Emergence of Population Growth Models: Fast Migration and Slow Growth

We present aggregation and emergence methods in large-scale dynamical systems with different timescales. Aggregation corresponds to the reduction of the dimension of a dynamical system which is replaced by a smaller model for a small number of global variables at a slow timescale. We study the couplings between fast and slow dynamics leading to the emergence of global properties in the aggregated model. First, we study the case of a single population in a patchy environment. Growth rates are assumed to be linear on each patch. Individuals can migrate from one patch to another at a fast timescale. We choose different density dependent migration processes. In each case, we use aggregation methods to obtain the corresponding growth equation for the total density of the population at a slow timescale. We look for particular density dependent migration processes leading to an aggregated logistic-like equation. Second, we study the case of two interacting populations. A particular choice of density dependent migrations leads to an aggregated competition model.