneza and Obrycki (1997) studied the functional response of Coleomegilla maculata De Geer (Coleoptera: Coccinellidae) preying on

Leptinotarsa decemlineata (Say) (Coleoptera: Chrysomelidae) eggs in the laboratory, greenhouse and field. Although the authors always found a type II functional response, attack rates and handling times differed among laboratory, greenhouse, and field conditions. The highest attack rate was found in the laboratory and the lowest in the field, whereas handling time was higher in the field than in laboratory conditions. Interestingly,

the authors found a differential relationship between the number of egg consumed per egg mass and the number of egg masses. In the laboratory, the relationship was linear while in the greenhouse and field the number of eggs consumed per egg mass fluctuated around an average as egg masses increased. This result may provide evidence for a switch in the foraging behavior of C. maculata when the density of prey decreases.

Within Petri dishes, C. maculata may exhibit a foraging pattern to exploit an environment in which prey is distributed uniformly, whereas in greenhouse or field conditions it may switch to a more complex behavioral pattern of patch exploitation.

O'Neil (1997) reached similar conclusions after assessing the functional response of Podisus maculiventris (Say) (Heteroptera: Petatomidae) attacking third-instar

Colorado potato beetles in the laboratory and potato fields. He found little correspondence between functional response parameters measured in the laboratory and the field. As in the study above, attack rates were lower and handling times higher in the field than in the laboratory, supporting previous studies on the same predator

(Wiedenmann and O'Neil 1991). O’Neil (op. cit.) hypothesized that differences in predator performance in the field might be due to the “artificially high” prey densities offered in laboratory experiments. While predators in the laboratory experiment showed an increased number of attacks with prey density, the number of attacks remained constant in the field. Furthermore, he found that the area searched by P. maculiventris declined as the Colorado potato beetle density increased. The author concludes that P. maculiventris employs a search strategy that minimizes the area searched, maintaining a

relatively constant rate of attacks. This might be the result of changes in predator foraging behavioral patterns under low densities and aggregated distribution of prey.

Similar results were found in studies with the same predator preying on Epilachna varivestis Mulsant (Coleoptera: Coccinellidae) (O’Neil 1988). The study concludes that it is not appropriate to rely on attack rates derived from laboratory experiments since they are focused merely on “consumptive behaviors”, which are irrelevant if attack rates are significantly reduced (more than a half) due to search strategies that only arise under field conditions.

In addition to changes in predation rates, attack rates and handling times, changes in the type of functional response model between field and laboratory conditions have also been reported. For example, Nachman (2006b) found that the consumption rates of

Phytoseiulus persimilis Athias-Henriot (Acari: Phytoseiidae) feeding on Tetranychus urticae Koch (Acari: Tetranychidae) differ depending on the scale of observation, and that these changes were associated with changes in the type of functional response

(Nachman 2006a). Similar results have been reported for phyto- and zoo-plankton

(Morozov 2010, Cordoleani et al. 2013) and are explained by the interaction between the effect of prey patchiness and predator aggregative behavior.

In summary, three main problems can be identified when predation data obtained from laboratory experiments are compared with data collected in the field: (1) predation rates are higher and handling times lower in laboratory settings than in the field, (2) the general shape of the functional response curve may vary under field and laboratory

conditions, and (3) several behavioral patterns arise in patchy habitats that are not observed in conventional laboratory functional response bioassays. Estimation of functional responses at the field or whole plant scale is difficult because experimental settings are often complicated, time-consuming and expensive. Such limitations are even more evident when the species under study are small and conventional techniques for measuring predation in the field might not be applicable. To develop theory on predator- prey systems, therefore, it is critical to make laboratory settings useful to infer reliable estimates of predation rates at more realistic spatial and temporal scales.

Scale transition and functional responses

The problem of scale transition in ecology has long been recognized (O'Neil

1990, Ives et al. 1993, Englund and Leonardsson 2008). Like most ecological processes, predation is non-linear with respect to its explanatory variable, which implies that expected values from functional responses at spatial scales with patchy distributions of resources should generally be different from the responses estimated at smaller spatial scales where the distribution of resources is often random (Welsh et al. 1988). Since a constant probability of encounter of prey items, indicating random spatial distribution of the prey, is one of the assumptions of functional responses estimated in the laboratory or other small-scale settings, the direct extrapolation of these results to patchy environments can lead to biased model predictions (Englund and Leonardsson 2008).

Small-scale environments often neglect variation due to aggregated prey distributions. The nature of plant architecture leads insect herbivore populations to be distributed in patches within leaves and/or leaflets. However, such patches of insect herbivores might be distributed in a nonrandom manner within the plant canopy. The spatial arrangement of leaves along the plant main stem often reflects a gradient of leaf tissue ages, which in turn is associated with changes in leaf nutrient content, moisture, and chemical and structural defenses (Coley and Kursor 1996, Guiboileau et al. 2010).

Young leaves are valuable to plants because they have greater potential to contribute to future fitness (Matsuki et al. 2004). Young leaves also tend to have higher concentrations of defensive compounds than their mature counterparts (Herms and Mattson 1992). Yet, young leaf tissue is also a particularly attractive food source for insect herbivores because of its relatively high concentrations of water and nutrients (Coley 1980). Numerous studies have documented that leaf age preference by insect herbivores vary in predictable ways based on their degree of specialization (e.g. Cates 1980, Raupp and Denno 1983,

Blüthgen and Metzner 2007). Specialist herbivores are expected to prefer young leaves because they contain higher nutrient and water concentrations, even though these leaves tend to be highest in concentrations of qualitative defenses (e.g. toxins). Generalist herbivores, being more vulnerable to plant defenses, are usually deterred by the most defended plant parts and prefer mature leaves. These leaf preference patterns contribute to defining herbivore within-plant spatial aggregations, and make certain plant regions consistently more likely to be infested (Raupp and Denno 1983).

Small-scale environments also neglect variation in predator behaviors associated with finding food. In particular, optimal foraging theory predicts that predators will increase their search effort with prey encounters when prey is aggregated to concentrate their search in high resource density areas. The mechanism is known as ‘area- concentrated’ or ‘success-motivated’ search, and is maladaptive if resources are randomly distributed (Pyke et al. 1977, Bell 1991, Benhamou 1992). ‘Area-concentrated’ search has been described for various insect species, especially for Coccinellids (e.g.

Carter and Dixon 1982, Nakamuta 1985, Ferran et al. 1994, Guershon and Gerling 2006).

This behavior is often described in terms of an increase in turning angle (direct klinokinesis) along with a reduction in walking speed (inverse orthokinesis) of predators in two-dimensional arenas with aggregations of prey, so that more time and energy are invested in prey patches within which prey are randomly distributed (Bell 1991, Turchin

1998). However, prey distribution within laboratory arenas is often random and, therefore, an increase in search effort motivated by prey encounter would not provide, in theory, any benefit for the predator. In contrast, in pacthy environments, an increase in number of prey patches visited or in area explored motivated by the encounter of prey would result in a higher chance of finding new prey items in the nearby area.

Several alternatives have been introduced for combining small-scale observations from the laboratory with field measurements of heterogeneity. The earliest attempts introduced spatial heterogeneity into host-parasitoid systems implicitly using probability distribution functions (Bailey et al. 1962, Hassell 1980). More recently, this approach has

also been applied to predator-prey systems (Nachman 2006a). More general mathematical methods of scale transition have been applied to functional responses of aquatic predator-prey systems by Bergstrom et al. (2006) and Englund and Leonardsson

(2008). The general idea is to approximate the deviation due to heterogeneity by Taylor’s expansion series, performed to the second order for functional responses: the first to approximate the deviation from prey spatial variance, and the second to assess the spatial covariance between prey and predator populations.

However, I am not aware of any attempts to scale-up functional responses by combining small-scale observations with explicit measurements of spatial heterogeneity.

This is surprising because predation is largely the result of individual behaviors, which are influenced by spatial non-uniformities and variation. Yet, models that do not explicitly incorporate spatial heterogeneity can only account for interactions at the population level, limiting the examination of key individual behavioral traits that can affect predator performance in biological control programs. Models that incorporate spatial heterogeneity only implicitly may also provide biased predictions if predator population density is low (i.e. less than 1 per prey patch), which is often the case in agricultural systems. Moreover, these models often assume that any prey patch is equally likely to be found and that predators redistribute themselves in response to changing prey distributions. However, predators often search more extensively within some plant regions than others, favoring non-uniform predator-prey interactions at the spatial scale of the plant that could potentially impact predation rates (e.g. Costamagna and Landis

2011, Hodek et al. 2012, Reynolds and Cuddington 2012), and can affect population dynamics and ecosystem function (DeAngelis and Mooij 2005).

Research foci

Ideally, predation should be measured through direct observation for which laboratory settings are the more feasible alternative. However, the development of modeling approaches to combine measurements of spatial heterogeneity with laboratory observations are a necessity if predator-prey models are to be used to help fine-tune protocols of biological control programs. My thesis is that predation rates within whole plants are driven by the interaction between prey distribution, individual predator patch- to-patch behavior and consumption rates within patch units. I propose that results derived from simple laboratory settings can be useful to predict predation rates within whole plants if they are combined with spatially explicit descriptions of prey distribution and predator movement patterns. I assume that the leaf is the smallest patch unit within a plant within which prey and predators behave as in laboratory settings.

Study system

My system consisted of the silverleaf whitefly, Bemisia tabaci biotype B

(Gennadius) (Hemiptera: Aleyrodidae), and the predator Delphastus catalinae (Horn)

(Coleoptera: Coccinellidae), inhabiting tomato plants in a greenhouse environment. I chose a greenhouse system because the application of ecological models is simpler, given

the reduced chance of unfavorable weather events, emigration and intra-guild predation.

What follows is a more detailed description of the study system.

The tomato plant

The tomato, Solanum lycopersicum L. (Solanaceae), is one of the most important vegetable crops in the U.S.A. and the state of Ohio. The U.S.A. is the second world’s tomato producer, with more than 165.000 ha harvested and 12.5 million tons produced during 2008 (FAOSTAT 2009). Ohio is the fourth state in the country planting and producing tomatoes, with an average of 5,000 ha harvested and 300,000 tons of tomatoes produced yearly (USDA 2010).

Tomato plants are typically made of a main stem with a terminal bud from which leaves grow indeterminately. Lateral (secondary) stems can grow out from buds located in the axil of leaves but, in tomato production systems, plants are typically grown as single stems by removing all lateral buds, known as suckers, to increase fruit yield and quality (Cockshull et al. 2001). Tomato leaves are compound and consist of an odd number of leaflets of about the same area, distributed along the leaf rachis. The number of leaflets differs among leaves depending on their nodal position. The first four leaves contain between 3-9 leaflets on average, whereas the rest typically contain 13 (Sarlikioti et al. 2011).

The silverleaf whitefly, Bemisia tabaci biotype B

The silverleaf whitefly, B. tabaci biotype B, has a long and complex taxonomic history. Bemisia tabaci was first described in 1889 by Gennadius as a pest in tobacco in

Greece. Since then, biotypes have been assigned to different strains based on distinguishable characteristics as agricultural pest, mostly after the invasion of the B biotype to at least 54 countries in the late 80s (De Barro et al. 2011). Such invasion lead to the designation of Bemisia argentifolii Bellows and Perring (Bellows et al. 1994) to the

B biotype, but such designation at the species level was criticized due to the insufficient molecular or biological data available to support it (De Barro et al. 2005). Since then, most literature has been referring to the SWF as one of the several B. tabaci biotypes.

However, new developments on species delimitation have shown that B. tabaci is a cryptic species complex containing 11 higher genetic groups and at least 24 morphologically indistinguishable species, the B biotype as a monophyletic group among them (Dinsdale et al. 2010). At least 10 more homogeneous groups have been added to the B. tabaci complex since such delimitation was first published, four new groups added in a survey in China (Hu et al. 2011) and six more found in a survey in Argentina

(Alemandri et al. 2012). This trend is expected to continue given the worldwide distribution of the B. tabaci complex. For example, several geographically related genetic distinctions were made in Colombia and Israel from B. tabaci collections (Wool et al.

1991, Wool et al. 1994) that are waiting to be analyzed under the new parameters of classification. I will refer to the SWF as B. tabaci biotype B in this thesis because, to

date, no formal complete taxonomic revision of the B. tabaci complex has been published.

The silverleaf whitefly is a haplodiploid species and a polyphagous herbivore that is considered one of the most important pests of a number of greenhouse and field crops, including tomato (Capinera 2001). Both adults and nymphs feed primarily on the underside of leaves, by piercing and sucking sap phloem. Feeding might cause reduction in plant vigor, and honeydew produced by nymphs and adults serves as a substrate for sooty molds. Indirect effects of SWF infestations include induction of growth disorders and irregular fruit ripening in tomatoes (Schuster et al. 1990, Schuster 2001). The SWF also vectors many plant pathogenic viruses from seven groups including: geminiviruses, closteroviruses, carlaviruses, potyviruses, nepoviruses, luteoviruses and a DNA- containing rod-shaped virus (Oliveira et al. 2001). In practically all countries where tomatoes are grown, geminiviruses (Begomovirus spp.) affect production significantly.

Among the most economically relevant whitefly-transmitted viruses are the Tomato yellow leaf curl virus and the Tomato mottle virus, but at least 17 more of these viruses have been identified only in the western hemisphere (Polston and Anderson 1997).

The silverleaf whitefly goes through four nymphal stages that are all sessile, except for neonates, which move a few millimeters before settling sometime during the first few minutes after hatching (Byrne and Bellows 1991, Simmons 2002). Adults are winged and females lay their eggs as they feed on the underside of leaves (Byrne and

Bellows 1991). As an ectothermic organism, the length of the nymphal phase and both

adult longevity and fecundity largely depend on temperature and also on the host-plant

(Tsai and Wang 1996, Drost et al. 1998, Gruenhagen and Perring 2001, Nava-Camberos et al. 2001, Takahashi et al. 2008). In tomato plants grown in greenhouses at ~20-25°C, females lay on average ~150-190 eggs. The complete immature phase (from egg to adult) is about 17-22 days long, which represents about 50% of the SWF life cycle (Tsai and

Wang 1996, Takahashi et al. 2008). Most predators and parasitoids used for biological control of SWF infestations are specialized on nymph or egg consumption, mainly because the sessile phase (egg and nymphal stages) is particularly vulnerable to predation

(Gerling et al. 2001).

The predator Delphastus catalinae

The predator D. catalinae, as the SWF, also has an interesting taxonomic history.

This obligated whitefly predator was first noted preying on B. tabaci in Florida (U.S.A.) during the late 1980s and identified as Delphastus pusillus (LeConte). Since then, a number of biological and efficacy studies were carried out and several massive cultures were developed. The distribution of “D. pusillus” expanded since then in agricultural systems in response to high demand that resulted from its popularity as a natural enemy of B. tabaci (Hoelmer and Pickett 2003). In 1994, Gordon (1994) made a major taxonomic revision of the genus where several species were re-described, some new species were described, and their known geographic distribution was updated. Delphastus dejavu Gordon and D. sonoricus Casey, previously considered conspecific with D.

pusillus, were designated as separate species, and the distribution of D. pusillus was restricted to the southeast of the U.S.A. In turn, the distribution of D. catalinae was reported from northern Colombia through Mexico into the southeastern coast of the

U.S.A. The new designations raised questions on the identity of the specimens collected during the late 1980s, and those studied and cultured in Florida in the early 1990s.

Samples from populations collected in the 1990s were re-examined and were all re- identified as D. catalinae, including all samples collected from commercial and non- commercial cultures. Older collections from 1960-1980 were also identified as D. catalinae, but all specimens collected prior to 1960 were identified as D. pusillus. It is now accepted that both D. catalinae and D. pusillus coexist in Florida today, and that D. catalinae is native to Colombia and introduced to Florida through plant material during the early 1980s. Therefore, it is believed D. pusillus was misidentified for a while and that most (if not all) the studies of D. pusillus biology and behavior preying on Bemisia spp. published in the early 1990s actually refer to D. catalinae (Hoelmer and Pickett

2003).

The predator D. catalinae typically goes through four larval stages that also prey upon whiteflies and pupate in the base of the stem of plants and the underside of low leaves (Hoelmer et al. 1993). The development of immature life stages, and adult longevity and fecundity vary with temperature and diet (Hoelmer et al. 1993, Kutuk and

Yigit 2007, Legaspi et al. 2008). In the greenhouse at temperatures of ~20-25°C, eggs hatch in approx. 4.5 days, and the larval and pupal stages last about 10 and 5 days,

respectively. Adults live for ~77-138 days and each female lays approx. 260 eggs; most eggs are laid during the first 100 days after the 9th day post-emergence (Simmons and

Legaspi 2004, Legaspi et al. 2008, Simmons et al. 2012). Both D. catalinae larvae and adults are obligate whitefly predators. Although they occasionally feed on the honeydew excreted by B. tabaci, they do not seem to be attracted to it and its consumption does not replace nutritionally a whitefly diet (Simmons et al. 2012).

Adults and larvae of the predator generally feed only on the internal contents of their prey leaving the collapsed cuticle of nymphs or shed of eggs after feeding. Adults can eat up to 160 SWF eggs in a day, although consumption rate decreases with the size of the prey to about 11 4th instar nymphs in a day, mainly due to increases in handling time (Hoelmer et al. 1993, Legaspi et al. 2006). Given the observed predation rates, D. catalinae is considered the best commercially available predator of Bemisia spp. (Hunter

1998, Obrycki and Kring 1998). However, high voracity shown in laboratory settings does not translate to SWF suppression in the field. Heinz et al. (1999) found that SWF suppression was > 50% within field cages, but they did not find any significant reductions in SWF populations in the open field. Significant reductions in SWF populations were also observed in greenhouse poinsettias with the release of D. catalinae individuals, and even damage recorded in plants treated with chemical insecticides was not different from that observed in areas where D. cataliane was released. However, the cost associated with biological control of the SWF was approx. five times greater than the insecticide-based SWF control program (Heinz and Parrella 1994b).

Research objectives

Significant reductions in costs associated with biological control of greenhouse pests could be achieved if timing and release rates of predators are optimized. An important approach for generating information-based application methods for the SWF-

D. catalinae system is the development of accurate ecological models. However, critical parameters of the system, such as predation rates, have to be estimated through laboratory experiments because of the difficulty of calculating them directly at realistic spatial and temporal scales. Thus, the main objective of this research was to develop a model to predict predation rates at a large spatial scale (plant), from data collected at a small spatial scale (laboratory). In Chapter 2, I present an algorithm developed to generate spatial counts of SWF eggs and nymphs within tomato plants. The algorithm considers the per-node infestation probabilities that result from SWF leaf preference and the aggregation patterns observed within tomato plant canopies. In Chapter 3, I examine the spatial interaction between the distribution of SWF nymph patches and the foraging pattern exhibited by D. catalinae within tomato plants. I found that the spatial interaction between the SWF and D. catalinae is not uniform within tomato plants and that predation rates and the functional response depend on the spatial scale at which the observation is made. In Chapter 4, I present a spatially explicit individual-based model that simulates the interaction between the SWF and D. catalinae and predicts predation rates at the whole plant spatial scale. The model combines laboratory measurements of predation rates with the spatial distribution of the SWF nymphs and observations on the patch-to-

patch movement patterns of D. catalinae. I show that simple measurements of spatial heterogeneity at relevant spatial scales can be used to scale up information obtained from small-scale laboratory experiments. Using this model, I analyzed the predator behavioral traits that affected both the reduction in predation rates and the shift in type of functional response that results from the increase in the scale of observation. In Chapter 5, I summarize my findings and analyze the implications in light of the stated and tested hypotheses. I suggest the future directions for research on predator-prey interactions in greenhouse environments and the development of models that can be used to optimize the use of natural enemies to control insect pests, with emphasis on annual crops grown in controlled environments.

Research hypotheses

Hypothesis 1: Silverleaf whitefly distribution within plants is consistent and predictable.

Prediction: Probability of infestation of silverleaf whitefly adults and eggs will

increase towards the mid third of tomato plants, where mature leaves are

(evaluated in Chapter 2).

Hypothesis 2: Delphastus catalinae adults exhibit search behaviors that are responsive to patchy distributions of silverleaf whitefly populations.

Prediction 1: The distribution of individuals of Delphastus catalinae among leaflets will be associated with the local density of silverleaf whitefly nymphs

(evaluated in Chapter 3).

