Ecological stability, evolutionary stability and the ESS maximum principle
 T. L. Vincent,
 M. V. Van,
 B. S. Goh
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Since the fitness of each individual organism in a biological community may be affected by the strategies of all other individuals in the community, the essential element of a ‘game’ exists. This game is an evolutionary game where the individual organisms (players) inherit their strategies from continuous play of the game through time. Here, the strategies are assumed to be constants associated with certain adaptive parameters (such as sunlight conversion efficiency for plants or body length in animals) in a set of differential equations which describe the population dynamics of the community. By means of natural selection, these parameters will evolve to a set of strategy values that natural selection, by itself, can no longer modify, i.e. an evolutionarily stable strategy (ESS). For a given class of models, it is possible to predict the outcome of this evolutionary process by determining ESSs using an ESS maximum principle. However, heretofore, the proof of this principle has been based on a limited set of conditions. Herein, we generalize the proof by removing certain restrictions and use instead the concept of an ecological stable equilibrium (ESE). Individuals in a biological community will be at an ESE if fixing the strategies used by the individuals results in stable population densities subject to perturbations in those densities. We present both necessary and sufficient conditions for an ESE to exist and then use the ESE concept to provide a very simple proof of the ESS maximum principle (which is a necessary condition for an ESS). A simple example is used to illustrate the difference between a strategy that maximizes fitness and one that satisfies the ESS maximum principle. In general they are different. We also look for ESEs in Lotka—Volterra competition and use the maximum principle to determine when an ESE will be an ESS. Finally, we examine the applicability of these ideas to matrix games.
 Title
 Ecological stability, evolutionary stability and the ESS maximum principle
 Journal

Evolutionary Ecology
Volume 10, Issue 6 , pp 567591
 Cover Date
 199611
 DOI
 10.1007/BF01237708
 Print ISSN
 02697653
 Online ISSN
 15738477
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 evolutionarily stable strategies
 ecological stability
 evolutionary games
 Authors

 T. L. Vincent ^{(1)}
 M. V. Van ^{(1)}
 B. S. Goh ^{(2)}
 Author Affiliations

 1. Aerospace and Mechanical Engineering, University of Arizona, 85721, Tucson, AZ, USA
 2. Department of Mathematics, University of Western Australia, 6009, Perth, WA, Australia