Does Migration Stabilize Local Population Dynamics? Analysis of a Discrete Metapopulation Model MATS GYLLENBERG, GUNNAR SGDERBACKA AND STEFAN ERICSSON zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH Department of Applied M athematics, Luleii University of Technology, S-95187 Luleii, Sweden Receioed 3 August 1992; revised 25 November 1992 ABSTRACT A discrete model for a metapopulation consisting of two local populations connected by migration is described and analyzed. It is assumed that the local populations grow according to the logistic law, that both populations have the same emigration rate, and that migrants choose their new habitat patch at random. Mathematically this leads to a coupled system of two logistic equations. A complete characterization of fixed point and two-periodic orbits is given, and a bifurcation analysis is performed. The region in the parameter plane where the diagonal is a global attractor is determined. In the symmetric case, where both populations have the same growth rate, the analysis is rigorous with complete proofs. In the nonsym- metric case, where the populations grow at different rates, the results are obtained numerically. The results are interpreted biologically. Particular attention is given to the sense in which migration has a stabilizing and synchronizing effect on local dynamics. 1. INTRODUCTION The classical models of single-species population growth such as Malthus’s law of exponential growth and the logistic model of Verhulst are based on the assumption of homogeneous interactions among individuals. Most natural populations are not homogeneous but have a hierarchical structure. Several local populations occupying different habitat patches constitute a metapopulation. The local populations of a metapopulation are connected by migration. The first mathematical metapopulation model was introduced by Levins [15, 161. It is a simple phenomenological model consisting of a single ordinary differential equation describing how the fraction of occupied patches changes in time. The Levins model ignores local dynamics; in particular it tacitly assumes that emigration and immigra- MATHEMTICAL BIOSCIENCES 118:25-49 (1993) OElsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas, New York, NY 10010 25 0025-5564/93/$6.00