Prediction 2: A type II functional response will be observed at the intra-leaflet scale because this functional response is typically the result of uniform, well- mixed predator-prey interactions (evaluated in Chapter 3).

Prediction 3: A type III functional response will be observed at the intra-plant scale, because this type of functional response is observed when prey distribution is patchy, as a result of the establishment of co-aggregations between prey and predator populations (evaluated in Chapters 3 and 4).

Prediction 4: Delphastus catalinae adults will display ‘area-concentrated’ search in response to aggregated prey populations (evaluated in Chapter 4).

Chapter 2: Generating within-plant spatial distributions of an insect herbivore based on aggregation patterns and per-node infestation probabilities

Abstract

Most predator-prey models extrapolate functional responses from small-scale experiments assuming spatially uniform within-plant predator-prey interactions.

However, some predators focus their search in certain plant regions, and herbivores tend to select leaves to balance their nutrient uptake and exposure to plant defenses.

Individual-based models that account for heterogeneous within-plant predator-prey interactions can be used to scale up functional responses, but they would require the generation of explicit prey spatial distributions within plant architecture models. The silverleaf whitefly (SLW) Bemisia tabaci biotype B is a significant pest of tomato crops worldwide that exhibits highly aggregated populations at several spatial scales, including within the plant. As part of an analytical framework to understand predator-SLW interactions, the objective of this research was to develop an algorithm to generate explicit spatial counts of SLW nymphs within tomato plants. The algorithm requires the plant size and the number of SLW individuals to distribute as inputs, and includes models that describe infestation probabilities per leaf nodal position and the aggregation pattern of the SLW within tomato plants and leaves. The output is a simulated number of SLW 23

individuals for each leaf and leaflet on one or more plants. Parameter estimation was performed using nymph counts per leaflet censused from 30 artificially-infested tomato plants. Validation revealed a substantial agreement between algorithm outputs and independent data that included the distribution of counts of both eggs and nymphs. This algorithm can be used in simulation models that explore the effect of local heterogeneity on whitefly-predator dynamics.

Introduction

One important component of predator-prey models is the functional response, which is a mathematical description of the consumption rate of predators as a function of prey density (Juliano 2001). Often, functional responses are derived from small-scale experiments using small, uniform environments (e.g. Petri dishes) as experimental arenas, where prey densities far exceeding economic thresholds are evaluated (Fenlon and Faddy

2006). These functions are then extrapolated to larger natural systems, often leading to overestimations on the search and predation capabilities of natural enemies.

This error in extrapolation from laboratory experiments has long been recognized by ecologists (O'Neil 1990, Ives et al. 1993, Pitt and Ritchie 2002, Fenlon and Faddy

2006, Englund and Leonardsson 2008) and several modeling approaches have been used to scale up functional responses (Oaten 1977, van Roermund et al. 1997, Walde and

Nachman 1999, Nachman 2006a, Hemerik and Yano 2011). These models assume that functional responses derived from laboratory experiments can be used to approximate

within-patch predation rates at the most homogeneous prey patch unit (e.g. leaf lamina).

The overall functional response is estimated by modeling the interaction between within- patch predation rates, inter-patch predator movement rules (e.g. giving-up thresholds and inter-patch travel times), and some description of prey distribution across patches. Prey distribution is often defined in terms of a negative binomial probability distribution function (PDF), where a probability is assigned to the different possible patch sizes to stochastically generate realistic prey counts [but see van Roermund et al. (1997) for an individual-based approach]. As a result, the modeled environment is often a two- dimensional structure where prey patches are identical except for the variation in number of prey, equidistant from each other, and the number of prey on them is irrespective of their location within the plant canopy. Beyond a number of limitations of the negative binomial PDF for describing animal distributions (Taylor et al. 1979), this approach has been useful to extrapolate functional responses in some systems where it is reasonable to assume that predators or parasitoids search the entire plant canopy with the same intensity (e.g. van Roermund et al. 1997, Hemerik and Yano 2011).

Many insect predators (e.g. Coccinellidae), however, search more intensively within certain regions of the plant canopy, resulting in uneven predation pressures at within-plant scale (Bond 1983, Costamagna and Landis 2011, Hodek et al. 2012,

Reynolds and Cuddington 2012). Moreover, the spatially uneven predation pressures lead to the formation of prey refuges that may impact prey population growth rates either by

reducing overall predation rates, by favoring removal of relatively fecund prey individuals located in high-quality plant parts, or both (Costamagna et al. 2013).

Aggregations of insect herbivores (i.e. prey patches) can also be distributed in a nonrandom manner within the plant canopy. The spatial arrangement of leaves along the plant main stem often reflects a gradient of leaf tissue ages, which in turn is associated with changes in leaf nutrient content, moisture, and chemical and structural defenses

(Coley and Kursor 1996, Guiboileau et al. 2010). Young leaves, located closer to the apical meristem, are valuable to plants because their photosynthetic rates and nitrogen concentrations tend to be higher (Mooney and Gulmon 1982) and have greater potential to contribute to future fitness (Ruohomaki et al. 1997, Matsuki et al. 2004). Young leaf tissue is also a particularly attractive food source for insect herbivores because of its high concentrations of water and nutrients (Coley 1980, Coley and Kursor 1996). As a result, young leaves tend to have higher concentrations of defensive compounds than their mature counterparts (McKey 1974, McCall and Fordyce 2010). Numerous studies have documented that leaf age preference by insect herbivores vary in predictable ways based on their degree of specialization (e.g. Cates 1980, Raupp and Denno 1983, Ikonen 2002,

2014). These leaf preference patterns contribute in defining herbivore within-plant spatial aggregations, and make certain plant regions consistently more likely to be infested

(Raupp and Denno 1983).

More realistic scaling up of functional responses could be developed if the spatially non-uniform local predator-prey interactions that occur at the within plant scale are modeled explicitly. Individual-based models have been used to examine population and community-level effects of non-uniform local interactions, because they can account for the movement and distribution of discrete individuals (Judson 1994, DeAngelis and

Mooij 2005). However, these models require realistic representations of prey spatial patterns within plant structures and detailed descriptions of predator movement rules.

The silverleaf whitefly (SLW), Bemisia tabaci biotype B (= Bemisia argentifolii)

(Gennadius) (Hemiptera: Aleyrodidae), is a generalist herbivore and a significant worldwide pest of a variety of field and greenhouse production systems, including tomato

(Stansly and Naranjo 2010). Adults are good flyers and females lay eggs as they feed on the underside of leaves (Byrne and Bellows 1991). All four immature life stages

(nymphs) are sessile except neonates (also known as crawlers), which move a few millimeters before settling during the first few minutes after hatching (Byrne and Bellows

1991, Simmons 2002). Because the sessile phase (egg and nymphal stages) is particularly vulnerable to predation and represents approximately 50% of the SLW life cycle (Tsai and Wang 1996), most predators and parasitoids used for biological control are specialized on nymph or egg consumption (Gerling et al. 2001).

At least 25 species of predators have been associated with the SLW, most of them coccinellids (Gerling et al. 2001, Hodek and Honek 2009). Despite the fact that functional responses of several of these predators have been investigated under laboratory conditions (Legaspi et al. 1996b, Guershon and Gerling 1999, Liu and Stansly 1999), their predation capability in more realistic environments has not been studied.

Specifically, functional responses involving coccinellid predators within single plants are important for SLW control models, because this is the spatial scale at which coccinellid prey refuges are often formed (Costamagna and Landis 2007, 2011, Reynolds and

Cuddington 2012). However, quantifying predation on SLW eggs or nymphs within plants through direct observation is difficult, due to the size of these insects and the unreliability of available methods to distinguish consumed prey (Hoelmer et al. 1993,

Lucas et al. 2004). Even when experiments are performed in small plants (e.g. Heinz and

Zalom 1996), extrapolations to larger plants are questionable.

As part of an analytical framework to study predator-SLW interactions at the plant scale, the main objective of this research was to develop and evaluate a stochastic algorithm to generate realistic spatial distributions of SLW nymphs within tomato plants.

The process was based on the aggregation pattern of SLW nymphs and the probability of them infesting the different leaf nodal positions of tomato plants of different sizes (i.e. number of leaves). Validation was conducted by comparing algorithm outputs with datasets from the literature that included distribution of both eggs and nymphs, and from

plants used in my research that were not included in algorithm development and calibration.

Materials and methods

Plant-insect system

Most tomato cultivars are decumbent and indeterminate. The structure of tomato plants is typically made of a main stem with a terminal bud at the apex, and lateral

(secondary) stems that grow out from buds located in the axil of leaves. In tomato production systems, however, plants are typically grown as single stems by removing all lateral buds (known as suckers) to increase fruit yield and quality (Cockshull et al. 2001).

Tomato leaves are compound and consist of a petiole and an odd number of leaflets of about the same area distributed along the leaf rachis. Some leaflets are compound as well

(secondary leaflets) and smaller leaflets may occur in between leaf rachis segments

(intercalary leaflets) (Figure 1).

Silverleaf whitefly populations are highly aggregated within tomato plots (Byrne and Bellows 1991, Liu et al. 1993) and plants (Schuster 1998, Muniz et al. 2002, Arno et al. 2006). The distribution of SLW nymphs within plants is the result of a process that involves leaf selection, aggregation patterns and oviposition of adults, and dispersal and survival of nymphs. Silverleaf whitefly adults tend to form aggregations that enhance their fecundity and fertility (Ruan et al. 2007), and prefer mature leaves for feeding and oviposition when associated with tomato plants (Ohnesorge et al. 1980, Schuster 1998,

Arno et al. 2006). Dispersal of neonate nymphs should induce differences between egg and sessile nymph distributions within leaflets, but not among leaves or leaflets because nymphs usually settle on the same leaflet upon which the egg was laid (Byrne and

Bellows 1991). The distribution of the subsequent (sessile) nymphal stages among leaves is progressively shifted down with the appearance of new tomato leaves (Schuster 1998), although the shape of the nymph distribution function may change as nymphs develop if there are different nymph survival rates among leaves within plants.

Figure 1. Schematic representation of a tomato plant (a) and leaf (b), showing how leaf nodal positions and leaflets were numbered to record the location of the silverleaf whitefly nymphs.

Data collection

Silverleaf whitefly individuals were obtained from a colony maintained on tomato plants (Solanum lycopersicum L. var. Trust) in greenhouses at the Ohio Agricultural

Research and Development Center (OARDC), Wooster, OH (USA). Plants were fertilized at each watering, using Miracle Gro® water soluble tomato plant food at a rate of 250 mg/L of nitrogen (18-18-21). Insects and plants were maintained at 25±2°C and a photoperiod of 14:10 (light: dark).

Thirty tomato plants (S. lycopersicum var. Trust) with 6-10 leaves (7-9 weeks old) were grown in 15 x 14.5 cm (diameter x height) pots and PRO-MIX BX ® soilless media and placed individually in 130 x 60 x 60 cm (high x width x deep) mesh cages. As typically recommended for production systems (Cockshull et al. 2001) and because the distribution of SLW nymphs among tomato leaves is similar along main and lateral stems

(Schuster 1998), plants were grown as single stems by removing all lateral buds as they appeared. Each plant was infested with different SLW adult densities that ranged between

6 and 25 3-day-old male-female pairs, to generate densities similar to those found by

Setiawati (2009) in greenhouse tomatoes. The number of adults released was chosen to obtain a range of population densities according to the number of adults found at initial phases of a SLW infestation. The infested cages were kept in a greenhouse room at

25±2°C and a photoperiod of 14:10 (light: dark).

After 96 hours, the SLW adults were removed from the plants, which were then transplanted to 21.6 x 21.6 cm pots. The youngest fully formed, but not fully expanded,

leaf of each plant was marked with a permanent marker to quickly identify the point in the plant below which leaves were infested with SLW. The plants were then kept in cages in the same greenhouse room for 15 days to let eggs develop into 3rd and 4th instar nymphs. These larger nymphs are easier to see and count. Leaves were numbered from the oldest to the youngest (bottom to top), and leaflets from the terminal to the most basal

(Figure 1). All the leaflets of each plant were cut to count the number of SLW nymphs on the underside of each leaflet using a dissecting stereomicroscope. The marked leaf was considered the youngest, and only the marked leaf and all the older leaves below were inspected for nymphs. Because very few nymphs were found in intercalary leaflets, only primary and secondary leaflets were inspected.

Algorithm description

The goal of the algorithm I developed was to generate a set of numbers of SLW individuals on each leaflet within one or more plants of a given size. The required input is the total number of nymphs to distribute and the plant size (total number of fully formed leaves). The algorithm was built in such a way that it stochastically distributes the total number of individuals on tomato plant structures in two hierarchical steps: (1) among leaves within the plant and (2) among leaflets within leaves. The algorithm was developed in R language using R software version 3.1.1. (R Core Team 2013) ( see R code in Appendix A).

The procedure to distribute a given number of SLW nymphs among a given number of leaves of a plant is summarized as follows (Figure 2a):

1. The required input is the number of SLW nymphs, W, and the total number of

fully formed leaves in the plant, nv.

2. Determine the number of infested leaves, nv’. This number is drawn at random

from a binomial PDF with parameters Pv and nv. The parameter Pv represents the

proportion of leaves infested and is estimated as a function of nymph density per

leaf (from model f in figure 2).

3. Generate the set of infested nodal positions, E. Each nodal position is randomly

drawn without replacement from all the possible nodal positions (leaves) [1, …,

nv], using the set of infestation probabilities of the nodal positions (model F in

figure 2) as weights.

4. Generate the set of numbers of nymphs for the infested nodal positions, V. The

numbers of nymphs for each of the leaf nodal positions generated in the previous

step are drawn at random from a multinomial PDF with a set of probabilities H

and W number of trials. The vector H is the set of probabilities of infestation for

the nodal positions generated in the previous step and is calculated using function

F solved for E.

5. The output vector consists of nv number of elements, ordered by nodal position

with the first element corresponding to the oldest leaf, and includes the elements

in V assigned to the infested nodal positions and zeros assigned to the rest. 33

Once the number of nymphs per leaf is generated, numbers of nymphs are then assigned to leaflets within leaves in a procedure that is summarized as follows (Figure

2b):

1. The input required to generate the number of SLW nymphs among leaflets is the

output on the distribution among leaves, including the leaf’s nodal position

(according to figure 1a), j, and the number of nymphs on it, Tl.

2. Determine the number of leaflets in the leaf, nl. Tomato composite leaves

typically have 10-13 leaflets (Sarlikioti et al. 2011), but the oldest 4-5 leaves may

have fewer. I used the mode of the number of leaflets in each nodal position

observed in the 30 plants used for data collection: 3, 5, 5, 9, and 9 leaflets for the

first to fifth oldest leaves, respectively, and 13 for the rest.

3. Determine the number of infested leaflets, nl’. This number is drawn at random

from a binomial PDF with parameters Pl and nl. In this case, the parameter Pl

represents the proportion of leaflets infested and is estimated as a function of

nymph density per leaflet (from model g in figure 2).

4. Generate the set of numbers of nymphs for the infested leaflets, B. The number of

nymphs on each infested leaflet was drawn at random from a multinomial PDF

with a set of probabilities A and Tl number of trials. The vector A is the set of

probabilities derived from a PDF (model G in figure 2) solved for a sequence of

numbers from zero to nl’-1.

5. The output vector consists of nl number of elements including the elements in B

and zeros assigned to the rest, all arranged randomly among leaflets.

Calibration of models

Calibrated equations and associated parameter estimates are listed in table 1.

Distribution of whiteflies among leaves. Wilson and Room’s (1983) ‘model 2’ was used to describe proportion of infested leaves within plants, Pv(I), as a function of the mean number of nymphs per leaf per plant (function f in figure 2). This model assumes that a negative binomial PDF provides an appropriate characterization of the frequency distribution of SLW nymphs per tomato leaf and is given by:

(Eq. 1). where av and bv are the constants from the Taylor’s variance-mean model (Taylor 1961), and 푥̅푣 is the mean number of nymphs per leaf in one plant. The estimation of av, and bv was performed using maximum likelihood (see Table 1 for parameter estimates).

The probability of infestation per leaf nodal position (function F in figure 2) can be described as a multinomial process where each event (leaf infestation) may take several outcomes (i.e. as many as there are available leaves). I used a beta-binomial PDF to assign the probabilities of infestation to each leaf because it required only three parameters, compared with as many parameters as available leaves are in a given plant

for the multinomial, and its shape can reflect the pattern of nymphs among leaves along

the plant main axis.

A common parameterization of the beta-binomial PDF involves three parameters:

the average per-trial probability, the overdispersion parameter, and the number of trials

per sample (Bolker 2008). I denote the average per-trial probability and the

overdispersion parameter as pv and θv, respectively. Parameter pv sets the location of the 36

PDF along the main plant axis, while θv describes how close the distribution of SLW nymphs is to uniform across all leaf nodal positions. The number of trials per sample is denoted here as nv and represents the total number of fully-formed leaves in the plant.

Parameters pv and θv were estimated by maximum likelihood (see Table 1) using 30 sets of SLW nymphs counts per leaf nodal position (from 30 infested plants) and the respective total number of leaves in the plant. A multinomial PDF was used to estimate the likelihood of each set of SLW counts and the respective number of fully-formed leaves per plant by deriving the multinomial probabilities from a beta-binomial PDF with parameters pv and θv. The aim was to find the values for pv and θv that maximize the likelihood of the multinomial PDF, given the sets of counts from each plant, and allowing the size parameter, nv, to vary according to the respective number of fully-formed leaves per plant. The likelihood function is given by:

(Eq. 2). where Xi = [0,…, xi,j,…,nv,i - 1], Si = [si,1,…,si,n_i] and si.=∑si,j. The parameter N is the total number of plants in the sample, Si is the vector with the number of nymphs si,j on the leaf nodal position j, si. is total number of nymphs, nv,i is the number of fully formed leaves, and Xi is a vector with the nodal positions, all of the above on the ith plant. Notice that 1 has to be subtracted from the sequence of nodal positions in X and from nv in the

beta-binomial PDF, so that the numeration of nodal positions fits with the probability estimates, which starts from zero in the beta-binomial PDF.

Distribution of whiteflies among leaflets. Similar to the proportion of infested leaves, the proportion of infested leaflets, Pl(I), was described as a function of the mean number of nymphs per leaflet in each leaf, using ‘model 2’ in Wilson and Room (1983) (function g in figure 2):

(Eq. 3). where 푥̅푙 is the mean number of nymphs per leaflet per leaf and al and bl are the constants of Taylor’s variance-mean model (Taylor 1961). The estimation of al and bl was performed by maximum likelihood (see Table 1 for parameter estimates).

The location of infested leaflets within a leaf was chosen at random, and numbers of nymphs were assigned to them according to an empirical distribution of proportions

(function G in figure 2). The proportion of SLW nymphs among infested leaflets within leaves was described using a beta-binomial PDF with parameters pl and θl (see Table 1).

The procedure was as described for the distribution of nymphs among nodes, but in this case each data vector was the set of numbers of nymphs on the infested leaflets (i.e. excluding zero values) within each of the 120 tomato leaves used for the analysis. A multinomial PDF was used to estimate the individual likelihood of each set of values, but the values in each vector were arranged in a decreasing order and did not represent the location of the infested leaflets. Thus, the likelihood function was given by: 38

(Eq. 4), where Wk = [0,…, wk,c,…,nl,k' – 1], Vk = [vk,1,…,vk,n_(l,k)') ] and vk. = ∑ vk,c. The parameter

M is the total number of infested leaves used in the analysis, nl,k’ is the number of infested leaflets within the kth leaf and Vk is a vector with the number of nymphs per leaflet c arranged decreasingly (i.e. leaflet location not modeled explicitly). The parameter vk. is the total number of nymphs on leaf k (i.e. size parameter of the

multinomial PDF), and W is a vector with a sequence of numbers from zero to nl,k’-1.

Models’ calibration was conducted using the R package “emdbook” (Bolker 2013).

