A reduced model for spatially structured predator–prey systems with fast spatial migrations and slow demographic evolutions

Abstract

We consider a spatially structured predator-prey model where fast migrations occur inside a given spatial domain, while slow predator-prey interactions prescribe the demographic evolution. The unknowns of our model are the numbers of predators and prey at each time t and each site x of the domain. In the idealized limit where migrations are infinitely fast, we show one can approximate the global dynamics using the mere two unknowns corresponding to the total number of preys and predators, irrespective of their respective spatial repartition. Besides, the error term induced by this approximation can be made exponentially small with respect to the natural asymptotic parameter. In doing so, we completely characterize how migrations do modify both the qualitative and quantitative properties of the global demography. Our analysis relies on a convenient version of the central manifold theorem, in conjunction with a spectral gap estimate on the involved migration operator.