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Detection and Assessment of Ecosystem Regime Shifts from Fisher Information

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Detection and Assessment of Ecosystem Regime Shifts from Fisher Information Copyright © 2008 by the author(s). Published here under license by the Resilience Alliance. Karunanithi, A. T., H. Cabezas, R. Frieden, and C. Pawlowski 2008. Detection and assessment of ecosystem regime shifts from Fisher information. Ecology and Society 13(1): 22. [online] URL: http://www. ecologyandsociety.org/vol13/iss1/art22/ Research Detection and Assessment of Ecosystem Regime Shifts from Fisher Information Arunprakash T. Karunanithi 1, Heriberto Cabezas 1, B. Roy Frieden 2, and Christopher W. Pawlowski 3 ABSTRACT. Ecosystem regime shifts, which are long-term system reorganizations, have profound implications for sustainability. There is a great need for indicators of regime shifts, particularly methods that are applicable to data from real systems. We have developed a form of Fisher information that measures dynamic order in complex systems. Here we propose the use of Fisher information as a means of: (1) detecting dynamic regime shifts in ecosystems, and (2) assessing the quality of the shift in terms of intensity and pervasiveness. Intensity is reflected by the degree of change in dynamic order, as determined by Fisher information, and pervasiveness is a reflection of how many observable variables are affected by the change. We present a new robust methodology to calculate Fisher information from time series field data. We demonstrate the use of Fisher information to detect regime shifts on a model for a shallow lake. Next, we use Fisher information to analyze marine ecosystem response to physical changes using real time-series data of a coastal marine ecosystem, the North Pacific Ocean. Key Words: ecosystems; Fisher information; marine ecosystem; regime shifts; resilience; sustainability. INTRODUCTION in a particular dynamic regime with a particular degree of dynamic order. But they are capable of The study of regime shifts in complex dynamic shifting abruptly to different dynamic regimes with systems has been quickly gaining importance different degrees of dynamic order. Further, during because of the implications that regime shifts can the transition between two different regimes, there have for modern society. Of particular importance is typically a temporary loss of dynamic order. to the United States Environmental Protection Afterwards, the system might shift to a different Agency (U.S. EPA) is the fact that sustainability is regime with either higher or lower dynamic order. about finding and maintaining a regime suitable to Regime shifts can be defined as substantial, long- the social and economic development of humans term reorganizations of complex systems such as without major consequences to the environment, e. societies, ecosystems, and climate (Carpenter and g., land, atmosphere, and ocean, and the ecological Brock 2006). Regime shifts have been documented services on which people depend. Sustainability, and studied extensively in ecological systems applies to complex integrated systems comprising (Steele 1998, Scheffer et al. 2001, Folke et al. 2004, human components, e.g., society, economy, etc., Carpenter and Brock 2006, Mayer et al. 2006), and natural components such as ecosystems, social systems (Wood and Doan 2003, Kinzig et al. climate, etc. Kates and Parris (2003) provide an 2006), and climate systems (Miller et al. 1994). insightful review on the changes and long-term Quantifying and characterizing regime shifts is trends in socioeconomic, environmental, and critical for understanding dynamics in ecosystems. ecological systems and their relation to Regime shifts, in the context of ecosystem sustainability transition. These complex, diverse resilience, could be a shift to an alternative basin of systems have a common characteristic, namely attraction by a stochastic event (Scheffer and dynamic order, when looked at from an information Carpenter 2003). In ecology, some examples of point of view, in that they are orderly and well- regime shifts include eutrophication of lakes and organized dynamic systems. Over time, they exist coastal oceans, shifts among grassy and woodland 1U.S. Environmental Protection Agency, Office of Research and Development, National Risk Management Research Laboratory, Cincinnati, OH 45268, 2University of Arizona, Optical Sciences Center, Tucson, AZ 85721, 3RD Zande and Associates, Cincinnati, OH 45249 Ecology and Society 13(1): 22 http://www.ecologyandsociety.org/vol13/iss1/art22/ cover types in rangelands, degradation of coral sustainable regime hypothesis that formally relates reefs, and regional climate change (Carpenter and Fisher information, dynamic order, and regime Brock 2006). Quantification and classification of shifts to sustainability. We believe