Algorithm verification and validation

Distribution of whiteflies among leaves. The ability of the model to generate both the mean proportion of SLW nymphs and their respective coefficient of variation (CV) for each leaf nodal position was tested by comparing generated nymph counts with empirical datasets. Ideally, this analysis is to be performed using independent datasets (not used for parameter estimation) where both model inputs (total number of individuals to distribute and fully formed leaves per plant) and outputs (number of individuals for each leaf) are explicitly provided. Unfortunately, this information is rarely presented in the literature, despite several studies reporting the distribution of SLW individuals within tomato plants

(e.g. Schuster 1998, Tsueda and Tsuchida 1998, Muniz et al. 2002, Arno et al. 2006).

Instead, SLW density is often estimated by sampling leaflets from leaves along the main axis of randomly chosen plants, without reporting plant sizes (total number of leaves) or estimates of the total number of individuals per plant.

A verification analysis (comparison with the dataset collected for parameter estimation) was conducted to evaluate the performance of the model in producing realistic CV’s about the proportion of nymphs on each leaf nodal position within tomato plants. I decided to perform a verification analysis for the algorithm’s generated CV’s (as opposed to validation) because preliminary algorithm runs showed that small changes in the input data impact significantly the distribution of the variability (e.g. CVs) among leaf nodal positions (data not shown). Thus, by using the same dataset that I collected for model calibration, I guaranteed reliable input data (plant sizes and nymph density per plant) that is not easily found in the literature. Because the variance about the generated means depends on the number of runs, the verification criterion was a comparison of variation about each proportion relative to that obtained for other leaf nodal positions (i.e. by checking for systematic bias according to leaf nodal position). To compare observed data with the model output, I generated 30 sets of leaf counts using the observed plant sizes and their respective number of nymphs as inputs. The output of each simulated plant was a matrix with proportions of SLW nymphs per leaf, with plants in the rows and leaves in the columns (a 30 x 12 matrix). Thus, each data generation gave a vector with the CV for each nodal position. The process was run 1000 times and the results were

summarized in a vector with the generated mean CV for each of the 12 observed nodal positions.

A validation analysis (comparison with an independent dataset) was conducted to evaluate the performance of the algorithm in generating realistic mean proportions of

SLW nymphs for the different leaf nodal positions of tomato plants. I chose the dataset reported by Schuster (1998) because it provides enough information to approximate the size of the sampled tomato plants and the total number of individuals per plant.

Schuster (1998) counted the number of SLW (Biotype B= B. argentifolii) eggs and nymphs on the lower surfaces of the terminal 3 leaflets of each fully formed leaf along the main axis of 35 randomly selected tomato plants from a field tomato crop. The sampling was conducted weekly by inspecting 5 plants for 7 weeks starting 3 weeks after transplanting. The sizes of the sampled tomato plants (number of leaves) were derived from the highest plant size reported by Schuster (15 leaves), the conventional age at which tomato plants are transplanted, 6-7 weeks old (i.e. 5-6 leaves) (Vavrina and

Orzolek 1993); and the leaf appearance rate for tomato plants. At an average rate of 1 fully formed leaf/week (de Koning 1994, Najla et al. 2009), I approximated the plant sizes for each sample week. I estimated the total number of nymphs per plant by multiplying the mean number of nymphs per leaflet at each sampling date by the corresponding estimated number of leaflets.

I simulated Schuster’s experiment (op. cit.) by generating 35 sets of leaf counts, based on Schuster’s plant sizes and their respective total numbers of nymphs. The output

for each simulation was a matrix with proportions of SLW nymphs for each leaf on each plant, with plants in the rows and nodal positions in the columns (a 35 x 15 matrix). The output matrix from each run was summarized in a vector with the mean proportion of

SLW nymphs in each nodal position. The process was run 1000 times and the results were summarized in separate vectors with the generated grand mean proportions of nymphs on each of the 15 leaf nodal positions reported by Schuster (1988) in his original experiment. Because the model assumes that the youngest leaf on the plant is the last fully formed leaf at the moment of oviposition, the elements in the output vector of the generated distribution of nymphs were reduced by two positions (assigning “0” nymphs to the two youngest leaves) to correct for the estimated number of leaves formed while eggs developed into nymphs.

To examine how well nymph distribution reflects egg distribution, I simulated

Schuster’s experiment 1000 more times using the estimated number of eggs per plant as input. The procedures for both estimating the number of eggs per plant and conducting the algorithm runs were similar to those described above for nymphs. All the data extracted from Schuster (1998), was read from Schuster’s figures 1 and 2 using the R package ‘digitize’ (Poisot 2010).

Model outputs were compared with the respective observed datasets visually and by comparing the best-fit line between observed and generated values with perfect concordance, using the concordance correlation methodology (CCM) (Lin 1989, Madden et al. 2007). The CCM is based on the concordance correlation coefficient, ρc, which

ranges from -1 to 1 and equals 1 when there is a perfect agreement between observed and generated values. The coefficient ρc results from the product of the generalized bias parameter, Cb, and the coefficient of correlation, r, of the observed and generated values.

The first reflects the accuracy (or bias) and the second the precision (or the variability) of the model, and range from 0 to 1 and from -1 and 1, respectively.

Distribution of whiteflies among leaflets. Of the total number of infested tomato leaves obtained from the 30 plants, 120 were used for parameter estimation and 76 for validation. To compare the output of the model with the observed distribution of SLW nymphs within the tomato leaves, I generated counts on 76 leaves using the corresponding number of leaflets and nymphs per leaf of the 76 leaf samples as inputs.

The vectors of generated numbers of nymphs per leaflet were summarized as frequency distributions with a regular bin size of 2. The process was run 1000 times and the results were summarized in a vector with mean frequency distribution, which was compared with the corresponding frequencies in the observed values. The frequency distributions were compared using a chi-square test.

Individual parameter value sensitivity analysis

For the distribution among leaves, I assessed the effect of varying parameter values on both the generated mean proportion of SLW individuals per leaf nodal position and their respective CV’s. The output vector obtained from simulating Schuster’s (1998) dataset with the distribution of SLW eggs among tomato leaves was used as reference.

For the distribution among leaflets, the sensitivity analysis was conducted on the mean frequency distribution of SLW nymphs per leaflet. For this analysis, the output vector obtained from generating the 76 infested leaves used for validation was used as reference.

For the distribution among both leaves and leaflets, the output sensitivity to changes in parameter values was first screened by calculating elasticity as the relative change in algorithm output, divided by the relative change in the parameter value

(Caswell et al. 1984):

(Eq. 5). where eϒ is the elasticity coefficient of the parameter ϒ, and Yϒ, Yϒ+ and Yϒ- are the mean output vectors from 1000 runs for the unchanged parameter value, the parameter value raised by 20%, and the parameter value lowered by 20%, respectively. Only the parameters with an eϒ greater than 0.3 (i.e. > 0.3% change in output for every 1% change in the parameter) were examined more closely by comparing the output vectors visually with changes of + and – 20% and 50%, with 1000 runs for each set of parameter values.

Table 1. Algorithm parameters and associated models, estimates and standard errors (SE). See text for details.

Function Parameter Estimated function Estimates f Proportion of av= 24.500 (SE= 푃̂푣(퐼) = 1 − exp (−푥̅푣 infested 11.186) ∗ [푙푛(24.50 ∗ 푥̅(0.566)) leaves, Pv 푣 bv=1.566 (SE= (0.566) −1 0.120) ∗ (24.50 ∗ 푥̅푣 − 1) ]) (Eq. 6) F Infestation 푃푟표푏(푗|푛 ) pv= 0.561 (SE= probabilities 푣 0.002) 푛푣 − 1 퐵(푛푣 − 푗 + 3.279, 푗 + 3.195) per leaf nodal = ( ) θv= 7.475 (SE= position, j 푗 − 1 0.011 0.200) (Eq. 7) g Proportion of al= 35.658 (SE= 푃̂푙(퐼) = 1 − exp (−푥̅푙 infested 20.973) (0.347) leaflets, Pl ∗ [푙푛(35.66 ∗ 푥̅푙 ) bl= 1.347 (SE= (0.347) −1 0.210) ∗ (35.66 ∗ 푥̅푙 − 1) ]) (Eq. 8)

G Proportion of 푃푟표푏(푐|푛푙′) pl= 0.243 (SE= nymphs per 푛푙′ − 1 퐵(푛푙′ − 푐 + 2.102, 푐 − 0.324) 0.003) = ( ) infested 푐 − 1 0.852 θl= 2.778 (SE= leaflet, c (Eq. 9) 0.091) Note: nv is the number of fully formed leaves in a given plant and nl’ is the number of infested leaflets in a given leaf. See figure 2 for function notations.

Results

Algorithm verification and validation

Distribution of whiteflies among leaves. The verification analysis (comparison with the dataset collected for parameter estimation) showed that the algorithm produced reasonable representations of the relative variation about the mean proportion of SWF nymphs across leaf nodal positions within tomato plants (Figure 3a). The estimated concordance correlation coefficient, ρc, between observed and generated values is 0.782,

with a generalized bias parameter Cb= 0.845 and a coefficient of correlation r= 0.925.

However, a further analysis revealed that the quadratic term in the regression is significant (quadratic term= 0.003, SE< 0.001, t= 4.314, P= 0.001), violating the assumption of linearity in the relationship between observed and generated values for the

CCM methodology. The relationship is linear if the outlier derived from the particularly large difference between the observed and predicted CV in the 12th leaf nodal position is dropped (Figure 3a, regression plot). With the linearity assumption met, this result suggests that the deviation between generated and observed CVs is mostly due to a systematic deviation rather than to low precision in the model output. A systematic variation can be produced by either a “location shift”, which is when the slope of the best-fit line between generated and observed values is 1 but the intercept is not 0; a “scale shift”, when the intercept is 0 but the slope is not 1; or both. The parameter Cb is made of two coefficients, u and v, which measure the “location shift” and the “scale shift” by the difference between their estimates and 0 and 1, respectively. The coefficients u and v were estimated as 0.591 and 0.880, which means that the bias is mainly due to predictions that are consistently higher or lower than the observed values (Madden et al. 2007). This result is supported by figure 3a where the best-fit line in the regression plot is consistently above the concordance line, but the bias does not depend on the magnitude of the generated value. Therefore, the relative variation about the mean proportion of nymphs per leaf is generated reasonably by the algorithm, even though its overall magnitude is lower due to factors, such as variations in tomato plant architecture or

whitefly behavioral polymorphisms (e.g. Byrne 1999), that are necessarily left out by the models in the algorithm.

The validation analysis [comparison with Schuster’s dataset (op. cit.)] showed that the algorithm generated realistic mean proportions of SLW nymphs across leaf positions of infested tomato plants (Figure 3b). The concordance correlation coefficient between observed and generated values was ρc= 0.972, and the generalized bias parameter Cb= 0.996, indicating high accuracy/low bias, whereas r= 0.975, indicating some variability in the regression fit (Madden et al. 2007). The algorithm also generated mean proportions of SLW eggs across leaf positions of tomato plants with about the same level of accuracy and precision (Figure 3c). The concordance correlation coefficient was

ρc= 0.968, the generalized bias parameter Cb= 0.999 and the coefficient of correlation r =

0.969.

Distribution of whiteflies among leaflets. The validation analysis (comparison with a

dataset not used for parameter estimation) showed that the algorithm generated realistic

distributions of SLW nymphs among leaflets within tomato leaves (Figure 4). I found no

significant differences between the frequency distributions of observed and generated 48

2 numbers of nymphs on leaflets (X = 9.943, df= 14, P= 0.766). The estimated ρc is 0.999

with a Cb of 0.999 and an r of 0.999. This suggests both low bias and high precision of

the model in predicting frequency of nymph abundance in leaflets within tomato leaves.

Model elasticity and individual parameter value sensitivity analysis

Only parameters pv and θv yielded elasticity values > 0.3 for the generated mean

proportions of SLW eggs among leaves. As described in material and methods, the

parameter pv sets the location of the PDF along the main plant axis, while θv describes

how close the distribution of SLW is to uniform across all leaf nodal positions (Equation

7). A further description of the effect of changing pv and θv is shown in figure 5a.

Although pv yielded the highest elasticity [e(prop)pv= 2.320], it is estimated from data with great precision (SE= 0.002). The change in pv proportionally impacts the leaf nodal position in which most insects are predicted to be located. As pv increases, the leaves located closer to the top of the plant become those with the highest infestation probabilities. In contrast, the change of θv is evident in the shape of the infestation probability distribution across leaf nodal positions, although it’s impact is smaller than that of changing pv [e(prop)θv= 0.334]. When θv is large, the distribution becomes peaked and, when is near zero, it becomes flattened (i.e. close to uniform) [Figure 5a (θv)].

As for the sensitivity of the CV’s generated by the model, only av, bv and pv yielded elasticity values > 0.3. The effect of varying parameters av, bv and pv on the generated CV’s about the mean proportion of SLW eggs is shown in figure 5b. The change in both av and bv affects the model output by varying proportionally the CV’s across all leaf nodal positions, although the elasticity of bv is greater [e(CV)bv= 0.765] than that of av [e(CV)av= 0.455] [Figure 5b (av and bv)]. Parameters av and bv are used as part of the function that describes the proportion of leaves infested within single plants

(Equation 6). Parameter bv represents the gradient b from Taylor’s mean-variance model, which is well known as an aggregation index and has been argued to be specific for the interaction between species’ behavior and environment (Taylor 1984, Taylor et al. 1998,

Park et al. 2013). Parameter av represents the scale parameter of Taylor’s variance-mean

model, which is known to be sensitive to the ecological scale of sampling and as such, an important determinant of optimal sample size for sampling programs in agriculture

(Taylor et al. 1998).

Similar to the sensitivity for the mean proportions of eggs, the parameter pv yielded the highest elasticity [e(CV)pv= 1.252] for distribution of CV’s across leaf nodal positions. The effect of varying pv on the distribution of CV’s is closely related to its effect on the distribution of mean proportions of eggs, by turning the highest CV’s to the leaf positions with the lowest infestation probabilities and vice versa. For instance, as pv increases, leaves located closer to the top of the plant become those with the lowest CV’s

[Figure 5b (pv), □], matching the leaf positions with the highest infestation probabilities

[Figure 5a (pv), □].

Only the parameter bl yielded an elasticity value greater than 0.3 (ebl= 0.597) in generating whitefly distributions among leaflets. As bl increases, the number of empty leaflets within leaves increases proportionally, and the frequency distribution of nymph numbers becomes more skewed (Figure 5c). The parameter bl (Equation 8) has an interpretation similar to that described above for bv.

Discussion

The algorithm I developed generates realistic distributions of both SLW eggs and

3rd-4th instar nymphs among leaves and leaflets within tomato plants. Moreover, the relative between-plant variation about the mean proportions of individuals per leaf nodal position was realistically generated. The only similar algorithm I are aware of is the one developed by van Roermund et al. (1997) to simulate distributions of the greenhouse whitefly Trialeurodes vaporariorum (Westwood) (Hemiptera: Aleyrodidae) within tomato plants. However, this model distributes all T. vaporariorum individuals in equal proportions only among the three youngest leaves, and the resulting distribution and between-plant variation was not validated with independent datasets.

The fact that my algorithm generated within-plant distributions of both SLW eggs and 3rd-4th instar nymphs with about the same level of accuracy and precision indicates that it could be used to generate realistic distributions of SLW eggs or any immature life stages, as long as the assumptions hold. Moreover, because the distribution of adults reflects that of eggs within tomato plants (Arno et al. 2006), algorithm parameters may represent indices of aggregation and leaf selection of SLW adults.

Assumptions

The within-plant distribution of SLW nymphs results from a number of processes that includes leaf selection, aggregation patterns and oviposition of adults, and mortality

(survival) of nymphs. The degree to which these processes vary according to different

ecological scenarios would ultimately determine both accuracy and precision of the algorithm outputs. In this respect, the algorithm has two potentially restrictive assumptions: (1) mortality rate is constant among all tomato plant leaves and leaflets and independent of SLW local density; and (2) leaf selection pattern for oviposition by SLW adults and the resulting nymph aggregation pattern on tomato plants are independent of variation in environmental conditions.

Non-homogeneous mortality may cause systematic shifts in nymph distribution if

SLW individuals die at a higher rate in the most crowded tomato leaves (density- dependent mortality). Naranjo and Ellsworth (2005) analyzed the natural mortality of

SLW populations in cotton crops and found little evidence of direct or delayed density dependence in any mortality factors. Although Naranjo and Ellsworth (2005) focused their analysis at the plot scale, similar conclusions were reported by McGeoch and Price

(2005) for Euura lasiolepis (Hymenoptera: Tenthredinidae), and by Stiling et al. (1991) for Prokelisia marginata (Hemiptera: Delphacidae) using a multi-scale approach, including at the plant scale. In contrast, Freeman and Smith (1990) who studied

Liriomyza commelinae (Diptera: Agromyzidae), or Underwood (2010) with Spodoptera exigua (Lepidoptera: Noctuidae), detected density-dependent mortality factors at the plant scale. Unlike E. lasiolepis, P. marginata and B. tabaci, feeding by L. commelinae and S. exigua results in leaf tissue removal, which should strengthen intra-specific competition at relatively low population densities per plant. Thus, it is possible that density-dependent mortality factors are evident in SLW populations only at extremely

high densities that reduce the availability or quality of nutrients for phloem feeders and that are rarely observed in commercial tomato crops. Moreover, most mortality in SLW populations on cotton and tomato occurs during the 4th instar, after most of the immature life-stages have been completed (Naranjo and Ellsworth 2005, Yang and Chi 2006), which would reduce the chance that density dependent mortality would affect the precision of the model.

Leaf selection is generally constant for insect-plant associations despite variation in environments. Cates (1980) reviews the pattern of leaf selection across lepidoterans and coleopterans, and concludes that herbivores with restricted diets tend to prefer young leaves, whereas generalists often prefer more mature leaf tissue. Sessile insects, such as whitefly nymphs, rely on oviposition behaviors of mobile adults to reach their preferred leaf tissue. Bentz et al. (1995) found that SLW adults prefer plants and leaves with higher nitrogen content for feeding and oviposition, and that their leaf preference within plants was independent of the overall plant nitrogen content. Similar results were reported by

Jauset et al. (2000) for T. vaporiarorum feeding on tomato plants. Mobile herbivores, on the other hand, tend to disperse toward leaves of their preferred age as the crop matures, regardless of environment or plant age. For instance, the vertical distribution of larvae and adults of Thrips tabaci Lindeman (Thysanoptera: Thripidae) within leek plants is described by a single statistical distribution over all plant development stages

(Theunissen and Legutowska 1991). Similar results showing preference for leaves of particular ages were found for Spodoptera frugiperda (J. E. Smith) (Lepidoptera:

Noctuidae) within corn plants (Labatte 1993), Pieris rapae L. (Lepidoptera: Pieridae) within cabbage plants (Hoy and Shelton 1987), Tuta absoluta (Meyrick) (Lepidoptera:

Gelechiidae) within tomato plants (Torres et al. 2001), and spider mites within cotton plants (Wilson et al. 1983). However, some mobile insects have been found to change their leaf selection over time or as a function of microclimate. McCornack et al. (2008) found that Aphis glycines Matsumura (Hemiptera: Aphididae) colonizes young leaves first, but it moves progressively to mature leaves through time. Cabbage looper larvae,

Trichoplusia ni (Lepidoptera: Noctuidae) (Hübner), move either onto the youngest leaves, or down to the oldest leaves as a function of plant canopy microclimate (Hoy et al. 1989).

The presence of insecticides has been observed to affect leaf selection and within- plant distribution of several insect herbivores. This is generally the case when herbivores are able to detect and escape from high concentrations of insecticides, as documented for over 100 cases of behavioral responses of insects to insecticides [reviewed by Hoy et al.

(1998)]. However, Wen et al. (2009) examined the effect of foliar or systemic applications of azadirachtin on oviposition behavior of the SWF, and found no significant differences in the number of eggs laid on treated compared with untreated cucumber leaves. Similar results were found by Isaacs et al. (1999) for SWF adults feeding and ovipositing on cotton leaves treated with foliar or systemic applications of imidacloprid.

The algorithm presented assumes that plants are static structures (no plant growth). When the model is applied to sessile insects such as SLW nymphs, the elements

in the output vector were corrected for leaf formation while eggs developed into nymphs

(e.g. assigning “0” nymphs to the two youngest leaves). However, it is relatively straightforward to build plant development into the model structure. Graph-based architectural crop models such as GREENLAB-tomato (Dong et al. 2007) explicitly simulate the 3D structure of tomato plants, and include a dynamic phenology component.

