Dynamics of a population in two patches with dispersal. (English) Zbl 1427.92073
Summary: A two-dimensional discrete system of a species in two patches proposed by T.J. Newman, J. Antonovics and H.M. Wilbur [“Population Dynamics with a Refuge: Fractal Basins and the Suppression of Chaos”, Theor. Popul. Biology 62, No. 2, 121–128 (2002; doi:10.1006/tpbi.2002.1584)] is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton-Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.
MSC:
| 92D25 | Population dynamics (general) |
| 92D40 | Ecology |
| 39A30 | Stability theory for difference equations |
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