that this work regime shifts is extremely difficult because they will improve our understanding of dynamic regimes involve multiple causes; and hence, one must track and regime shifts in ecosystems and other complex multiple variables simultaneously over a long systems and help establish Fisher information as a period of time. Not only is the study of regime shifts new and very powerful tool for sustainable after its occurrence important but prediction of environmental management and in policymaking. regime shifts through leading indicators is also critical. Recently there has been some significant work on the development of leading indicators for METHODS regime shifts (Kleinen et al. 2003, Carpenter and Brock 2006, van Nes and Scheffer 2007). Methods Theory used for detection of dynamic regime shifts work very well on models, but the development of The statistician Ronald Fisher (Fisher 1922) methods for real field data are lacking because of developed a measure of indeterminacy, now called the large number of variables involved, noise in the Fisher information. This is the information about field data, and sparseness of the datasets. There is a the value of a parameter that is present in data y. great need for indicators of regime shifts, Specifically, the Fisher information I(θ) is defined particularly methods that are applicable to real by, ecosystems. In this context, it would be of immense value to develop methods having sound theoretical basis in order to detect and quantitatively characterize dynamic regimes and regime shifts. (1) Fisher information (Fisher 1922) is a fundamental quantity from which many known laws of nature can be derived (Frieden 2004). Our form of Fisher information has been used as a measure of dynamic θ order (Fath et al. 2003, Mayer et al. 2006). In this Here p(y| ), often called the ‘likelihood law,’ is the probability density for obtaining a particular value work we propose this latter form of the Fisher θ information along with a new robust calculation of y in the presence of . Starting with Fisher’s work, methodology as a measure of dynamic order in the Fisher information concept has been used as a complex systems. The new method for calculating unifying principle of physical laws (Frieden 2004). Fisher information from time series data is robust It also has been used as a measure of dynamic order and applicable to sparse data sets with noise. We in complex systems (Fath et al. 2003, Mayer et al. illustrate the principle with application to the 2006). detection and quantification of dynamic regime shifts in lake and marine ecosystems. Through When used as a measure of order in an environmental or ecological system, numerical Fisher Information, we also provide ways to classify θ dynamic regime shifts in terms of intensity and values of the Fisher information I( ) are required. pervasiveness. Intensity is reflected by the degree However, Eq. 1, as it stands, is impractical for these purposes. This is because computing I(θ) requires of change in dynamic order, and pervasiveness is a θ reflection of how many observable variables of the numerical knowledge of the derivative dp/d , and this depends upon the numerical value of the system are affected by the change. These θ quantitative classifications of regime shifts are unknown parameter . We develop below a means of addressing this issue as follows. Let the extremely important for ecosystem management. θ θ Furthermore, regime shifts often have large parameter be the ergodic mean = <y> of y given ecological and economical consequences, e.g., by losses of water quality, declining of fish stocks, breakdowns of dryland agriculture (Carpenter and Brock 2006). Hence it is clear that, detecting and accessing the quality of dynamic regimes and regime shifts is very critical in identifying sustainable regimes. Hence, we provide a Ecology and Society 13(1): 22 http://www.ecologyandsociety.org/vol13/iss1/art22/ (2) (5) where the integration is conducted over time T This expression has the benefit of eliminating the included in all data observations. Thus, I(θ) is division by p(s),which can be numerically specifically the information about the ergodic mean problematic if p(s) happens to be a small number. of the data. Define a new variable s = y-<y> The expression in Eq. 5 will be used in all our representing fluctuations in y from the mean. For calculations. the many cases common in environmental systems, it can be assumed to a reasonable approximation Order in dynamic systems is conceptually that within any particular dynamic regime the associated with repeatability of observations. fluctuations s are independent of the mean <y>, i. Hence, for a dynamic system that is perfectly e., the fluctuations about the mean are independent orderly, repeated observations of the variables over of the value of the mean. Note, however, that as a time yield
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