In this 3D model, the plant axis is made of a succession of growth units produced in successive growth cycles. Growth units appear at a constant rate driven by accumulation of degree-days as described by de Koning (1994). Thus, implementation of plant phenology may also be coupled with SLW oviposition pulses and life stage-specific mortality rates to develop dynamic models of tomato plant structures infested with the

SLW. Such models might be of value to improve crop simulation models, to evaluate insecticide application strategy, or biological control programs.

My algorithm also assumes that leaflet selection pattern within leaves by SLW adults is random. The quality of leaflets, however, may not be the same within a leaf.

Bentz et al. (1995) found that leaf acceptability for oviposition by the SLW in poinsettia plants is driven by plant cues that reflect nitrogen content and that are recognized by adult females. Distribution of nutrients within tomato leaves might not be homogeneous if there is a differential fertilization of above-ground plant parts. Orians et al. (2002) found that tomato plant sectors with direct connections to more fertilized lateral roots had larger leaves and leaflets, and lower concentrations of phenolics (defensive compounds).

I used potted tomato plants that were uniformly irrigated, where the distribution of

nutrients is likely to be homogeneous. In field or greenhouse tomato crops uneven distribution of soil nutrients is more likely to occur and the infestation probability of leaflets within leaves may vary.

Ecological Interpretation of Parameters

Leaf selection pattern. I suggest that parameter pv represents a constant that describes leaf preference for specific insect-plant associations, because it sets the location of the PDF describing the distribution of infestation probabilities among leaf nodal positions

[Equation 7, Figure 5a (pv)]. This study and others (e.g. Schuster 1998, Arno et al. 2006) observed that SLW adults focus their oviposition towards the middle third of tomato plants, on leaves of intermediate age, giving an intermediate value for pv (0.561).

However, SLW oviposition is focused on the youngest leaf tissue of preferred host plants, such as zucchini (Cardoza et al. 2000), where pv may take a higher value (i.e. closer to 1).

Therefore, pv may vary depending on the characteristics of a given host plant, and particularly how plant nutrients and toxins are distributed (Liang et al. 2007, Yan et al.

2011).

A different aspect of leaf preference is explained by θv (Equation 8), the intensity of leaf preference. The concept of intensity of preference is borrowed from economics

(Farquhar and Keller 1989) and involves the comparison of preferences. When θv is large, the distribution becomes more peaked and, when it is small, the distribution becomes close to uniform [Figure 5a (θv)]. Large values of θv denote a strong preference for some

leaf nodal positions, lower values describe less discriminating insect herbivores. For instance, over 70% of the eggs of T. vaporariorum are deposited in the youngest two leaves of tomato plants, which results in a skewed, peaked distribution of infestation probabilities among nodal positions (Arno et al. 2006). In contrast, barely 40% of SLW eggs are laid in their two most preferred leaves in the same plant species, resulting in a relatively flat distribution of infestation probabilities (Schuster 1998, Arno et al. 2006).

Aggregation pattern. The key parameter describing the proportion infested at either plant or leaf scale is the exponent b (bv and bl) from Taylor’s variance-mean model. The parameter bv impacts the overall variability about the mean proportion of SLW individuals among leaf nodal positions [Figure 5b (bv)], while bl determines degree of right-skew in the frequency distribution of nymphs numbers among leaflets (Figure 5c).

The exponent b is considered an aggregation index specific for relationships between pests and crops, and commonly used for the design of sampling programs (Taylor 1984,

Binns and Nyrop 1992). One ecological interpretation of b suggests that the spatial pattern of populations results from the balance between attractive and density-dependent repulsive behaviors among individuals (Taylor and Taylor 1977, Taylor 1981). In theory,

I could have used the same equation to describe the aggregation pattern of SLW nymph at both plant and leaf scales. In fact, I found that the parameter b for SLW nymphs was not significantly different between the spatial scales of plant and leaf, 1.566 (95% CI=

[1.326, 1.801]) and 1.346 (95% CI= [0.837, 1.725]) for bv and bl, respectively. However, variations in the parameter b may affect model output significantly, at both leaf and plant

scale [Figures 5b (bv) and c] and greater precision with less sensitivity in parameter values may be achieved if two separate functions for proportion infested in both plant and leaf are estimated.

Application of the algorithm

My ultimate goal for the model is incorporating detailed description of spatial variation of prey into individual-based models for the estimation of functional responses of insect predators. The simulated numbers of SLW individuals among leaflets, along with architectural attributes of tomato plants (e.g. Najla et al. 2009), can be used to generate ‘plant’ objects in an individual-based modeling context. These ‘plant’ objects would represent the structure where predators forage, providing realistic distances and connections between prey patches, with realistic prey population densities within patches

(leaves and/or leaflets).

I hypothesize that the functional response of predators within plants could be derived more accurately from simulations of the behavior of individual predators across a nested set of relevant scales (plant, leaf, and leaflet) and varying SLW densities. Using the model presented here, SLW densities can be given as number of individuals per fully formed leaf, so that the functional response can be extrapolated to different tomato plant sizes. This approach could be applied to a variety of associations between annual plants and sap-sucking insects and almost any predators that search for prey within plants by walking. However, it may be especially useful for predators that perceive their prey

patches at the plant spatial scale, such as most oligophagous coccinellid predators

(Schellhorn and Andow 2005, Bianchi et al. 2009). Functional responses within plants could then be scaled up to greenhouses or fields using existing analytical models (e.g.

Oaten 1977, Nachman 2006a), or explicit geostatistical simulations of between-plant whitefly and predator distributions (e.g. Pérez M. et al. 2011, Zhao et al. 2011).

Ultimately, such advances in modeling predator-prey interactions could improve biological control of insect pests.

Chapter 3: Intra-plant spatial interaction between Delphastus catalinae (Coleoptera: Coccinellidae) and Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) and its effect on predation rates.

Abstract

One critical limitation of augmentative biological control is that its efficacy is often unpredictable. Predictive models developed to provide recommendations about release rates and application methods often assume a constant probability of encounter between prey and predators within plants. I tested the assumed uniform probability of encounter between the coccinellid predator Delphastus catalinae and nymphs of the silverleaf whitefly (SWF), Bemisia tabaci biotype B (Hemiptera: Aleyrodidae), on tomato plants. I hypothesized that D. catalinae non-random search patterns would result in a spatially non-uniform intra-plant probability of predator-prey interaction that ultimately ends up affecting predation rates. Tomato plants infested with a range of SWF densities were placed in cages and D. catalinae individuals marked with different colors were released on them. The number of SWF nymphs before and after predator release and the predator visit frequency on each leaf and leaflet were recorded. I found that the distribution of predator visits and prey mortality is concentrated towards the mid and mid-low region of the plant main stem and follow a different within-plant distribution

than that of the SWF. The resulting non-uniform probability of prey encounter presumably affected the magnitude of predation rates and the shape of the functional response at the intra-plant scale. Results show that the assumption of uniform probability of encounter among leaves within a plant could be misleading at least for predictive models of the tomato-SWF-D.catalinae system, and that spatially explicit or individual- based modeling approaches are needed for accurate predictions.

Introduction

Despite the high consumption rates often measured for predators and parasitoids in laboratory settings, the success rate of augmentative biological control remains remarkably low (Collier and Van Steenwyk 2004). One contributing factor seems to be the incorrect selection of release rates and methods, which leads to unsatisfactory pest suppression and increases unpredictability of augmentation approaches (Orr and Fox

2012). Ecological models have been developed to assist the implementation of biological control programs either by increasing understanding of the system or by providing useful predictions for management, including agent selection and/or potential outcomes.

However, biological control models applied to specific systems rest upon assumptions that often restrict their application and sometimes compromise their performance (Barlow

1999).

One common assumption of biological control models is that the probability of encounter between pest and natural enemy is uniform throughout the occupied area (i.e.

populations are well-mixed within or among plants). Most models are written as a set of difference or differential equations that could represent insect life stages or entire populations, necessarily assuming random mixing among individuals within a given location. Spatially explicit simulation models divide space into discrete locations and apply these equations separately for each simulated location. Alternatively, predator-prey models may account for some degree of spatial heterogeneity in their functional responses. Variation in number of prey in patches that are visited by an average predator is defined in terms of a probability distribution function, but is irrespective of their location within the plant canopy (e.g. Nachman 2006a, Hemerik and Yano 2011).

However, non-uniform local predator-prey interactions that can affect population dynamics, ecosystem function, and system stability can only be accounted for by spatially explicit or individual-based modeling approaches (Judson 1994, DeAngelis and Mooij

2005, Fryxell et al. 2007).

Habitat structural complexity is considered a fundamental property of ecosystems that influences population dynamics and predator-prey interactions (McCoy and Bell

1991, Denno et al. 2005). Increased habitat structural complexity tends to result in spatially non-uniform predator-prey encounter probability distributions. First, structural complexity often leads to increased patchiness in the distribution of both prey and predators, which might result in profound differences in predation rates within a given patch and over all patches (Nachman 2006a, Fryxell et al. 2007). Second, habitat structural complexity can affect efficiency of predator foraging in various ways.

Efficiency of active predators often decreases as habitat structural complexity increases, because more refuges are available for prey, predator movement may be reduced, and prey detection may become more difficult (Legrand and Barbosa 2003, Denno et al.

2005, Klecka and Boukal 2014).

The differences in physical and chemical properties among leaves within single plants make the intra-plant scale particularly complex in terms of habitat structure (Raupp and Denno 1983, Langellotto and Denno 2004). There are at least four spatial scales that are relevant from an insect perspective: landscape scale (among different habitats), habitat scale (among plants, within a specific habitat), plant scale (among different plant parts, within single plants), and patch unit (within specific plant parts). The relative structural complexity is quantified by the number of different structural elements per unit volume (McCoy and Bell 1991). How the structural complexity is measured in terms of the diversity of structural elements depends on the spatial scale at which the ecological interactions are being examined (Langellotto and Denno 2004, Tokeshi and Arakaki

2012). For instance, within monocultures, the structural complexity at the plant scale is often considered the highest because without diversity in plant species the primary source of variation is the difference in structural elements (plant parts) within plant canopies.

The silverleaf whitefly (SWF), Bemisia tabaci biotype B (= Bemisia argentifolii)

(Gennadius) (Hemiptera: Aleyrodidae), is a significant worldwide pest of a variety of production systems including field and greenhouse tomato (Stansly and Naranjo 2010).

Damage in plants is caused by reduction in plant vigor and production of honeydew,

which serves as a substrate for sooty molds. The SWF also vectors many plant pathogenic viruses (Oliveira et al. 2001) and may induce growth and ripening disorders in vegetables (Schuster 2001). Although adults are good flyers, all immature life stages

(nymphs) are sessile except neonates (also known as crawlers), which move from a few millimeters to a few centimeters before settling during the first few minutes after hatching (Byrne and Bellows 1991, Simmons 2002). Most eggs are laid on the underside of leaves where, upon hatching, they undergo four sessile nymphal instars. These two major life stages together represent about 50% of the entire life span (Tsai and Wang

1996). Not surprisingly, most predators and parasitoids used for SWF biological control specialize on egg and/or nymph consumption (Gerling et al. 2001).

Predacious Coccinellidae are widely used as biological control agents, partly because they often show high consumption rates in laboratory settings. However, their rate of success in controlling pests in the field is highly variable (Obrycki and Kring

1998). Most Coccinellids are active predators of immature hemipterans, including whiteflies, aphids, mealybugs and scales, all of which exhibit low or no mobility (Hodek and Honek 2009). The interaction between coccinellid predators and their herbivorous prey can be complicated, however, by differences in intra-plant spatial distributions. On the one hand, insect herbivores are often distributed in a nonrandom manner within the plant canopy, largely because their leaf preferences patterns make certain plant regions consistently more likely to be infested (Cates 1980, Raupp and Denno 1983, McCornack et al. 2008). On the other hand, coccinellid predators tend to restrict their intra-plant

distribution to certain plant nodes to reduce interspecific competition, often leading to poor correlations between prey and predator distributions (Frazer and McGregor 1994,

Musser and Shelton 2003). Not surprisingly, prey refuges within plants have been reported frequently for predator-prey systems involving coccinellid predators (e.g. Clark and Messina 1998, Costamagna and Landis 2011, Hodek et al. 2012, Reynolds and

Cuddington 2012).

The oligophagous predator Delphastus catalinae (= pusillus) (Horn) is one of the more than 25 species of coccinellid predators that have been associated with the SWF

(Gerling et al. 2001, Hodek et al. 2012). Delphastus catalinae is currently the only

Coccinellid predator that is commercialized in the USA for SWF control and is considered one of the best commercially available predators of Bemisia spp. (Obrycki and Kring 1998). Despite the relatively high predation rates that have been reported for

D. catalinae in laboratory settings (Hoelmer et al. 1993, Liu and Stansly 1999), releases in the field have showed variable results (Heinz and Parrella 1994b, Heinz et al. 1999) and have been effective only when SWF densities are high (Heinz and Parrella 1994a), a significant drawback considering the ability of SWF to vector plant pathogenic viruses.

Understanding the interaction between the SWF and D. catalinae within plants will ultimately help fine-tune D. catalinae release rates and application methods and, if necessary, assess the development of complementary biological control strategies.

In previous research I showed that the distribution of SWF immatures within tomato plants can be predicted using per-node infestation probabilities and aggregation

patterns (Rincon et al. 2015). In this study, I hypothesized that D. catalinae would exhibit non-random search within the tomato plant canopy, which results in a non-uniform intra- plant predator-prey spatial interaction. Thus, the objectives of this study were: 1) to test the null hypothesis of uniform encounter rates between D. catalinae and SWF nymphs regardless of location within the canopy, and 2) to estimate the effect of a non-uniform interaction among prey and predator individuals, if present, on predation rates at the plant scale. To determine the nature of the spatial interaction between SWF and D. catalinae individuals, I examined the intra-plant search and predation pattern of adults of D. catalinae and compared them with the distribution of SWF nymphs within tomato plants.

To examine the effect of non-uniform predator-prey interactions on predation rates, I compared the functional response of D. catalinae preying on SWF nymphs at two intra- plant spatial scales with contrasting levels of habitat structural complexity, namely within leaves and among leaves. I predicted that a type II (saturating) functional response would be observed at the intra-leaflet scale because this functional response is typically the result of uniform predator-prey encounter rates (Holling 1961, Nachman 2006a, Morozov

2010). However, I predicted that a non-uniform, heterogeneous spatial interaction at the intra-plant scale resulting from prey patchiness and localized higher predation rates would result in a type III (sigmoid) functional response (Nachman 2006a, Morozov

2010).

Materials and Methods

The study was carried out at The Biological Sciences Greenhouse Facility of The

Ohio State University, Columbus, OH (USA). Silverleaf whitefly individuals were obtained from a colony maintained on potted tomato plants (Solanum lycopersicum L. var. Trust) and fertilized at each watering event using Miracle Gro® water soluble tomato plant food at a rate of 250 mg/L of nitrogen (18-18-21). Insects and plants were maintained in greenhouse conditions at 25±2°C and a photoperiod of 14:10 (light: dark).

Adult D. catalinae individuals were purchased commercially (Rincon-Vitova Insectaries

Inc., Ventura, CA, USA).

Forty seven tomato plants (S. lycopersicum var. Trust) with 6-12 leaves (7-10 weeks old) were grown in 15 x 14.5 cm (diameter x height) pots and PRO-MIX BX ® soilless media (Premier Tech Ltd., Rivière-du-Loup, Québec, Canada) and placed individually in 130 x 60 x 60 cm (high x width x deep) mesh cages. As typically recommended for greenhouse tomato production systems (Cockshull et al. 2001), plants were grown as single stems by removing all lateral buds as they appeared. Thirty five of the plants were infested with different densities of 3-day-old adult SWF male-female pairs, as follows: 6 plants with 14 pairs, 4 with 20, 9 with 30, 11 with 40 and 5 with 50; and the remaining 12 plants were left uninfested. The infested and uninfested cages with plants were kept in a greenhouse room at 25 ± 2°C and a photoperiod of 14:10 (light: dark).

After 48 h, the SWF adults were removed from the infested plants and the youngest fully formed, but not fully expanded, leaf of each plant was marked with a permanent marker to quickly identify the point in the plant below which leaves were infested with SWF. The plants were then kept in cages in the same greenhouse room for

12 days to let eggs develop into 2nd and 3rd instars. Leaves were numbered from the oldest to the youngest (bottom to top), and leaflets from the terminal to the most basal

(Chapter 2, Rincon et al. 2015). The number of nymphs in each leaf and leaflet was counted using a 10X hand lens. The main stem of infested plants was cut right above the marked leaf to remove all the leaves that appeared after the initial infestation. The next day, ten D. catalinae adult individuals were released in each plant cage onto the cage walls allowing the predators to arrive on the plant by their own means, usually by flight.

Each predator was marked with a dot of ink of a different color extracted from TexPen ® markers (ITW Dykem, Olathe, KS, USA) previously diluted at 5:1 (paint : thinner) with turpentine substitute, following the procedure described in Bates and Sadler (2004). A small ink dot was applied to one of the elytra of each beetle using a No. 0 paint brush, without painting the elytral suture to make sure the beetle’s ability to expand its wings and fly normally was not impaired. During the marking process beetles were not slowed down with CO2 or cold. The ink used has been shown to produce durable marks

(Wineriter and Walker 1984) and to have no effect on longevity of adult beetles, at least when applied to elytra (Bates and Sadler 2004). For the next 4 days, plants were observed each 2h during photophase (9am to 6pm) to record the number and identification of

individual predators that visited each leaf and leaflet. At the 4th day after predators were released, all the leaves and leaflets of each plant were cut to re-count the number of SWF nymphs on the underside of each leaflet using a dissecting stereomicroscope. I decided on this observation time period because it was sufficient for most of the predators to have either died or reached a plant and found prey, considering that less than 10% of D. catalinae adults survive without food after 4 days (Simmons et al. 2012). Also, a longer observation time would have complicated the distinction between SWF nymphs consumed by predators and those that completed their nymphal stage. The number of prey consumed by D. catalinae individuals was estimated as the difference between the number of SWF nymphs before and after the 4-day D. catalinae-SWF interaction period.

Data analysis

The increase in the proportions of leaves within single plants infested with SWF nymphs and visited by D. catalinae as a function of SWF nymph density were compared by fitting modified monomolecular models (Madden et al. 2007), where the intercept is assumed to be zero and the maximum level is a parameter to be estimated. Thus, the proportion of leaves either infested by SWF nymphs or visited by D. catalinae is given by:

(Eq. 10)

where rm is the rate parameter (1/SWF/plant), ymax represents the maximum proportion of leaves either infested by SWF nymphs or visited by D. catalinae within tomato plants, and w is SWF population density per plant. I chose this model because it requires few parameters and allows comparison of maximum proportion of leaves infested by SWF nymphs and visited by D. catalinae individuals as a function of SWF nymph density.

Equation 10 was also used to determine the maximum proportion of released beetles that arrived on the plants as a function of SWF nymph density.

The recorded leaf nodal positions within each plant were standardized by dividing them by the total number of nodes on the plant. The distribution of SWF nymphs along the plant axis was compared with the distribution of proportions of both SWF nymphs consumed and D. catalinae individuals that visited each plant. The comparison between distributions was performed using the modified t test for the association between two spatial processes (Clifford et al. 1989), with the standardized nodal positions as spatial coordinates. The intra-plant distributions of SWF nymphs before predators were released, number of predators that visited the plants, and proportion of SWF nymphs consumed were modeled across the standardized nodal positions using polynomial logistic models.

Parameter estimation was performed with iteratively reweighted least squares using a logit link function (Venables et al. 2002). To determine if the number of SWF nymphs per leaf was a better predictor of the proportion of D. catalinae individuals per leaf than leaf nodal position, I compared the fit of logistic models using the Akaike Information

Criterion (AIC), defined as:

(Eq. 11) where L is the log-likelihood and k is the number of parameters of the model. Small AIC values represent better fit to data, such that adding a parameter with a negligible improvement in fit, penalizes the AIC by 2 log-likelihood units (Bolker 2008).

