Symmetry Breaking Effects on Equilibria and Time Dependent Regimes in Adaptive Lotka -Volterra Systems

We study the properties of a n2-dimensional Lotka-Volterra system describing competition among n different species with adaptive skills, i.e. whose interaction coefficients are time averages of the species level of interaction over their past. Starting by the case of adaptive competition among species all having the same carrying capacities, we focus our attention on the model obtained on perturbing the carrying capacity of a fixed species, which is made more or less disadvantaged. We prove the existence of a certain class of invariant subspaces and introduce a seven-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the system. The relevance of this reduced model to the complete one has also been found when the time dependent regimes have been investigated. Ecologically relevant questions, i.e. species survival and the time dependent behavior of the system have also been analyzed focusing on the role of behavioral adaptation. In particular, we have found that competitive exclusion cannot occur but coexistence is possible in various forms (i.e. competitive stable equilibria and different periodic oscillations).