How do cross-migration models arise?

In this paper we present a general method to derive spatio-temporal population models mechanistically. We consider a system of multiple species living in a patchy habitat in which the local population of each species consists of some behavioural groups. We then formulate a continuous-time model where a small positive parameter is present, measuring the time scale of behavioural transitions relative to that of giving birth, death and migration among patches. By the singular perturbation method the model is reduced to a lower dimensional one in which the migration terms are, in general, nonlinear and related to the reaction terms describing the local dynamics. Two examples demonstrating the emergence of cross-migration models, i.e., the models in which the per-capita migration rate of one species depends on the density of some other species, are given.