Perturbations of the classical Lotka-Volterra system by behavioral sequences

Abstract

The complexity and the variability of parameters occurring in ecological dynamical systems imply a large number of equations.

Different methods, more or less successful, have been described to reduce this number of equations. For instance, in the paper of Auger and Roussarie (1993), the authors describe how to obtain a reduction by considering different time-scales. They consider a system which can be sub-divided into sub-systems such that the strengths of the intra-sub-systems interactions are much larger than those of the inter-sub-systems interactions. Using the Central Manifold Theorem, they obtain on the slow-manifold, a perturbation of the external dynamics by the internal dynamics.

In our paper, we apply the method of Auger and Roussarie to Lotka-Volterra's classical model. Then we bring to light two interesting consequences of this method. On the one hand, this method allows to remove one of the most important objection of the Lotka-Volterra's model. The dynamics obtained on the slow-manifold are a perturbation of the external dynamics, i.e. of the well-known center, and even if the external dynamics is degenerate, the perturbation cancels this degeneration. On the other hand, we can see that a variation, even a slight variation in some cases, of individual behaviour can modify global dynamics on the slow-manifold. We obtain Hopf-bifurcations on the central manifold by varying the intra-sub-systems parameters, so that it implies that a given stable focus becomes unstable and reciprocally. We briefly exhibit how the Perturbation Theory allows to describe the dynamics on the slow-manifold.