Evolutionary stability in Lotka-Volterra systems.
The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5. rcressma@wlu.ca
Ross
R
Cressman
József
J
Garay
D000818:Animals; D005075:Biological Evolution; D017753:Ecosystem; D005716:Game Theory; D008957:Models, Genetic; D010641:Phenotype; D011157:Population Dynamics; D013598:Systems Theory