To determine the potential effect of a non-uniform spatial interaction between D. catalinae and the SWF on predation rates, the functional response of D. catalinae preying on SWF nymphs was estimated at two spatial scales with contrasting degrees of structural complexity: within plants and within leaflets. To distinguish between type II and type III responses, the fit of both models to data was compared using the Likelihood

Ratio Test (LRT) (Okuyama 2013). The “random predation equation” (Rogers 1972) was used to describe the functional responses, because they allow for prey depletion during the course of the experiment. Rogers’s type II response describing the number of prey eaten, Ne, by a given number of predators, D, is given by:

(Eq. 12) where T, the total time of the interaction between predators and prey, was set as one; N is prey density; a is a constant known as “attack rate” expressed in terms of area searched during time T; and Th is the handling time, or the time spent handling one prey by single predators, expressed in terms of fraction of T. The main difference between type II and

type III responses is that a is constant in the first but is function of N in the second, which yields:

(Eq. 13) where b and c are the slope and the intercept, respectively, of a linear model describing attack rate as a function of SWF density. Among the several mathematical forms that have been used to describe the relationship between attack rate and N (Juliano 2001), I chose a linear function because it allowed me to compare type II and type III models as the first nested into the second, i.e., Eq. 13 becomes Eq. 12 if b= 0.

To express consumption on a per predator basis, I divided the number of SWF nymphs consumed at plant and leaflet scales by the number of individuals of D. catalinae that visited the respective plant or leaflet. Parameters of monomolecular and functional response models were estimated by Maximum Likelihood. The Lambert W function was used to compute a closed-form solution of Rogers’s equations for predation rates (Bolker

2008). All analyses were conducted using R software version 3.1.1 (R Core Team 2013).

The R package “SpatialPack” was used for the spatial association analyses (Osorio and

Vallejos 2014) and “emdbook” for parameter estimation of functional response equations

(Bolker 2013).

Results

The proportion of leaves infested with SWF nymphs increased at a rate of 0.008 ±

0.002 (95% CI, n = 35) per SWF nymph, whereas the proportion of leaves visited by D. cataliane increased by 0.013 ± 0.005 (95% CI, n = 47) per SWF nymph. Although the two rates are not significantly different (Z = 1.369, P = 0.170), the maximum proportion of leaves infested with SWF nymphs was significantly higher than the maximum proportion of leaves visited by D. catalinae (Z = 5.831, P < 0.001) (Figure 6a). The maximum proportion of leaves infested by SWF nymphs was 0.900 ± 0.062 (95% CI, n =

35), whereas the maximum proportion of leaves visited by D. catalinae individuals was

0.601 ± 0.072 (95% CI, n = 47). The proportion of predators that arrived on the plant increased by 0.008 ± 0.002 (95% CI, n = 35) per SWF nymph and reached a maximum proportion of 0.851 ± 0.064 (95% CI, n = 35) (Figure 6b). Thus, the proportion of predators observed on plants increased as a function of SWF density, at a rate similar to the increase in the proportion of infested leaves. However, predators only visited a maximum of 60% of the leaves of a given plant, even though more than 80% of them reached the most infested plants, indicating that predators kept visiting a relatively small portion of the plant.

I found that the intra-plant spatial distribution of SWF nymphs was significantly different from the distribution of mortality rates due to predation (modified t test: F =

5.094, n = 322, P = 0.025). I also found that the intra-plant distribution of SWF nymphs was significantly different from that of D. catalinae individuals (modified t test: F =

38.150, n = 322, P < 0.001). Interestingly, I found that the intra-plant distribution of D.

catalinae individuals and SWF mortality rates due to predation was also significantly

different (modified t test: F = 38.366, n = 322, P < 0.001).

A closer look at the differences between SWF and D. catalinae distributions

revealed that predators consistently concentrated their search in lower nodal positions 76

regardless of the location of SWF nymphs. I found that the leaf nodal position is a better predictor than the number of SWF nymphs per leaf for the proportion of D. catalinae individuals per leaf, as revealed by the difference in fit between the polynomial logistic and the logistic model (Table 2). Silverleaf whitefly nymphs were found more frequently in the middle to upper section of the tomato plant canopy (Figure 7a), whereas most SWF mortality by D. catalinae predation and most of the D. catalinae individuals that visited the plants were observed towards the mid-low and middle section of the plant canopy, respectively (Figures 7b and 7c), with corresponding differences in the coefficients of the polynomial logistic models that used the standardized nodal positions as predictor (Table

3). The proportion of nymphs eaten and the number of nymphs and predators are all low in the lowest part of the tomato plant canopy, as can be interpreted from the low intercept estimates. The linear coefficient describes the rate of increase in each variable from the lowest to the highest part of the plant, being lowest for the distribution of SWF nymphs and highest for the distribution of predators. The quadratic coefficient, which was negative for all three distributions, describes how fast the curve descends or how

“bumpy” the distribution peak is. The closer the estimate is to zero, the flatter the distribution peak. This coefficient is highest for the SWF distribution and lowest for the predator distribution. My interpretation is that, although SWF nymphs are clustered towards the mid-high part of the plant canopy, they are more evenly distributed among nodal positions than the predators. The cubic coefficient, in this case, describes the rate at which the variable increases after the mid peak. The cubic coefficient is not significantly

different from zero for the SWF distribution (Z = 1.710, P = 0.087), but it is significantly greater than zero for the distribution of both predators and proportions of SWF eaten

(Table 3). This describes the difference in location of the distribution peak of SWF nymphs, of predators, and of proportions of SWF nymphs eaten.

I found that a type II functional response provided a better fit than the type III model for the number of SWF consumed per predator within leaflets (LRT: X2 = 4.618, P

= 0.032) (Table 4). The range of initial numbers of SWF nymphs per leaflet was not wide enough to accurately estimate the asymptote of the model (Figure 8a), which resulted in an underestimated ‘handling time’ parameter. In contrast, the type III model provided a better description than the type II model of the number of SWF nymphs consumed per predator within plants (LRT: X2 = 18.236, P < 0.001) (Table 4). In this case, the range of initial numbers of SWF nymphs allowed the detection of a realistic plateau (and associated ‘handling time’ parameter) (Figure 8b).

Figure 7. Intra-plant distribution of silverleaf whitefly nymphs (a), the predator Delphastus catalinae (b) and silverleaf whitefly mortality (c).

Table 2. Comparison between logistic models to describe the number of Delphastus catalinae individuals found per tomato leaflet.

Model ∆AIC k Nodal positions (polynomial logistic) 0 4 Silverleaf whitefly nymph density (logistic) 59.7 2 ∆AIC = Difference in Akaike Information Criterion, k = number of parameters.

Coefficient SWF† Predator distribution SWF Mortality distribution Intercept -5.352 (0.113)* -12.217 (1.322) -4.596 (0.345)* Linear 11.422 (0.671) 50.929 (6.790) 22.345 (1.954) Quadratic -9.405 (1.212) -71.397 (10.979) -40.456 (3.477) Cubic 1.148 (0.671) 31.266 (5.600)* 20.622 (1.929)* †SWF = Silverleaf whitefly nymph.

Table 4. Parameter estimates of functional response models of the predator Delphastus catalinae at two spatial scales. Parameters were estimated by Maximum likelihood.

Spatial Parameter Estimate SE Z value P scale Intra-leaflet a 0.2618 0.0127 20.546 < 0.0001 (type II) h 0.0052 0.0061 0.8632 0.388 Intra-plant b 0.0001 < 0.0001 4.038 < 0.0001 (type III) c 0.0173 0.005 3.449 0.0005 h 0.0596 0.004 15.074 < 0.0001

Figure 8. Functional response of Delphastus catalinae preying on silverleaf whitefly nymphs at the intra-leaf (a) and the intra-plant (b) spatial scales.

Discussion

This is one of the first studies to examine predator-prey spatial interactions within the complex architectural structure of a plant canopy. Many predator-prey models assume a uniform spatial interaction among prey and predator individuals at this spatial scale.

Here, I present evidence that contradicts this assumption. First, I found that the proportion of leaves visited by D. catalinae individuals increased as fast as the proportion of leaves infested with SWF nymphs, as a function of SWF density per plant. However,

D. catalinae adults only visited a maximum of 60% of the plant canopy on average, which is considerably lower than the maximum average proportion of leaves infested by

SWF nymphs (over 90%). This relatively low proportion of the plant searched by D. catalinae is surprising, considering that predators visited plants more frequently as SWF density increased, reaching a maximum of 80% of the predators released and suggesting an aggregative response of D. catalinae to SWF nymphs, at least at the plant scale.

Second, I found that the spatial distributions of predators and SWF nymphs among leaf nodes were significantly different, and that the number of SWF nymphs is a poor predictor, when compared with the nodal position, of predator distribution within plants.

Moreover, the misalignment between predator and prey distributions was reflected in the unequal distribution of SWF mortality among leaves. Third, I found that a non-uniform predator-prey encounter rate and a patchy prey distribution might cause a significant reduction in predation rate per plant, and a shift in the type of functional response, compared with that expected for uniform, well-mixed populations.

The non-uniform intra-plant search pattern exhibited by D. catalinae adults implies that SWF patches in the top and bottom parts of plant canopies are consistently ignored by D. catalinae individuals, regardless of the availability of prey. However, this pattern does not fit in the ecological definition of refuge, since the low predation risk in the top and bottom parts of tomato plants canopies is not the result of the existence of locations that are inaccessible to predators. Furthermore, because immature stages of

SWF are all sessile, behavioral avoidance of predators or active seeking of intra-plant

locations to reduce predation risk would only be possible for SWF adults. In this experiment, SWF adults were allowed to mate and oviposit freely for 48 h without the presence of predators. The distribution of SWF adults is mostly driven by feeding and reproductive patterns, at least in absence of predators or parasitoids, and determines that of their immature offspring (Chapter 2, Bentz et al. 1995, Rincon et al. 2015). I observed that the non-uniform predation pattern resulted from the search behavior of D. catalinae rather than from responses of SWF individuals to the presence of predators.

Although the examination of adaptive responses of SWF individuals to predation was not part of this study, SWF choice of intra-plant locations may have been selected over time to reduce predation risk from D. catalinae and other predators. The intra-plant distribution I found in this research is comparable to previous descriptions of SWF distribution within tomato plants in absence of natural enemies (Chapter 2, Schuster

1998, Muniz et al. 2002, Arno et al. 2006, Rincon et al. 2015). However, SWF adults tend to move upward within the canopy of cucumber plants when D. catalinae adults are present, apparently to reduce the predation risk of their progeny, since predators seem to remain in lower plant strata (Lee et al. 2014). Releases of D. catalinae are commonly performed as part of augmentative biological control programs, mostly in controlled environments when SWF infestations are high, or at least when the SWF is already present. Thus, it is unlikely that D. catalinae individuals released under augmentative regimes are exposed to intra-plant SWF distributions affected by predators. However, migration of SWF individuals into upper plant strata may result in significant population-

level effects if SWF females are forced to lay their eggs in leaves that reduce survival, fecundity rates, or both. Zhang and Wan (2012) found that mortality of SWF nymphs is significantly greater and that females produce fewer eggs when they feed on tomato leaves that are less than two weeks old. Unfortunately, the extent to which reductions in survival and fecundity of this kind are translated into SWF population-level effects is poorly understood, but it has been shown to be significant in other predator-prey systems involving Coccinellids (Costamagna et al. 2013). Understanding the relative contribution of consumptive and non-consumptive predation effects on SWF populations may lead to the development of better predator release protocols and complementary control strategies. Moreover, predator-prey models intended to simulate D.catalinae-SWF systems should incorporate spatial non-uniformity in consumption rates, survival, and fecundity at the intra-plant scale.

I found that the average predation rate decreased as the scale of observation increased from intra-leaflet to intra-plant. In the context of scale transition, predation rates have been reported to be affected by the area (or volume) of observation and by the increase of structural complexity. Kaiser (1983) found that predation rates of the predatory mite Phytoseiulus persimilis Athias-Henriot (Acari: Phytoseiidae) on the spider mite Tetranychus urticae Koch (Acari: Tetranychidae) increased with the area of the experimental arena. Similar results were found in several aquatic predator-prey systems, using similar experimental conditions (Bergstrom and Englund 2002, 2004). In the above mentioned instances, predators and prey preferred the same parts of the arena (i.e. high

co-aggregation), and thus the area of the preferred proportion of sub-habitat decreased as the total arena size increased. My results show that aggregative responses of predators at the intra-plant scale can be impaired by predator preferences for certain canopy regions, which can result in a mismatch between the patches visited by the predators and those with the highest number of prey (i.e. less co-aggregation). Nachman (2006a) concludes that prey patchiness reduces the feeding rate of a predator, unless there is a strong predator aggregative response that leads to high co-aggregation between prey and predator individuals. Predation rate can also decrease due to dilution effects if sufficiently large areas are examined, even in highly co-aggregated predator-prey systems, as shown by Bergstrom and Englund (2002) using simulations.

Structural complexity often interacts with arena size to either positively or negatively affect predation rates. However, the most commonly reported pattern is a reduction of predation rates with increased structural complexity because of the increase in number of prey refuges (Clark and Messina 1998, Costamagna and Landis 2011,

Reynolds and Cuddington 2012). Although I observed a non-uniform predation pattern of

D. catalinae on SWF nymphs within tomato plants, I do not have evidence of plant structures that allow prey to hide or become inaccessible to predators. An alternative mechanism is provided by Kaiser (1983), who showed that predation rate might decrease with increased structural complexity when predators and prey show preference for similar physical structures. Predator-prey encounters are more likely in simple arenas where preferred structures tend to be densely populated, than in more complex ones that are

richer in those structures. Moreover, structural complexity may directly affect a predator’s ability to capture and subdue prey. This effect can be particularly significant for active predators, such as D. catalinae, because structural complexity may obstruct predator movement, which favors prey escape, increase predator searching time, or both

(Legrand and Barbosa 2003, Hauzy et al. 2010, Klecka and Boukal 2014). I showed that predation risk for SWF individuals is not uniform across the tomato plant canopy, so the reduction of SWF mortality in the top and bottom of the tomato plant canopy may explain, at least partially, the disagreement between the rates found at the intra-leaflet scale, where predator and prey mixing is relatively uniform, and the intra-plant scale.

In this research I assume that predation risk of SWF nymphs is uniform at the intra-leaflet scale. However, Keiser et al. (2013) showed that aphids located closer to the petiole, often the first place visited by Coccinellid predators once in the leaf, were at higher predation risk than their clone mates located along the leaf edges or veins.

Apparently, aphids feeding on high-risk leaf regions help inform others about predator position and reduce the overall impact of predation. This pattern was not observed by

Keiser et al. (2013) with flying parasitoids, which seemed to attack aphids regardless of their location within leaves. The type II functional response of D. catalinae preying on

SWF nymphs that I found at the intra-leaflet scale supports the assumption of uniform encounter probability between predators and prey at that spatial scale. Moreover, the petiole is not consistently the first leaflet region visited by D. catalinae individuals, because they often arrive on leaflets by flying or hopping (D. F. Rincon, per. obs.),

similar to parasitoids and different from aphidophagous Coccinellids that search plants by walking (Hodek et al. 2012).

My data supports my prediction that a non-uniform, heterogeneous spatial interaction between D. catalinae and the SWF at the intra-plant scale will result in a type

III functional response. I found that a type II response describes predation in homogenous, small-scale (local) environments within leaflets, but a type III emerges in more complex scenarios within plants. In particular, I found that the type of functional response of a predator is scale-dependent, as a result of structural complexity and the interaction between patchy prey distributions and localized higher predation rates caused by non-random predator search patterns. However, from my data, I can only speculate on a causal relationship between such interaction and a type III functional response.

Traditional explanations, such as learning, refuges for prey, or prey switching, are unlikely under the conditions of my experiment.

Morozov (2010) observed that zooplankton feeding on algae exhibited a type III functional response in water columns when local functional responses were non-sigmoid

(i.e., type I or II), due to the interplay between heterogeneous algae distribution and the active zooplankton foraging behavior. The exponential growth in predation rate, typical of type III responses, resulted from a pronounced vertical algae gradient in the water column with algae density (due to self-shading), followed by a strong aggregative response from zooplankton to such increasing food gradients. This does not seem to be the case for the SWF-D. catalinae system. First, I cannot think of a mechanism that may

increase intra-plant SWF nymph aggregation with SWF density, as algae do in water columns. In fact, Rincon et al. (2015) describes and validates an algorithm to generate

SWF distributions over a wide range of densities within tomato plants using a constant aggregation parameter. Moreover, my results showed a relatively weak aggregative response of D. cataliane to SWF nymph patchiness, because intra-plant predator distribution was better described by leaf nodal position rather than by SWF density within leaflets.

Scale-dependent shifts in functional response shapes have also been explained by the presence of prey refuges at certain spatial scales. Messina and Hanks (1998) found that the host plant may alter the shape of the functional response of an aphidophagous coccinellid predator. They found that a type II response from small-scale lab experiments turns into a type III response when predators are released in single plants that apparently provide refuges for a given number of aphids. While I found that SWF nymphs located at the top or bottom of tomato plants have lower predation risk than individuals located in the middle third of the plant canopy, nymphs in these parts of the canopy are still susceptible to being attacked. Moreover, refuges will only produce type III functional responses if they provide protection to a constant number of prey, rather than a proportion of the prey population (Berryman et al. 2006). In the case of a non-uniform predation pattern, as I describe here for the SWF-D. catalinae system, it is more likely that the reduction in predation rate in certain plant regions is proportional, rather than numeric, at least for realistic SWF densities.

I have demonstrated that a non-uniform predation pattern, in combination with a patchy prey distribution, can produce a sigmoid (type III) functional response. I propose the following generic mechanism of emergence of a type III response. When the SWF density is low, the patchy distribution along with the localized search pattern of D. catalinae causes a very low predation rate. As SWF density increases, the probability of encounter increases, which makes D. catalinae individuals stay longer in plants and visit more leaflets on them. In fact, I found that search rate is a linear function of SWF density, as can be inferred analytically from Eq. 13 and empirically from the positive linear relationship between the SWF density and mean number of leaflets visited per plant by

2 D. catalinae individuals (F1, 45 = 80.88, R = 0.642, P < 0.0001). The increase in search effort motivated by prey encounter resembles the ‘area-concentrated search’, defined as an increase in search effort after detecting a prey item because of the high probability of encountering more prey nearby (Benhamou 1992). Indeed, this adaptation is the mechanism predicted by optimal foraging theory when prey is aggregated, but it is maladaptive if resources are sparsely (randomly) distributed (Bell 1991). ‘Area- concentrated search’ has been described for various species of coccinellid predators (e.g.

Carter and Dixon 1982, Nakamuta 1985, Ferran et al. 1994, Munyaneza and Obrycki

1998), including D. catalinae (Guershon and Gerling 2006). However, this behavior is often described in terms of an increase in turning angle (direct klinokinesis) along with a reduction in walking speed (inverse orthokinesis) of predators in two-dimensional arenas, so that more time and energy are invested in prey patches within which prey are

randomly distributed. However, prey distribution within leaves is often considered to be random (Nachman 2006a, Hemerik and Yano 2011, Rincon et al. 2015). Thus, an increase in search effort motivated by prey encounter at the intra-leaflet scale would not provide, in theory, any benefit to the predator. As such, even if D. catalinae search effort increases with SWF density at the intra-leaflet scale (as described by Guershon and

Gerling 2006), such increase alone would not result in a significant increase in encounter rate, and therefore a type II (saturating, with constant search rate) emerges. In contrast, an increase in the number of leaflets visited motivated by the encounter of infested leaflets would result in a higher chance of finding new infested leaflets in the same plant. In this case, the increase in search effort represents a benefit for the predator, which is reflected by an exponential increase in predation rates as a function of prey density. Thus, a functional increase in search effort may only result in an exponential increase in predation rate when prey are aggregated, and such a pattern is strengthened by a low probability of encountering prey patches when prey density is low. Interestingly, the mechanism described above provides a sound explanation for the change in functional response from type II to type III found analytically by Nachman (2006a) when prey distribution was changed from random to aggregated.

The evidence for non-uniform predation patterns and their effects on predation rates in this paper is of vital importance at both fundamental and applied levels. On the one hand, I show that the common assumption of constant rates of encounter between predator and prey individuals within fields or greenhouses does not necessarily hold.

