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 dy namics of a system, describing the interactions between two populations, ge nerically called 'population N' and 'population P'. Our model may be applie d to prey-predator systems as well as to certain host-parasite or parasitoi d systems. A situation in which population P affects, the spatial distribut ion of population N is considered. We deal with a heterogeneous environment composed of two spatial patches: population P lives only in patch I, 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-popul ations and can migrate according to different migration laws. We make the a ssumption 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 e volution of the total populations, at a slow time scale. At first, a migrat ion 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 co nditions, population P always gets extinct. Then, the same model, but in wh ich individuals belonging to population N leave patch 1 proportionally to p opulation 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 additio n of an aggregative behavior of population N on patch 1. This leads to a mo re complicated situation in which, according to initial conditions, the glo bal system is described by two different aggregated models. Under certain c onditions on parameters a stable limit cycle occurs, leading to periodic va riations of the total population densities, as well as of the local densiti es on the spatial patches. (C) 1999 Elsevier Science Inc, Ail rights reserv ed.