Lotka-Volterra's model and migrations: Breaking of the well-known center

This paper is devoted to the study of the effect of individual behavior on the Lotka-Volterra predation. We assume that the individuals have many activities in a day for example. Each population is subdivided into subpopulations corresponding to different activities. In order to be clear, I have chosen the case of two activities for each population. We assume that the activities change is faster than the other processes (reproduction, mortality, predation…). This means that we consider population in which the individuals change their activities many times in a day while the reproduction and the predation effects are sensible after about ten days, for example. We use the aggregation method developed in 1] to obtain the global dynamics. Indeed, we start with a micro-model governing the micro-variables, which are the subpopulation densities; the aggregation method permits us to obtain a simpler system governing the macro-variables, which are the global population densities. Furthermore, this method allows us to observe emergence of the dynamics. Indeed, the method implies that the dynamics of the micro-system is close to an invariant manifold after a short time. We show that the dynamics on this manifold is a perturbation of the well-known center of the Lotka-Volterra model. Finally, we prove that a weak change of behavior can lead to a subcritical Hopf bifurcation in the global dynamics.