In this paper, a metapopulation model composed of patches distributed in two geographic scales is proposed in order to study the stability of the synchronous dynamics. The first scale is a metapopulation consisting of equal patches while the second scale is composed of an arbitrary number of metapopulations. During each time step, we assume that there are three processes involved in the population dynamics: a) the local dynamics, which consists of reproduction and survival and depends only on function chosen to calculate the density of each patch before migration; b) the dispersal of individuals between the patches of the first scale; and c) the movement between the metapopulations. We examine two kinds of synchronizations: when each patch has the same density (both scales are synchronized), and when each metapopulation has the same density patches with its patches not necessary synchronized, that is, only the second scale is synchronized. A stability criterion is obtained based on the computation of the transversal Lyapunov number of attractors on the synchronous invariant manifold. For case where both scales are synchronized, the criterion is determined by two terms: the Lyapunov number that depends only on the local dynamics of each isolated patch and a term that depends on the migration process. For the case where only the second scale is synchronized, the transversal Lyapunov number is calculated numerically.
Metapopulation; synchronization; Lyapunov number
