Behavioral choices based on patch selection: a model using aggregation methods

The aim of this work is to study the influence of patch selection on the dynamics of a system describing the interactions between two populations, generically called 'population N' and 'population P'. Our model may be applied to prey-predator systems as well as to certain host-parasite or parasitoid systems. A situation in which population P affects the spatial distribution of population N is considered. We deal with a heterogeneous environment composed of two spatial patches: population P lives only in patch 1, while individuals belonging to population N migrate between patch 1 and patch 2, which may be a refuge. Therefore they are divided into two patch sub-populations and can migrate according to different migration laws. We make the assumption that the patch change is fast, whereas the growth and interaction processes are slower. We take advantage of the two time scales to perform aggregation methods in order to obtain a global model describing the time evolution of the total populations, at a slow time scale. At first, a migration law which is independent on population P density is considered. In this case the global model is equivalent to the local one, and under certain conditions, population P always gets extinct. Then, the same model, but in which individuals belonging to population N leave patch 1 proportionally to population P density, is studied. This particular behavioral choice leads to a dynamically richer global system, which favors stability and population coexistence. Finally, we study a third example corresponding to the addition of an aggregative behavior of population N on patch 1. This leads to a more complicated situation in which, according to initial conditions, the global system is described by two different aggregated models. Under certain conditions on parameters a stable limit cycle occurs, leading to periodic variations of the total population densities, as well as of the local densities on the spatial patches.