Complex Features in Lotka-Volterra Systems with Behavioral Adaptation
politecico of torino
Department of Mathematics, University of Lecce
Last modified: August 16, 2006
Lotka-Volterra systems are a large class of models for interaction among species. Depending on such interactions competition, cooperation or predator-prey situations can occurr, giving rise to further classifications. The dynamics depends on parameters intrinsic to the species, tipically growth rate and carrying capacity, and on the interaction coefficients, which however are often more difficult to specify. We focus here on competition among species and, differently from the classical case, we consider for them a kind of “adaptive skills”: the ability to compete is proportional to the average number of contacts between species in their past, however with a “fade-out” memory effect. For the general case of N-species a system of NxN nonlinear ordinary differential equations is obtained, investigated according to bifurcation theory. The investigation takes advantage of the existence of reduced systems, where N appears as a parameter, accounting for all the equilibria and their stability. Such results are very useful in moving toward higher dimensional cases, for which not many results are available. Adaptation is shownn to be a mechanism able to establish the appearance of a variety of behaviors, different from equilibria, as distinct kinds of oscillations and chaotic patterns. In particular we provide here an example of species coexistence in the form of complicated alternance between chaotic behavior and periodic one, in both cases with multiplicity of attractors.