Fast oscillating migrations in a predator-prey model

The aim of this paper is to give a method which permits us to describe how individual properties can emerge at the population level, in population dynamics. We consider interacting populations. In order to take into account the spatial or behavioral heterogeneity, we subdivide each population into subpopulations. A given subpopulation corresponds to those individuals having the same behavior and who are in a homogeneous environment. Further more, we assume that the migration process is faster than the growth and interaction processes. Therefore, we must study models with many variables coupled together into large scaled differential systems. Firstly, our method permits us to reduce these complex systems into simpler ones. which will be called reduced systems. Secondly, these reduced systems give the population dynamics and contains informations about individuals' behavior, that is we can explain how individual dynamics emerge in the population dynamics. We already investigated the case where the fast dynamics reach an equilibrium in previous works. In this paper, we are interested in those models where the fast dynamics oscillates because it is ecologically relevant.