Conventional models, written as a set of difference or differential equations, cannot mechanistically represent local population non-uniformity, because they do not take into account discrete individuals (Judson 1994, DeAngelis and Mooij 2005). Here I show that predator movement patterns such as varying search effort as a function of local prey density, or the tendency of individuals to consistently visit some plant regions more often than others, may alter the shape and magnitude of functional responses and consumption rates across spatial scales. Thus, the interaction between non-random distributions of prey patches and predator patch-to-patch movement rules should be modeled using a spatially explicit individual-based simulation approach.

My results also imply that the tomato-SWF-D. catalinae system could be stable.

First, I show that the distribution of D. catalinae individuals among leaves is described more accurately by the leaf nodal position than by the number of SWF in the leaves. That is, predators are more likely to visit and predation rate is likely to be higher in leaflets or leaves located towards the middle and mid-lower region of the main stem, which are not necessarily the nodal positions with highest SWF nymph density. Predator aggregation can increase stability in predator-prey systems, especially if aggregation is independent of local (i.e. intra-patch) prey density (Murdoch and Stewart-Oaten 1989). Second, the emergence of a type III functional response at the intra-plant scale is of vital importance for mathematical modeling of SWF-D. catalinae dynamics, since this response type enhances the stability properties of the system (Berryman 1999, Berryman et al. 2006).

Moreover, the presence of regions where mortality due to predation is lower may help

predators to survive during periods of prey scarcity. However, this stability may not be economically important for greenhouse tomato crops if prey populations exceed economic thresholds. In particular, the extent to which increases in D. catalinae release rates, or the use of complementary control strategies will help maintain SWF populations below economic injury levels would need to be examined. Ultimately, these questions can be addressed using ecological models, provided restrictive assumptions, such as those identified in this research, are relaxed.

Chapter 4: Scaling up functional responses from Petri dish to plant scale based on predator patch residence times and patch visitation patterns

Abstract

Much of the information collected to understand ecosystem function is based on laboratory experiments. Consequently, extrapolation to the field, or scale transition, is necessary to develop ecosystem theory at realistic spatial and temporal scales. This is often the case for functional responses, which are important components of predator-prey models that are usually estimated in small-scale laboratory experiments. I present a spatially explicit individual-based model that estimates predation rates at the intra-plant scale, based on a functional response model estimated in the laboratory, predator patch residence times and intra-plant visitation patterns. The model explicitly incorporates the interaction between intra-plant prey infestation and predator visitation probabilities across leaf nodal positions. The model is parametrized with data from a predator-prey system consisting of the coccinelid predator Delphastus catalinae and the silverleaf whitefly (SWF) Bemisia tabaci biotype B (Hemiptera: Aleyrodidae) inhabiting tomato plants. Validation revealed that the model is useful in scaling up from laboratory functional responses to predation in whole tomato plants of varying sizes. Furthermore, the model shows that the observed reduction in predation rates and the shift in the shape 93

of the functional response from type II to type III during the scale transition is associated with imperfect alignment between intra-plant predator and prey distribution and predator foraging habits. I demonstrate that simple measures of prey distribution and predator foraging patterns can be used to scale up functional responses estimated through small experimental settings. This approach could be used to model population-level dynamics of the tomato-SWF-D. catalinae system, and could be easily applied in to other situations with active predators and low-mobility or sessile prey species.

Introduction

Functional responses are mathematical functions that describe the number of prey consumed by a predator based on the initial number of prey available in a given area (or volume) (Holling 1961, Juliano 2001). These functions are important components of predator-prey models, because they represent the link between predator and prey populations, and their shape is crucial in determining the dynamics and persistence of interacting populations (Berryman 1999). Because determining the number of prey eaten by individual predators in large, structurally complex environments is difficult, functional responses are often estimated in laboratory settings, where individual predators are given access to fixed numbers of prey for a given time period in a small arena. The outcome of such experiments is a response curve of the number of prey consumed by individual predators against the initial number of prey. The resulting model is often used to predict

predation rates and dynamics of much larger natural systems (Englund and Leonardsson

2008).

Ecological variables and processes are often scale-dependent in heterogeneous environments. Thus, extrapolation of results obtained at small scales to broader scales requires a means of measuring how heterogeneity patterns change with scale (He et al.

1994). Direct extrapolation of functional responses calculated in the laboratory to larger natural systems, without consideration of added heterogeneity, often leads to overestimations of natural enemy search and predation rates (O'Neill and Rust 1979,

O'Neil 1989, O'Neil 1990, Englund and Leonardsson 2008). This error in extrapolation has long been recognized and various attempts to combine small-scale observations in the laboratory with field measurements of heterogeneity have been incorporated into models.

Most of these models are based upon the assumption that functional responses derived from laboratory experiments can be used to approximate local predation rates and that scale transitions could be achieved by incorporating descriptions of prey distribution across patches and inter-patch predator movement patterns. The most common approaches define prey distribution and predator inter-patch behavior in terms of independent and conditional probability distributions, respectively (Bailey et al. 1962,

Hassell 1980, Nachman 2006a). Others incorporate heterogeneity using expressions of spatial variance in prey density and covariance between densities of predators and prey

(Englund and Leonardsson 2008). More sophisticated individual-based approaches estimate functional responses in heterogeneous environments based on observations of

individual behavioral rules of predators (or parasitoids) within and between prey patches, but without including laboratory measures of predation rates (van Roermund et al. 1997,

Hemerik and Yano 2011).

In most of the approaches described above the modeled environment is two- dimensional, where prey patches are equidistant from each other, and the number of prey on them is independent of their location [but see van Roermund et al. (1997)]. However, most insect predator-prey systems occur within plant canopies, where the high structural complexity and the differences in physical and chemical properties among leaves favor non-random distributions of prey patches (Raupp and Denno 1983). Moreover, many insect predators search more intensively within certain plant regions, resulting in uneven distributions of probabilities of encounter across leaves within single plants (Bond 1983,

Costamagna and Landis 2011, Hodek et al. 2012). Such uneven predation patterns have been reported to impact prey population growth rates either by reducing overall within- plant predation rates, by increasing predation risk of relatively fecund individuals, or both

(Costamagna et al. 2013).

In previous research I showed that the distribution of within-plant encounter probabilities between the coccinellid predator Delphastus catalinae (Horn) and the silverleaf whitefly (SWF), Bemisia tabaci biotype B (Gennadius) (Hemiptera:

Aleyrodidae), is not uniform and may affect the magnitude of predation rates and the shape of the functional response at the intra-plant scale (Chapter 3). I also presented an algorithm that generates spatial distributions of the SWF within tomato plant canopies of

different sizes (Chapter 2, Rincon et al. 2015). In this study, I present a spatially explicit individual-based model designed to scale up a laboratory functional response of D. catalinae preying on SWF nymphs to whole plants. The model I present includes descriptions of the within-plant foraging behavior of D. cataliane and the distribution of

SWF nymphs. The first is based on the observed residence times on leaflets and the location and number of leaflets visited by predators per plant. The distribution of SWF nymphs is incorporated into the model using an algorithm I described elsewhere (Chapter

2, Rincon et al. 2015). The model was validated using consumption data collected from the same tomato plants on which D. catalinae foraging patterns were observed. I show that the model predicts whole-plant predation rates and that low predation rates and type

III (sigmoidal) functional responses emerge at the intra-plant scale as a result of misalignment between predator and prey distributions resulting from predator foraging habits, even if local predation rates are high and functional responses are type II

(saturating).

Study system

My study system consists of the predator D. catalinae preying on the SWF inhabiting tomato plant canopies. The structure of tomato plants is typically a main stem with a terminal bud and lateral (secondary) stems that grow out from buds located in the axil of leaves. However, plants are usually grown as single stems in most tomato production systems; lateral buds are removed to increase fruit yield and quality

(Cockshull et al. 2001). Tomato leaves are compound and consist of a petiole and an odd number of leaflets of about the same size.

The silverleaf whitefly is a worldwide pest of a variety of greenhouse and field production systems, including tomato (Stansly and Naranjo 2010). Adults are able to fly and move within and among plants and feed on the underside of leaflets, where copulation occurs and females lay the eggs (Byrne and Bellows 1991). The immature stage consists of four nymphal instars, which are all sessile except the neonates that often move few millimeters before settling during the first minutes after hatching (Byrne and

Bellows 1991, Simmons 2002). Because the sessile phase (eggs and nymphs) is particularly vulnerable to predation or parasitization, and it represents ~ 50% of the life span (Tsai and Wang 1996), most predators and parasitoids used for biological control are specialized on nymph or egg consumption (Gerling et al. 2001). Silverleaf whitefly populations are highly aggregated within tomato plots (Byrne and Bellows 1991, Liu et al. 1993) and plants (Chapter 2, Schuster 1998, Rincon et al. 2015). Adults often prefer mature leaves to feed and oviposit when associated with tomato plants (Muniz et al.

2002, Arno et al. 2006). The within-plant distribution of adults reflects that of eggs, and the distribution of subsequent sessile nymphal instars among leaves and leaflets progressively shifts down with the appearance of new leaves (Chapter 2, Schuster 1998,

Rincon et al. 2015).

Of more than 25 species of predators that have been reported to be associated with the SWF, the Coccinellid D. catalinae is currently the only one that is commercialized in

the USA for whitefly control (Obrycki and Kring 1998, Gerling et al. 2001). However, the use of D. catalinae as a biological control agent has been limited because of the variable degree of success suppressing whitefly populations in the field (Heinz and

Parrella 1994b, Heinz et al. 1999). Moreover, D. catalinae is suggested to only be effective when SWF populations are high (Heinz and Parrella 1994a), a significant restriction, considering the ability of SWF to vector plant pathogenic viruses. Although the predation rates of D. cataliane have been estimated in laboratory settings (e.g.,

Hoelmer et al. 1993, Guershon and Gerling 1999, Liu and Stansly 1999), estimations of its consumption capacity in more realistic scenarios are needed to fine-tune release rates and application methods. This is especially important since recent research has shown that predation rates are significantly lower at the within-plant scale than those estimated in the laboratory (Rincon et al. In prep.).

Materials and methods

Experiments were carried out at The Biological Sciences Greenhouse Facility of

The Ohio State University, Columbus, OH (USA), as described in Chapter 3. Briefly, forty seven 7-10 week-old tomato plants (Solanum lycopersicum L. var. Trust) were grown in pots and placed individually in 130 x 60 x 60 cm (high x width x deep) mesh cages. Thirty five of the plants were infested with a variety of 3-day-old adult SWF male- female pairs: 6 plants with 14 pairs, 4 with 20, 9 with 30, 11 with 40 and 5 with 50; and the remaining 12 plants were left uninfested. Silverleaf whitefly adults were removed

after 48 h and, after 12 days, I recorded the number of nymphs in each leaf and leaflet.

The next day, ten D. catalinae adult individuals were released into the cages. Each predator was marked with a dot of ink of a different color extracted from TexPen ® markers (ITW Dykem, Olathe, KS, USA) previously diluted at 5:1 (paint : thinner) with turpentine substitute, following the procedure described in Bates and Sadler (2004). For the next 84 h (3.5 days), plants were observed every 2 h during photophase (9am to 6pm) to record the number and identity of individual predators that visited each leaf and leaflet.

At the 4th day after predators were released, all the leaflets of each plant were cut to re- count the number of SWF nymphs on their underside. The number of prey consumed by

D. catalinae individuals was estimated as the difference between the number of SWF nymphs before and after the 4-day D. catalinae-SWF interaction period.

Model description

The purpose of the model is to estimate the number of prey consumed by one or more predators within single tomato plants using a functional response model estimated in the laboratory as a reference, and sub-models of intra-plant prey distribution and predator patch-to-patch movement patterns. The model simulates the movement and prey consumption of one predator within a tomato plant canopy that is infested with different numbers of prey items on each leaflet. The model is spatially explicit in that it keeps track of the location of each leaf and leaflet within the plant, but not of the locations within leaflets. The basic principle is that predators consume a number of prey every

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hour, based on the number of prey available in the leaflet being visited and according to the functional response model estimated in the laboratory by Guershon and Gerling

(1999). The number of prey in leaflets is updated accordingly every hour, whereas the decision to stay, visit a new leaflet or leave the plant is made hourly based on the number of prey in the current leaflet and plant, respectively. Repeated simulations of individual predators provides an estimate of the mean and variance of various predation parameters such as prey consumed per predator. Following is a general description of the model

(Figure 9), which will be followed by more detailed descriptions of the associated algorithms, parameters and the sources of data for them.

The model was written in R language using the R software version 3.1.1 (R Core

Team 2013) (see flowchart in Figure 9, R code provided in Appendix B). The process starts with the generation of plant objects, which is described in detail in Chapter 2 [and in Rincon et al. (2015)] and requires the number of leaves in the plant and the number of

SWF nymphs on them as inputs. The model then simulates the activity of a single predator and generates three statistics based on the simulated pattern of predator movement and prey consumption: “leaves visited”, “leaflets visited” and “prey consumed”. The location of the leaves and leaflets visited and the number of prey consumed in each leaflet by the predator are stored in matrices. The number of leaflets visited by the predator is drawn randomly from a normal probability distribution function

(PDF) with mean based on the number of SWF on the plant and variance estimated from data (see “Model calibration” section and table 5) and rounded to the closest whole

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number. The location of the leaves visited by the predator is then randomly drawn with replacement from all possible nodal positions (leaves) in the plant using a set of probabilities of visitation per nodal position as weights. Leaflets visited within leaves are sampled randomly without replacement from all possible locations within the leaf. If the selected leaflet is infested, a residence time in hours is randomly drawn from an exponential PDF, with parameter λ based on the number prey on the leaflet, and a number of prey consumed is drawn from a binomial PDF, with parameters n, the number of prey on the leaflet, and pe, the proportion eaten based on the remaining number prey

(see “Model calibration” section and table 5). The number of nymphs and residence time is updated and, if more prey are available and the residence time has not been reached, the process repeats. If the number of prey on the leaflet or residence time remaining reaches zero, the process repeats on a new randomly selected leaflet. The instantaneous shift to a new leaflet simulates the movement pattern of D. catalinae foraging adults, which move to new leaves by short flights rather than by walking between leaflets. If more than one predators forage in the same plant, the model assumes that predators never feed at the same time in the same leaflet, because predators are added sequentially.

Model calibration

The calibration procedure of the sub-model used to generate plant objects is detailed in Chapter 2 [and Rincon et al. (2015)]. I used all the parameter estimates reported in Chapter 2 except the parameters of the beta-binomial PDF that produces the

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per-node infestation probability of SWF nymphs, which were estimated using the observed SWF nymph counts before predators were released into the plant cages. I used a beta-binomial PDF to assign the probabilities of a predator visiting each leaf, similar to the per-leaf probabilities of infestation that were described in Chapter 2 for the SWF on tomato plants. This allowed me to explicitly simulate the unique distributions of probabilities of SWF infestation and predator visits per nodal position within tomato plants I described in Chapter 3. The use of beta-binomial PDFs to describe probabilities of events that occur across nodal positions within plants is described in detail in Chapter

2. Briefly, this approach assumes that the probability of a predator visiting a leaf nodal position can be described as a multinomial process, where each visit could be on any of the available leaves. Parameters were estimated by Maximum likelihood using the 35 sets of counts of different predator individuals that visited each nodal position (from the 35 infested plants) and the number of leaves in the respective plant. A multinomial PDF was used to estimate the likelihood of each set counts and the respective number of leaves per plant by deriving the multinomial probabilities from a beta-binomial PDF. Thus, the likelihood function is given by:

(Eq. 14), where p denotes the average per-trial probability and θ the overdispersion parameter. The number of trials per sample (i.e. size parameter) is denoted here as ni and it represents the

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total number of leaves in the ith plant. The vector Si contains the numbers of predator individuals, si,j, on the leaf nodal position j (i.e. Si = [si,1,…, si,ni]), and si. is the sum of the number of different individuals that visited each leaf in the ith plant. The parameter N is the total number of plants in the sample (N = 35) and x is the nodal position, j, minus 1, so that the numbering of nodal positions fits with the probability estimates, which start from zero in the beta-binomial PDF. The aim is to find the values for p and θ that maximize the likelihood of the multinomial PDF, given the sets of counts from each plant, and allowing the size parameter, ni, to vary according to the respective number of leaves per plant.

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Figure 9. Flowchart describing the routine and decisions followed by a modeled predator in a four-day predator-prey interaction period. See text for details.

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The search effort predators invest within patches was measured in terms of leaflets visited for the between-plants scale, and in time for the between-leaflets scale.

The mean number of leaflets visited, f(x), and the mean residence time in leaflets, g(x), by individual predators were modeled as a function of the initial SWF nymph density within a given leaflet and plant, respectively, using saturation functions:

(Eq. 15a),

(Eq. 15b), where x represents the SWF density on a plant and z on a given leaflet, and VL, VT, DL and

DT are parameters. Parameters VL and VT are the maximum value of f(x) and g(x), respectively. Parameters DL and DT represent the x or y value at which the function equals one half of its maximum. Parameters were estimated by Maximum likelihood assuming a normal distribution for the mean number of leaflets visited by a predator [Eq. 15a;

2 variance about f(x), σL estimated from data, see table 5], and an exponential distribution for residence times of predators in leaflets [Eq. 15b; λ estimated as 1/g(z)]. Saturating functions such as those described above should provide a good description of D. catalinae patch-to-patch foraging patterns, as long as the variation of the predator’s search effort within prey patches follows the model described by Waage (1979) for

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parasitoids but often generalized for predators (Bell 1991, Biesinger and Haefner 2005).

Infested plants and leaflets within plants can be seen as prey patches at different spatial scales, where the time (or search effort) a predator will spend on each is determined by the interaction between two contrasting processes: stimulation provided by prey chemicals and encounter rate, and habituation to the response. As such, Waage’s (1979) model predicts that a predator’s search effort in a patch should increase with prey density until it reaches a critical prey density, after which no further increases in search effort would be expected.

The number of prey eaten by a predator in a 1-hour interval was modeled with a functional response model using the parameters reported by Guershon and Gerling (1999) for D. catalinae adults preying on “setose” SWF nymphs within “tomentose” tomato leaf discs. However, Guershon and Gerling (1999) estimated functional response parameters

(i.e. handling time and attack rate) for the ‘disc equation’ (Holling 1961), which assumes constant prey density (Juliano 2001). This means that, in Guershon and Gerling’s (1999) experiment, prey were replaced as they were eaten by predators within experimental units. Under the scenario simulated by my model, prey density declines over time without replacement, at least during the 1-hour interval before the prey density is updated in the leaflet being visited by the predator. Thus, I used the integral of the ‘disc equation’, known as ‘random predation equation’, to account for changing prey density which is defined as (Rogers 1972):

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(Eq. 16), where a is the attack rate expressed in terms of area searched per hour, Th is the handling time of predators for one prey in hours, and Ne is the number of prey eaten by a predator in an area with N prey available during an interval of time T (= 24 h).

Model validation

Ideally, the model output should be validated using an independent dataset describing the average prey consumption of D. catalinae individuals in tomato plants of known sizes (number of leaves) and initial prey densities. I compared model outputs with the mean consumption rates per predator observed in the experiments conducted to estimate residence times, number of leaflets visited and, both predator visitation and prey infestation probabilities per leaf nodal position. These data were independent of my model parameters in that I did not use them to estimate the functional response parameters, rather these were taken from the literature.

To compare the observed data with the model output, I simulated the experiment described in Chapter 3 1000 times, using the observed plant sizes and their respective initial number of SWF nymphs as inputs, with 10 predators foraging sequentially on each plant. The output of each simulation was a 10 x 35 matrix with the numbers of prey consumed by predators in the rows and individual plants in the columns. For the output of each simulation, the mean number of prey consumed by predators was calculated for each

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plant, and their averages from the 1000 simulations compared with the observed mean number of prey eaten by the predators in the respective plants. Comparison between model outputs and observed data was performed using concordance correlation methodology (CCM; Lin 1989, Madden et al. 2007). The CCM is based on the concordance correlation coefficient, ρc, which ranges from -1 to 1 and equals 1 when there is a perfect agreement between observed and simulated values. The coefficient ρc results from the product of the generalized bias parameter, Cb, and the coefficient of correlation, r, of the observed and simulated values. The first reflects the accuracy (or bias) and the second the precision (or variability) of the model, and range from 0 to 1 and from -1 and 1, respectively. Both Cb and r equal 1 when the best fitting line between observed and simulated values is identical to the perfect concordance line. The CCM was performed using the R package “epiR” (Stevenson et al. 2014)

To detect potential disagreements between the simulated and the observed functional response parameter values, I estimated functional responses from the simulated predation rates and compared the estimates with those observed in Chapter 3.

