Effects of ecological differentiation on Lotka-Volterra systems, for species with behavioural adaptationand variable growth rates

We study the properties of a n2-dimensional Lotka–Volterra system describing competing species that include behaviorally adaptive abilities. We indicate as behavioral adaptation a mechanism, based on a kind of learning, which is not viewed in the evolutionary sense but is intended to occur over shorter time scales. We consider a competitive adaptive n species Lotka–Volterra system, n ⩾ 3, in which one species is made ecologically differentiated with respect to the others by carrying capacity and intrinsic growth rate. The symmetry properties of the system and the existence of a certain class of invariant subspaces allow the introduction of a 7-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the complete system. The reduced model is effective also in describing the time-dependent regimes for a large range of parameter values. The case in which one species has a strong ecological advantage (i.e. with a carrying capacity higher than the others), but with a varying growth rate, has been analyzed in detail, and time-dependent behaviors have been investigated in the case of adaptive competition among four species. Relevant questions, as species survival/exclusion, are addressed focusing on the role of adaptation. Interesting forms of species coexistence are found (i.e. competitive stable equilibria, periodic oscillations, strange attractors).