Abstract: I will describe a model for populations that occupy several geographically separated patches of habitat, one for which there is no limit on the number of patches that can be occupied. Colonisation and extinction events are assumed to occur in distinct successive phases. When the expected number of patches colonised is proportional to the number currently occupied, the occupancy process is a branching process. However, allowing more general dependence on the number currently occupied, and introducing a threshold, permits a degree of regulation in the colonisation process. We present a large of large numbers for the occupancy measured relative to the threshold, which identifies an approximating deterministic trajectory, as well as a central limit law for fluctuations about that trajectory. We shall see that equilibrium behaviour is richer and more interesting than for standard finite-patch models, because now the limiting deterministic model can exhibit the full range of long-term behaviour, including limit cycles, and even chaos.
Acknowledgement: This work is support by the ARC
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Last modified: 15th June 2018