To describe whole-plant simulated predation values, Ne(sim), as a function of SWF per plant, N0(plant), I used the type III functional response model:

(Eq. 17), where b and c are the slope and the intercept of a linear model describing attack rate as a function of SWF density, respectively, and Th(sim) is the ‘handling time’ expressed in

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terms of fraction of the total time of interaction, T (set as one), estimated from simulated values. Functional response parameters were estimated by Maximum likelihood, using the Lambert W function to compute a close-form solution of Eq. 17 for Ne(sim), and parameter comparisons between observed and simulated models were performed using Z tests (Bolker 2008).

Sensitivity analysis

To determine the effect of predator patch-to-patch movement parameters on the resulting whole-plant functional response, I performed sensitivity analysis on key parameters controlling the number of leaflets visited, the residence time spent in leaflets, and the distribution of visits among leaf nodal positions of simulated predators within plants. I analyzed the resulting whole-plant functional response after changing the mean number of leaflets visited and the residence time spent in leaflets by varying values of VL and VT by +50% and -50% and also by setting f(x), the mean number of leaflets visited, and g(z), the mean time spent on leaflets, at arbitrarily chosen constants, f(x) = 3 and g(z)

= 10. The magnitude of the constants was chosen based on the approximate maximum of both mean number of leaflets visited and the residence time spent in leaflets by predators

(Table 5). This analysis allowed me to determine the effect of the predator search effort, in magnitude and prey density dependency, within both plants and leaflets on the resulting predation patterns.

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I analyzed the change in the distribution of predator visits by varying the parameter p by -60% and -80%. I selected p because it determines the location (median) of the probability distribution of predators visiting the different leaf nodal positions, allowing me to examine the effect of misalignment between intra-plant predator and prey distributions on the resulting functional response. In this case, the resulting whole-plant functional responses from reductions in p were compared with those resulting from the perfect alignment between intra-plant predator search patterns and prey distribution (i.e.

θv = θ and pv = p, Figure 10a).

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Each parameter change was evaluated by running 1000 simulations. Functional response parameters were estimated as described above for validation, using the mean number of prey consumed by predators calculated from each output. To distinguish between type II and III functional response models, the fit of Eq. 16 and Eq. 17 to simulation data was compared using the Likelihood Ratio Test (LRT), which should be adequate as Eq. 16 is nested into Eq. 17 (i.e. Eq. 17 becomes Eq. 16 if b = 0) (Okuyama

2013). When no significant differences were detected (α = 0.1), the simplest model (i.e. with fewer parameters, type II) was assumed to be the best. The R package “emdbook” was used for parameter estimation of all models (Bolker 2013).

Results

The saturating functions (Eq. 15a and 15b) provided a good description of the variation in predator search effort based on SWF density both within plants and within leaflets (Figure 11). Parameter estimates used in the simulation model are listed in Table

Model validation

The functional response models derived from simulated and observed values did not differ significantly (Figure 12a). I did not detect any significant differences between simulated and observed functional response parameter estimates (Table 6). The validation analysis also showed that the simulation model produces reasonable predictions of D.

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catalinae predation rates at the within-plant scale (Figure 12b). The concordance

correlation coefficient between observed and simulated values was ρc = 0.688, the

generalized bias parameter Cb = 0.996, and the correlation coefficient r = 0.691. These

results indicate that the departure of the calculated ρc from 1 (perfect concordance) is

mainly due to low precision (i.e. low r), rather than to bias in the prediction (Madden et

al. 2007).

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Table 5. Description of parameters used in the model.

Parameter Description Estimate SE Source Generation of plant objects av Intercept of the Taylor’s variance- 24.50 11.186 Rincon et al. (In mean model at plant scale. press.) bv Exponent of the Taylor’s variance- 1.566 0.120 Rincon et al. (In mean model at plant scale. press.) pv Mean per-trial probability of a beta- 0.6168 0.002 This research binomial PDF: SWF distribution among leaves θv Over-dispersion parameter of a 5.266 0.114 This research beta-binomial PDF: SWF distribution among leaves al Intercept of the Taylor’s variance- 35.658 20.973 Rincon et al. (In mean model at leaf scale press.) bl Exponent of the Taylor’s variance- 1.347 0.210 Rincon et al. (In mean model at leaf scale press.) pl Mean per-trial probability of a beta- 0.243 0.003 Rincon et al. (In binomial PDF: SWF distribution press.) within leaves θl Over-dispersion parameter of a 2.778 0.091 Rincon et al. (In beta-binomial PDF: SWF press.) distribution within leaves Predator patch-to-patch movement p Mean per-trial probability of a beta- 0.609 0.009 This research binomial PDF θ Over-dispersion parameter of a 8.876 1.126 This research beta-binomial PDF VL Max number of leaflets visited by a 2.984 0.182 This research predator DL SWF density within plants that 105.992 24.989 This research produces VL/2 2 σL Variance of mean number of leaflets 0.114 0.040 This research visited VT Max time in hours spent by 10.701 2.202 This research predators in leaflets DT SWF density within leaflets that 5.120 3.970 This research produces VT/2 Local (laboratory) functional response a Attack rate 0.091 0.011* Guershon and Gerling (1999) Th Handling time 1.116 0.086* Guershon and Gerling (1999) *Estimated from the reported 95% CI and sample size.

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Table 6. Whole-plant functional response parameter estimates obtained from observed and simulated values. Numbers in parenthesis are Standard Errors of the Estimates.

Parameter Observed Simulated Z-value P-value Handling time (Th) 0.0596 (0.0039) 0.039 (0.0121) 1.609 0.107 Slope (b) 0.0001 (< 0.0001) 0.00006 (< 0.0001) 0.859 0.390 Intercept (c) 0.0173 (0.005) 0.0075 (0.0072) 1.113 0.265

Figure 12. Validation analysis of an individual-based model developed to scale up laboratory functional responses to the scale of plant. a) Comparison between observed (○) and simulated (∆) predation rates, showing the observed (dashed curve) and simulated (solid curve) functional response models at the scale of plant. b) Relationship between observed and simulated predation rates, showing the best-fitting line (solid) and the perfect concordance line (dashed).

Sensitivity analysis

I found that variations in the maximum number of leaves visited, VL, and the

maximum time spent on leaflets, VT, by predators always resulted in whole-plant type III 116

functional responses, except when the number of leaflets visited by predators, f(x), is constant and with 50% reductions in either VL or VT, conditions that resulted in type II responses (Table 7). In general, changes in VL resulted in proportional changes in magnitude of predation rates, but prey density-dependence of f(x) impacted only predation rates at low prey densities which were higher when this variable was constant

(i.e. when f(x) is non prey density-dependent) (Figure 13a). The effect of changing VT was not as large as that of VL, because it was only evident towards the mid part of the ascending portion of the functional response model, where higher values [including constant the maximum time spent on leaflets by predators, g(z)] resulted in steeper s- shaped curves. In fact, a 50% reduction in VT resulted in the typical saturating (type II) response (Table 7, Figure 13b). I also found that changes in p do not have an effect on the type of the resulting whole-plant functional response (Table 7). However, lower predation rates were obtained with greater reductions in parameter p (Figure 13c), meaning that simulated predation rates were reduced as the distributions of predator and prey diverged.

Discussion

Animal and plant populations aggregate as a consequence of social interactions, distribution of resources or the presence of structural complexity in the environment.

However, much of the ecological theory on predation is based upon an assumption that individuals interact in a random, well-mixed fashion across simple two-dimensional

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environments. In fact, extrapolations from small-scale, short-term, simplified experiments to larger spatial and temporal scales are often unsuccessful because of they ignore key behavioral and physiological variables that are important at the larger scales. Methods to extrapolate results obtained in homogenous, small-scale settings to relevant larger spatial and temporal scales are needed. I found that the individual-based model I developed estimates predation rates of D. catalinae foraging on tomato plants infested with SWF nymphs reasonably well, using a functional response model derived from an independent, small-scale, laboratory setting. The model simulates the interaction between predator and prey within-plant distributions, and predator patch-to-patch foraging habits and hourly consumption rates, over a 4 d time period. The model can be easily parameterized for a number of plant-prey-predator systems with active predators that move by flying or hopping between prey patches (short transit time) and sessile or low-mobility prey inhabiting single-stem plant structures. The data required consists essentially of spatial counts of prey within plants, sufficient for estimating within plant spatial distributions, and observations of predator patch-to-patch movement sufficient to determine rules for movement patterns over time.

My model has a number of assumptions that may limit its application to different environments or to other plant-prey-predator systems. First, my model does not consider variations in predation rates due to temperature. Typically, the relationship between temperature and both attack and maximal consumption rate (i.e. inverse of handling time) is hump-shaped, at least under “biologically relevant temperature ranges” (Logan and

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Wolesensky 2007, Englund et al. 2011). In particular, Simmons and Legaspi (2004) found that D. catalinae predation rates on SWF eggs and nymphs increase with temperature until reaching a peak at about 30° C. Similar results have been found for other Coccinellid species and SWF parasitoids (e.g. Enkegaard 1994, Xia et al. 2003,

Jalali et al. 2010). However, my model was developed from and is intended to generate predictions for greenhouse environments, where temperature is relatively constant. In order to make the model useful for field environments, local functional responses might have to be adjusted to let both attack rates and handling times be a function of environmental temperature, and predator patch-to-patch movement patterns revised.

These changes are certainly feasible enhancements of the current model.

Functional Chi-square Parameter Change P-value response type value +50% III 4.558 0.032* V L -50% II 1.338 0.247 f(x) Constant II 0.620 0.430 +50% III 3.226 0.072* V T -50% II 1.777 0.182 g(z) Constant III 4.297 0.038* -60% III 3.832 0.050* p -80% III 28.964 < 0.001* Aligned with p and θ III 3.953 0.046* pv and θv *Significant differences (α = 0.1)

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Figure 13. Sensitivity analysis of an individual-based model developed to scale up laboratory functional responses to the scale of plant. a) Effect in functional response of changing parameter VL (max. number of leaflets visited by predators) by +50% (dashed line), -50% (dash-dot line) and when the number of leaflets visited per plant by predators, f(x), is set constant (solid line), compared with the output from unchanged parameters (dotted line). b) Effect in functional response of changing parameter VT (max. residence time in leaflets of predators) by +50% (dashed line), -50% (dash-dot line) and when the time spent by predators within leaflets by predators, g(z), is set constant (solid line), compared with the output from unchanged parameters (dotted line). c) Effect in functional response of changing parameter p by -40% (dash-dot line) and -60% (dotted line), compared with the output obtained when p = pv and θ = θv (dotted line). 120

The model I present here also assumes that there is no alternative prey, including eggs and different life stages of the SWF. However, it is unlikely that several potential prey species for D. catalinae are present on the same plants. Although D. catalinae preys upon several species of whiteflies (Obrycki and Kring 1998), the SWF is often dominant and its populations are rarely mixed, at least with different B. tabaci biotypes (Pascual and Callejas 2004, Luan et al. 2013). One important exception is the interaction between the SWF and the Greenhouse whitefly Trialeurodes vaporariorum (Westwood)

(Hemiptera: Aleyrodidae), whose populations can coexist under the right conditions, at least during the first few generations, after which the SWF usually excludes T. vaporariorum (Zhang et al. 2011). The silverleaf whitefly and T. vaporariorum have distinct intra-plant distributions when associated with tomato plants, which are not affected by the presence of each other in the same plant (Arno et al. 2006). Thus, it would be possible to determine the ratio of both species as a function of leaf nodal position, and generate spatial distributions of mixed infestations within tomato plants, following the same approach described in Chapter 2 (Rincon et al. 2015). Predation rates of D. catalinae do depend on whitefly species, and rates reported for T. vaporariorum (Lucas et al. 2004, Gonzalez et al. 2007) are lower than those reported for the SWF (Hoelmer et al. 1993, Guershon and Gerling 1999). Although more than one functional response model can be incorporated and predation rates for each prey species stored and analyzed, very little is known about preference of D. catalinae when choosing between the two whitefly species. Similarly, when more SWF life stages are present in the same plant,

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more sub-models (functional responses and stage distributions) can be incorporated for each, but preference for one or more SWF life stages has not been studied. Legaspi et al.

(2006) found that D. catalinae adults eat significantly more SWF eggs than small or large nymphs, when presented simultaneously in a single arena. However, from the data available, it is hard to conclude that D. catalinae prefers SWF eggs over nymphs, because the handling time for eggs is significantly lower (Hoelmer et al. 1993) and, therefore, more eggs might be consumed per time unit, even without a preference for eggs over nymphs.

My model assumes that the presence or consumption of prey byproducts (e.g. feces, honeydew, etc.) does not affect predator movement patterns or predation rates.

Previous research has shown that D. catalinae individuals rarely feed on the honeydew excreted by SWF nymphs and adults, at least when whitefly prey are present (Simmons et al. 2012). This report agrees with observations on other coccinelid predators that use honeydew excreted by their hemipteran prey as a significant food source only when prey is low-quality or absent (Lundgren 2009). Thus, accuracy of predictions of predation rates are not likely to be reduced by predators spending time feeding on honeydew.

Honeydew has also been reported to arrest and intensify the foraging of coccinellid predators that seem to use honeydew as an indication of the presence of prey on plants or plant parts, without necessarily using it as a food source (Ide et al. 2007, Pettersson et al.

2008, Lundgren 2009, Hodek et al. 2012). This behavior may explain the relationship I found between both the number of leaflets visited per plant and the time spent within

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leaflets by D. catalinae with the density of SWF nymphs within plants and leaflets, respectively.

My model does not consider intra-specific competition, mating behaviors or social interactions among predators. This assumption implies that predators visit plants sequentially, or if simultaneously, then the only interaction is indirect, through reduction in available prey. To my knowledge, there are no studies examining the mating or social behavior of D. catalinae and, in general, such studies are scarce in the literature for

Coccinellids (Hodek and Ceryngier 2000). However, evidence on the presence of short range non-volatile sex pheromones mediating sexual interactions between male and female Coccinelids has been reported previously (Obata 1988, Hemptinne et al. 1996).

Recently, volatile compounds released by females playing a role in remote attraction of males have been identified (Fassotte et al. 2014). The presence of such compounds may affect predation rates at both leaflet and plant spatial scales and, especially, that of male predators, which would spend time searching for female cues, even in the presence of prey (Obata 1997). In my experiments, I observed two or more D. catalinae individuals on the same leaflet relatively often and mating events were not rare, but the extent to which interactions among predators affect overall predation rates remains to be studied.

My model is similar to previous individual-based approaches developed to generate predictions of predation or parasitization rates that incorporate spatial effects at the intra-plant scale (e.g. van Roermund et al. 1997, Hemerik and Yano 2011). However, my model is different in that it does not explicitly model intra-patch movement patterns

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and encounter rate-dependent giving up times. Instead, I use a functional response model estimated from a laboratory setting to produce hourly predations rates, and the assignment of predators to leaflets as well as the time they spent on them is modeled as a function of initial SWF density within plants and leaflets, respectively. This simplification not only makes conventional laboratory functional responses useful to infer population-level consequences of the introduction of predators to new environments, but also reduces the amount of data required to feed predator-prey models that consider spatial effects.

My simulations were for 4 d, a time period that may not be sufficient to capture longer-term interaction between the SWF and D. catalinae. Although local (within- leaflet) predation rates were estimated hourly (from a laboratory functional response), the time spent within leaflets and the number of leaflets visited per plant by predators was estimated from an 84-h experiment. I chose this observation period because after 84 h most of the predators would have either died or reached a plant and found prey, considering that less than 10% of D. catalinae adults survive without food for 4 days

(Simmons et al. 2012). Furthermore, longer observation times would have resulted in more SWF completing the nymphal stage, and the resulting exuviae are difficult to distinguish from SWF nymphs that were consumed by predators. A linear relationship between time and the number of prey consumed by predators within plants could be assumed if the whole-plant functional response were to be used in a model of predator- prey interactions over longer periods of time. This assumption is reasonable given that

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predation rate of D. catalinae on SWF eggs and nymphs remains relatively constant from emergence to death, at least under laboratory settings (Hoelmer et al. 1993).

The model may have to be extended and enhanced to simulate predation at the field scale. Predator numerical response to SWF-infested plants, SWF development, and plant growth could easily be incorporated into my model. However, extending the simulation to larger spatial and temporal scales requires the assumption that the functional relationship I found between SWF density and both the time predators spent within leaflets and the number of leaflets they visit per plant (i.e. Eqs. 15a and 15b) applies at these scales. The maximum residence times within both leaflets and plants (i.e.

VT ~ 11 h, and VL ~ 3 leaflets ~ 33 h, respectively) were considerably shorter than the length of my observation periods, therefore, they would likely hold for even longer periods of time. Heinz and Zalom (1996) found that D. catalinae individuals remained for 4.8 h in tomato plants infested with 1000 SWF eggs, a time that is considerably lower than my estimate (~ 33 h). However, the authors warn that predators used in their study were obtained from a laboratory colony with a superabundance of prey, which can result in shorter residence times in response to lower population densities under experimental conditions. Moreover, the plants used by Heinz and Zalom (op. cit.) likely lacked honeydew because they were only infested with eggs and were only 4 weeks old, typically with just 3-4 leaves (de Koning 1994, Najla et al. 2009). In contrast, the plants I used in my experiment were infested with honeydew-producing SWF nymphs, which could arrest searching predators (Ide et al. 2007, Hodek et al. 2012), and were

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considerably larger in size, having between 6 and 12 leaves and an even greater difference in number of leaflets, considering that the first four compound tomato leaves contain 3-9 leaflets on average, while the rest typically contain 13 (Chapter 2, Sarlikioti et al. 2011, Rincon et al. 2015). Bias in the estimated number of leaflets visited by D. cataliane on tomato plants would be only evident when f(x) (Eq. 15a) is close to the plateau, that is, when predators forage on heavily infested plants (≥ 500 nymphs, Figure

11).

I used my model to determine the magnitude of predation rates and emergence of type III functional responses with the transition between spatial scales. The emergence of type III functional responses when local type II responses are scaled up to heterogeneous prey distributions has been reported previously for mites (Nachman 2006a) and phyto- and zooplankton (Morozov 2010, Cordoleani et al. 2013). In essence, these population- based models agree that type III responses result from the interaction between predator and prey aggregation patterns and few inferences could be made about the predator’s individual behavior. However, my model is the first one to incorporate explicitly realistic numbers of prey patches and their spatial distribution along with individual aspects of predator foraging behavior. These attributes allowed me to directly examine the causes of changes in the simulated functional responses when the scale of observation is increased.

In particular, I showed in Chapter 3 that the functional response of D. catalinae preying on SWF nymphs within tomato leaflets is type II, but it shifts into a type III response when the scale of observation is increased to the plant. I also found that predation rates

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within tomato plants are considerably lower than those observed within Petri dishes.

However, without the additional model results, I could only speculate on the causes of such changes in functional response. Here, using simulation, I examined the effect of the behavioral trend of predators in visiting certain plant regions, and the number of leaflets predators visit as well as the time spent searching within leaflets as a function of prey density within plants and leaflets, respectively.

I found that that type III responses emerge at the plant scale from local type II responses in most simulated situations, the exception being when the average number of leaflets visited by predators is either low or constant with respect to the prey population density, and when the average time spent within leaflets is low. These results agree with the mechanism speculated in Chapter 3, that the increase in search effort motivated by the encounter of prey or prey byproducts (i.e. ‘area-concentrated search’) only provides benefit for predators at spatial scales that are greater than the smallest patch unit (i.e. the leaflet). The result of this behavior is reflected in increasing predation rates at low prey densities, which is a key characteristic of sigmoid (type III) functional responses. I found that as SWF density per plant increases, more leaflets are visited by predators (Figure

11a, a measure of within-plant search effort) leading to increased probabilities of encountering prey. When the number of leaflets visited by predators is simulated to be constant [i.e. constant f(x)], or when it does not increase rapidly enough as a function of prey density (i.e. VL reduced by 50%), the probability of encountering prey does not increase with prey density and the functional response is type II (Figure 13a, Table 7). In

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contrast, the ability of predators to increase their search effort as a function of initial prey density at the within-patch scale (within leaflets), where prey are assumed to be randomly distributed, does not seem to translate into additional increases in encounter rates. This is evidenced by the type III functional response that results even when the time spent by predators within leaflets (Figure 11b, a measure of within-leaflet search effort) is simulated to be constant [i.e. constant g(z), Table 7]. However, if search effort within leaflets is too small, predators might not be able to take advantage of visiting more leaflets and predation rates do not increase enough with prey population density to produce type III functional responses (i.e. VT reduced by 50%, Table 7, Figure 13b).

