| Abstract | A general N-species Lotka-Volterra system is considered for which, in absence of interactions, each species is governed by a logistic equation. Interactions among species take place in the form of competition, which also includes adaptive abilities through a (short term) memory effect. As a consequence the dynamics of the model is governed by a system of N^2 non-linear ordinary differential equations. The existence of classes of invariant subspaces, related to symmetries, allows the introduction of reduced models of four equations, where N appears as a parameter, which are proven to account for existence and stability of the equilibria. Reduced models are found effective also in describing the transitions to time-dependent regimes. Such regimes exhibit remarkable properties of synchronization in the cases of both periodic and chaotic behavior, with multiplicity of attractors. In systems with few species, increasing the adaptation characteristic time, the strange attractors merge and synchronization is lost. On the contrary in larger communities multiplicity of attractors and synchronization persist also for very large adaptation time. |
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