Stochastic Identification of Stability of Competitive Interactions in Ecosystems

The problem of finding an optimum within a set of possibilities that represent the varying successfulness of numerous subjects competing with one another is highly relevant in the field of ecosystem interactions. We propose a method for solving this problem by the application of the Nash equilibrium concept, which is frequently used in ecology. The proposed model is based on the transformation of the initial payoff vectors of subjects that interact in different situations into a statistical set of symmetrical game matrices that consist of permutations of payoff values. The equilibrium solution is expressed as values of the probability of Nash equilibrium occurrence with uniform distribution over all possible permutations based on uncertainty of positions of payoff values in the matrix. We assume that this equilibrium solution provides information on the distribution of the degree of stability among individual situations and interacting subjects. In this paper, we validate this assumption and demonstrate its application to a dataset that represents interspecies interactions in plant ecology. We propose that the use of the Nash equilibrium in the analysis of datasets formalized according to the Pareto optimality scheme is applicable in numerous other contexts.

[1]  R. Gibbons.  Game theory for applied economists , 1992 .

[2]  E. David Ford,et al.  Informed multi-objective decision-making in environmental management using Pareto optimality , 2007 .

[3]  Peter Hammerstein,et al.  Game Theory in the Ecological Context , 1983 .

[4]  S. McNaughton.  Diversity and Stability of Ecological Communities: A Comment on the Role of Empiricism in Ecology , 1977, The American Naturalist.

[5]  Alex Coram,et al.  Evolutionary Stable Strategies , 2001 .

[6]  P. Pardalos,et al.  Pareto optimality, game theory and equilibria , 2008 .

[7]  M. Rejmánek,et al.  Connectance in real biotic communities and critical values for stability of model ecosystems , 1979, Nature.

[8]  J. Apaloo,et al.  Revisiting matrix games: the concept of neighborhood invader strategies. , 2006, Theoretical population biology.

[9]  E. David Ford,et al.  Using Multicriteria Analysis of Simulation Models to Understand Complex Biological Systems , 2011 .

[10]  J. Lepš.  Nutrient status, disturbance and competition: an experimental test of relationships in a wet meadow , 1999 .

[11]  H. Young,et al.  Handbook of Game Theory with Economic Applications , 2015 .

[12]  Joel s. Brown,et al.  Games roots play: effects of soil volume and nutrients , 2008 .

[13]  D. Tilman.  Biodiversity: Population Versus Ecosystem Stability , 1995 .

[14]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[15]  M. Rejmánek,et al.  Community stability, complexity and species life history strategies , 1982, Vegetatio.

[16]  L. Shapley.  A Value for n-person Games , 1988 .

[17]  Crow White,et al.  The value of coordinated management of interacting ecosystem services. , 2012, Ecology letters.

[18]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[19]  J. Guckenheimer,et al.  Random evolutionarily stable strategies. , 1978, Theoretical population biology.

[20]  T. Vincent,et al.  Evolutionary Game Theory as a Framework for Studying Biological Invasions , 2011, The American Naturalist.

[21]  J. Downing,et al.  Biodiversity and stability in grasslands , 1996, Nature.

[22]  B. Stengel,et al.  COMPUTING EQUILIBRIA FOR TWO-PERSON GAMES , 1996 .

[23]  Thomas L. Vincent,et al.  Evolutionary game theory: ESS, convergence stability, and NIS , 2009 .

[24]  L. Shapley,et al.  Stochastic Games* , 1953, Proceedings of the National Academy of Sciences.

[25]  Pedro Beja,et al.  Optimization of Landscape Services under Uncoordinated Management by Multiple Landowners , 2014, PloS one.

[26]  Jacob Johansson,et al.  Game theory sheds new light on ecological responses to current climate change when phenology is historically mismatched. , 2012, Ecology letters.

[27]  M. Nowak,et al.  Evolutionary Dynamics of Biological Games , 2004, Science.

[28]  L. Shapley,et al.  Equilibrium points in games with vector payoffs , 1959 .

[29]  G. McNickle,et al.  Game theory and plant ecology. , 2013, Ecology letters.

[30]  J. Schellberg,et al.  The Rengen Grassland Experiment: Plant species composition after 64 years of fertilizer application , 2007 .

[31]  J. Lovett-Doust,et al.  Plant strategies, vegetation processes, and ecosystem properties , 2002 .

[32]  Robert M. May,et al.  Dispersal in stable habitats , 1977, Nature.

[33]  J. Fletcher.  Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics , 2006, Journal of Mammalian Evolution.

[34]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[35]  S. Hart,et al.  Handbook of Game Theory with Economic Applications , 1992 .

[36]  Robert J. Weber,et al.  Probabilistic Values for Games , 1977 .

[37]  Matteo Convertino,et al.  Portfolio Decision Analysis Framework for Value-Focused Ecosystem Management , 2013, PloS one.

[38]  Michael A. Huston,et al.  Local processes and regional patterns : appropriate scales for understanding variation in the diversity of plants and animals , 1999 .

[39]  T. Vincent,et al.  A theory for the evolutionary game , 1987 .

[40]  John C. Harsanyi,et al.  Games with Incomplete Information Played by "Bayesian" Players, I-III: Part I. The Basic Model& , 2004, Manag. Sci..

[41]  D. Avis,et al.  Enumeration of Nash equilibria for two-player games , 2010 .

[42]  J. M. Smith,et al.  The Logic of Animal Conflict , 1973, Nature.

[43]  J. Harsanyi.  Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points , 1973 .