Interestingly, I found in Chapter 3 that the functional response of D. catalinae preying on SWF nymphs is type II at the leaflet scale. This may contradict the mechanism mentioned above because, as shown in the present research, search effort of

D. catalinae individuals within leaflets is a function of local SWF density (Figure 11b).

However, type III responses will emerge as long as such predator behavior is combined with aggregated prey distributions.

I found that the reduction in the magnitude of predation rates at the within-plant scale in comparison with those found in the laboratory described in Chapter 3 may be, at least in part, due to differences in the distributions of predator visits and prey patches across leaves within plants (i.e. parameter p). Changes in the simulated distribution of predator visits led to large changes in predation rates, but not the type of resulting functional response (Table 7, Figure 13c). This result suggests the concept of

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‘proportional refuge’ as described by Berryman et al. (2006). When prey refuges are established, they lead to sigmoidal (type III) functional responses only if a constant number of prey individuals are protected by the refuge (i.e. ‘numerical refuge’).

However, if the refuges protect a constant proportion of the prey population (i.e.

‘proportional refuge’) only the magnitude of predation rates is affected, as predation rates are density-independent (Berryman et al. 2006). In the case of the tomato-SWF-D. catalinae system, both SWF individuals and D. catalinae visits are distributed in constant proportions across leaves within plants (Chapter 2, Rincon et al. 2015), so that any misalignment is in terms of proportions of the two populations. Interestingly, the magnitude of predation rates were also affected considerably by changes in the maximum number of leaflets visited by predators (Figure 13a), but very little by the maximum time they spend within leaflets (Figure 13b). However, caution should be taken interpreting this result because my model does not simulate any waste of time or energy by predators while searching within empty or depleted leaflets, rather the predators are assumed to move off leaflets that do not have prey immediately. The model does not keep track of predator energy gain and consumption, and the number of leaflets they visit is independent of the time they spend searching within leaflets.

The present study helps determine the relative importance of the distribution of prey patches and predator search behavior on the resulting predation rates of predator- prey systems within whole plants, which are more structurally complex than Petri dishes.

Moreover, I showed that data collected in laboratory settings can be used to infer the

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behavior of processes at ecologically relevant spatial and temporal scales such as whole plants over 4 days, as long as it can be integrated with information related to behavior of the predator at these larger scales. The significant impact of the patch-to-patch behavioral parameters of predators on predation rates observed at the intra-plant scale shows that conclusions derived from models that assume random movement of predators might be misleading. This is especially true when prey distribution and predator movement patterns interact with the architecture of the structures they inhabit, which is the case in a number of different predator-prey systems involving Coccinellids (Musser and Shelton

2003, Costamagna et al. 2013). Ultimately, accurate evaluations of D. catalinae release strategies can be accomplished by incorporating the simulation model developed here, and perhaps enhanced as described above, to either existing analytical models (e.g. Oaten

1977) or to explicit geostatistical simulations of the SWF-D. catalinae system (e.g. Pérez

M. et al. 2011, Zhao et al. 2011).

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Chapter 5: Conclusions and future directions

My thesis is that predation rates within whole plants are driven by the interaction between prey distribution, individual predator patch-to-patch behavior and consumption rates within patch units. I propose that results derived from simple laboratory settings can be useful to predict predation rates within whole plants, if they are combined with spatially explicit descriptions of prey distribution and predator movement patterns. The major findings that support this thesis include: 1) the distribution of nymphs of the silverleaf whitefly (SWF), Bemisia tabaci biotype B (Gennadius) (Hemiptera:

Aleyrodidae), within tomato plants is non-random and can be predicted as a function of leaf nodal preference and aggregation patterns; 2) the leaf nodal preferences of the predator Delphastus catalinae (Horn) (Coleoptera: Coccinellidae) follow a distribution that is different from and independent of that of SWF; 3) the differences in functional response of D. catalinae (in magnitude and shape) between scales of observation are explained by the interaction between predator foraging behavior and prey distribution; 4) differences in functional responses and predation rates between spatial scales can be predicted as long as information on predator functional response at the smallest scales, and foraging habits and prey distribution at larger scales is available. In the following, I

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summarize and discuss the results that support the findings mentioned above, highlighting the conclusions that are relevant to the hypotheses and predictions stated in

Chapter 1.

Distribution of silverleaf whitefly nymphs within tomato plants

In Chapter 2, I showed that the distribution of SWF populations within tomato plants are aggregated and can be described in terms of probabilities of infestation per leaf nodal position. This result confirms my prediction that SWF adults will prefer to oviposit on mature leaves, presumably to balance their nutrient and water uptake with their exposure to plant defenses. I also demonstrated that accurate and precise spatial distributions of SWF eggs and nymphs within tomato plants can be generated through a probabilistic computer algorithm. Although the algorithm is computationally complex, its parametrization only requires the proportion of SWF individuals located on each leaf nodal position and conventional measures of spatial aggregation, in my case: Taylor’s variance-mean model parameters. Also, even though the algorithm parameters were estimated from plants infested artificially with SWF within cages, it was validated with data collected in the field (i.e. Schuster 1998) and even provides a good fit to results obtained in greenhouses with single SWF infestations and with SWF infestations mixed with other whitefly species (i.e. Arno et al. 2006) (data not shown). Together, these findings show that the algorithm represents a significant advance towards the

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development of inexpensive methods to simulate experiments without having to spend resources in costly facilities, materials and labor.

Differences in leaf nodal preference between the silverleaf whitefly and the predator

Delphastus catalinae

In Chapter 3, I found that D. catalinae adults consistently visit the same leaves within tomato plants, but their distribution is significantly different from the distribution of SWF nymphs among leaves. I also found that the distribution of both SWF mortality rates due to predation and D. catalinae individuals among leaves are significantly different from the distribution of SWF nymphs. Leaf nodal position is a better predictor of the number of D. catalinae that visit a given leaflet than local SWF density. Together, these results contradict my prediction that the distribution of D. catalinae individuals within the plant will be associated with the distribution of their prey. In contrast, I found differences in these distributions that would lead to non-uniform intra-plant predator-prey spatial interactions. The common assumption of uniform local interactions between predator and prey individuals, therefore, does not hold, and spatially explicit models are needed to accurately model the SWF-D. catalinae system at plant and greater scales.

Relationship between predator foraging behavior and type of functional response

In Chapter 3, I found fundamental differences in the functional response of D. catalinae preying on SWF nymphs when foraging within leaflets and within plants. The

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functional response found within leaflets was type II and resembles that found in the laboratory, whereas the functional response within plants was type III. This result matches two of my predictions: 1) that a type II functional response will be observed in uniform, well-mixed predator-prey interactions; and 2) that a type III functional response will be observed in the patchy environment of the plant canopy. The emergence of a type

III functional response has resulted from co-aggregations between prey and predator populations in previous research (Nachman 2006a, Morozov 2010, Cordoleani et al.

2013). In my simulations, a type III functional response seemed to result from the interaction between prey aggregation and search behavior of predators, which would lead to such co-aggregations. In Chapter 4, I found that D. catalinae individuals adjust their search effort within tomato plants based on the number of prey in the area, which matches my prediction that predators will display ‘area-concentrated’ search in response to aggregated prey populations. I demonstrated that type III functional responses may emerge when predators that exhibit ‘area-concentrated search’ forage in heterogeneous environments, even though the functional response within homogeneous patch units is type II. This result is remarkable because it represents a simplified measure of the behavioral mechanism behind predator ‘area concentrated search’ in comparison with more explicit descriptions of behavioral rules for turning angle and walking speed (e.g.

Turchin 1998). Also, the fact that the type III functional response model explains predation rates in the SWF-D. catalinae system is significant because type III responses,

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in contrast to those of type II, enhance the stability properties of predator-prey systems, meaning that long-term coexistence between D. catalinae and the SWF is feasible.

Prediction of predation rates within plant structures

In Chapters 2 and 3, I presented data that confirmed all my predictions, except one: that the distribution of D. catalinae individuals among leaflets will be associated with the local density of SWF nymphs. Instead, I found that the distribution of predators was independent of the distribution of prey across leaves, and that leaf nodal position is a better predictor of the number of D. catalinae visits to a leaflet than the local SWF density. Despite these differences in predictors of D. catalinae distribution from the envisioned local SWF as the best predictor, significant overlap between the distributions of SWF and D. catalinae were found. In Chapter 4, I presented a model that integrates my results, including the prey density-independent movement patterns of D. catalinae among leaves, its prey density-dependent search effort and intra-plant SWF distribution. I showed that laboratory measurements of predation rates combined with observations of prey spatial distribution and predator behavior can be useful to infer predation rates at larger spatial scales, and that my model can capture changes in both shape and magnitude of functional responses that result from the increase in scale of observation. The model that I present in Chapter 4 also represents the first attempt to use a functional response model estimated through a laboratory experiment in an individual-based model to predict predation rates at larger spatial scales. Previous individual-based attempts modeled

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predator movement patterns and encounter rates explicitly within prey patches, a process that is complicated to measure efficiently even in laboratory settings (Turchin 1998). My approach uses a laboratory functional response to produce hourly predations rates, and the arrestment of predators to plants or leaflets is modeled as a function of initial SWF density within plants and leaflets, respectively. Predator patch-to-patch movement patterns, observed to be short flights within the plant canopy, are modeled explicitly using observations on its visit frequency to the different leaf nodal positions within plants. This model not only makes conventional laboratory functional responses useful to infer population-level consequences of the introduction of predators to new environments, but also reduces the amount of data required to feed predator-prey models that consider spatial effects.

Future directions

The model presented in Chapter 4 could be applied to fine-tune release procedures of biological control programs as long as information on the heterogeneity at the field spatial scale is available. This includes the SWF distribution among plants and the numerical response of D. catalinae individuals to SWF-infested plants. The model could be extended or become part of an expanded SWF-D. catalinae dynamic model through at least two different approaches. First, both plant and insect development can be incorporated into the model structure. Graph-based architectural crop models, such as

GREENLAB-tomato, explicitly simulate the structure of tomato plants, and include a

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dynamic phenology component driven by accumulation of degree-days (Dong et al.

2007). Similarly, SWF and predator oviposition along with life stage-specific development and mortality rates can be used to model population dynamics of predators and prey. Second, the model I presented may also be combined with existing analytical population models (e.g. Oaten 1977), or explicit geostatistical simulations of the SWF and D. catalinae distributions (e.g. Pérez M. et al. 2011, Zhao et al. 2011), to evaluate short-term outcomes of predator releases. Altogether, my model overcomes current limitations in the extrapolation of data collected in the laboratory to the field by incorporating observations at realistic spatial and temporal scales.

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Appendix A: R code of the algorithm to generate silverleaf whitefly distributions within tomato plants

170

require(splancs) require(bbmle) require(emdbook) setClass("plant", representation(leaves = "list", size = "numeric", nymphs = "numeric", nymphsTot = "numeric")) setGeneric(name = "initialize_pla", def = function(NoNym, nnodes) standardGeneric("initialize_pla")) setMethod("initialize_pla", definition = function(NoNym, nnodes) {

intn <- seq(1, nnodes) # Sequence of numbers representing leaf nodal positions

# Function to estimate the proportion of infested leaves (Equation 6) WR2a <- function(x) { 1 - (exp( - x * ((log(24.5 * x ^ (0.566))) * ((24.5 * x ^ (0.566) - 1) ^ ( - 1))))) }

# Estimation of the number of infested leaves funInf <- function(NoNym, nnodes) { Nympl <- NoNym / nnodes # No. of nymphs per leaf propinf <- WR2a(Nympl) # Estimate the proporition infested from equation 6 if (propinf < 0) propinf <- 0 ninf <- rbinom(1, prob = propinf, size = (nnodes - 1)) ninf <- ninf + 1 if (ninf > NoNym) ninf <- NoNym ninf }

# Selection of the infested nodes in the plant, funlocs <- function(ninf, nnodes) { sample(seq(1, nnodes), size = ninf, replace = FALSE, prob = dbetabinom(seq(0, (nnodes - 1)), prob = 0.61685, theta = 5.2661474, size = (nnodes - 1)) }

# Generation of number of nymphs per node NymAm <- function(nnodes, infnodes, NoNym) { a <- rmultinom(1, size = NoNym, prob = dbetabinom((infnodes - 1), 171

prob = 0.61685, theta = 5.2661474, size = (nnodes - 1))) matrix(c(infnodes, a), length(a[, 1]), 2) }

# Integrating the functions

algorithm <- function(NoNym, nnodes) { ninfes <- funInf(NoNym, nnodes) inflocs <- funlocs(ninfes, nnodes) Result <- NymAm(nnodes, inflocs, NoNym) Result <- Result[order(Result[, 1]), ] matrix(Result, ncol = 2) }

# Function for integration and output arrangement in a matrix. Genplants <- function(NoNym, nnodes) { plantV <- algorithm(NoNym, nnodes) plantVFin <- matrix(NA, nnodes, 2) for (i in 1 : length(plantV[, 2])) { plantVFin[plantV[i], ] <- plantV[i, ] } plantVFin[is.na(plantVFin[, 2]), 2] <- 0 plantVFin[, 1] <- seq(1, nnodes) plantVFin[, 2] }

# Estimation prey vector preysVec <- Genplants(NoNym, nnodes)

# Initilalization of "Leaf" objects that made tha "Plant" leaves <- list() for (i in 1 : nnodes) { leaves[[i]] <- initialize_lfa(nposition = intn[i], NoNym = preysVec[i]) } return(new("plant", leaves = leaves, size = nnodes, nymphs = preysVec, nymphsTot = NoNym)) } )

# The class "plant" has the following attributes: # leaves = List of "Leaf" objects # size = Total number of nodes 172

# nymphs = Number of nymphs in each leaf # nymphsTot = Total number of nymphs

# Code to initialize "Leaf" objects. setClass("leaf", representation(nposition = "numeric", nymphs = "numeric", nymphsTot = "numeric")) setGeneric(name = "initialize_lfa", def = function(nposition, NoNym) standardGeneric("initialize_lfa")) setMethod("initialize_lfa", definition = function(nposition, NoNym) {

modells <- c(3, 5, 5, 9, 9)

if (nposition <= 5) { nlls <- modells[nposition] } else { nlls <- 13 }

WR2b <- function(x) { 1 - (exp( - x * ((log(35.658 * x ^ (0.347))) * ((35.658 * x ^ (0.347) - 1) ^ ( - 1))))) }

pinflls <- function(nlls, NoNym) { if (NoNym == 0) { res <- 0 } else { nympll <- NoNym / nlls res <- WR2b(nympll) } res }

llInf <- function(nlls, NoNym) { propinf <- pinflls(nlls, NoNym) ninf <- rbinom(1, prob = propinf, size = (nlls - 1)) ninf <- ninf + 1 173

ninf }

assnymph <- function(ninf, NoNym) { rmultinom(1, size = NoNym, prob = dbetabinom(seq(0, (ninf - 1)), prob = 0.243, size = (ninf - 1), theta = 2.778) }

GenLefVec <- function(nlls, NoNym) { a <- rep(0, nlls) infs <- llInf(nlls, NoNym) inds <- sample(seq(1, nlls), infs) a[inds] <- assnymph(infs, NoNym) a }

nymVec <- GenLefVec(nlls, NoNym)

leaflets <- list() for (n in 1 : nlls) { leaflets[[n]] <- initialize_llt(Loc = n, nposition = nposition, NoNym = nymVec[n]) }

return(new("leaf", nposition = nposition, nymphs = nymVec, nymphsTot = NoNym)) } )

# Command to initialize a plant ‘pla_1’, where ‘nnymph’ is the number of silverleaf # whiteflies in the plant and ‘nnode’ is the number of fully-formed leaves in the plant pla_1 <- initialize_pla(nnymph, nnode)

174

Appendix B: R code of a spatially explicit individual-based model to scale-up a functional response estimated in the laboratory for the system silverleaf whitefly- Delphastus catalinae inhabiting tomato plants.

175

require(splancs) require(bbmle) require(emdbook) setClass("predator", representation(consumed = "numeric", consumed_tot = "numeric", lf_visited = "numeric", llt_visited = "numeric", sex = "logical", history = "matrix")) setGeneric(name = "initialize_predator", def = function(sex) standardGeneric("initialize_predator")) setMethod("initialize_predator", definition = function(sex) { return(new("predator", consumed = 0, consumed_tot = 0, lf_visited = 0, llt_visited = 0, sex = sex, history = matrix(NA, 1, 1))) } ) setGeneric(name = "move", def = function(predator, plant) standardGeneric("move")) setMethod("move", signature(predator = "predator", plant = "plant"), definition = function(predator, plant) { nvisited <- round(rnorm(1, mean = mik_llts(plant@nymphsTot), sd = coef(m2)[3]))

if (nvisited == 0) { return(predator) } else { vis_locs <- function (nvis, nnodes) { sample(seq(1, nnodes), size = nvis, replace = TRUE, prob = dbetabinom(seq(0, (nnodes - 1)), prob = coef(mod_predators)[1], theta = coef(mod_predators)[2], size = (nnodes - 1))) } predator@lf_visited <- vis_locs(nvis = nvisited, nnodes = plant@size)

for (i in 1: length(predator@lf_visited)){ predator@llt_visited[i] <- sample(seq(1, length(plant@leaves[[(predator@lf_visited[i])]]@nymphs)), size = 1) } return(predator) }

} ) 176

setGeneric(name = "eat", def = function(predator, plant) standardGeneric("eat")) setMethod("eat", signature(predator = "predator", plant = "plant"), definition = function(predator, plant) {

if (sum(predator@lf_visited) == 0) { return(list(predator, plant)) } else { times_e <- rep(NA, length(predator@lf_visited))

for (i in 1: length(predator@lf_visited)) { if (plant@leaves[[(predator@lf_visited[i])]]@nymphs[(predator@llt_visited[i])] > 0) { times_e[i] <- round(rtrunc(1, "exp", rate = (1 / mik_times(plant@leaves[[(predator@lf_visited[i])]]@nymphs[(predator@llt_visited[i])]) ), b = 15)) } else { times_e[i] <- 0 }

}

if (sum(times_e) == 0) { predator@consumed <- rep(0, length(predator@lf_visited)) return(list(predator, plant)) } else { consumed <- matrix(NA, length(predator@lf_visited), max(times_e)) sizeT <- rep(NA, length(predator@lf_visited)) for (n in 1: length(predator@lf_visited)) { i <- 0 sizeT[n] <- plant@leaves[[(predator@lf_visited[n])]]@nymphs[(predator@llt_visited[n])] if ((times_e[n] == 0) || (sizeT[n] == 0)) { consumed[n, 1] <- 0 } else { while (times_e[n] > 0) { i <- i + 1 consumed[n, i] <- round(rbinom(1, prob = prop.eaten.llt1b(sizeT[n]), size = sizeT[n]) / 10) if(consumed[n, i] > sizeT[n]) break sizeT[n] <- sizeT[n] - consumed[n, i] 177

if (sizeT[n] == 0) break times_e[n] <- times_e[n] - 1 } } }

predator@consumed <- rowSums(consumed, na.rm = TRUE)

for (i in 1: length(predator@lf_visited)) { plant@leaves[[(predator@lf_visited[i])]]@nymphs[(predator@llt_visited[i])] <- sizeT[i] }

predator@consumed_tot <- sum(predator@consumed) predator@history <- consumed

return(list(predator, plant)) } } } )

# Command to initialize a predator ‘pr1’, where sex = T is female and sex = F is male # no distinction has been made between sexes yet. pr1 <- initialize_predator(sex = T)

# Command to generate the leaflets and leaves where the predator has been foraging. # Movement is stored in a different predator object pr1a <- move(predator = pr1, plant = pla_1)

# Command to generate the numbers of prey eaten by predators # Movement and prey consumed is stored in a new predator object. pr1b <- eat(predator = preds_1[[i]], plant = pla_1)

178