Interaction between habitat quality and an Allee-like effect in metapopulations

Many species exist as a collection of local populations occupying spatially distinct habitat patches. Such a collection of local populations is called a metapopulation. Metapopulations are constantly changing due to the processes of extinction and colonisation occurring at each habitat patch. The primary focus in the study of metapopulations is to determine if the metapopulation will persist and, if so, at what level. Mathematical models of metapopulations provide ecologists with tools for better understanding the dynamics of the metapopulation. A useful class of metapopulation models is the stochastic patch occupancy models (SPOMs). The characterising feature of a SPOM is that only the presence/absence of a population at each habitat patch is modelled. Classical metapopulation models such as Levins's model and the stochastic logistic model assume homogeneity of habitat quality throughout the metapopulation. However, it is known that species distribution patterns are effected by the quality of habitat available. Furthermore, it is necessary to incorporate habitat quality into the modelling in order to study the effect of habitat degradation and destruction on the persistence of the metapopulation. A review of some models that attempt to better reflect the ecological reality including incorporating variation in habitat quality is provided by Hanski and Ovaskainen [Theoretical Population Biology 64, 119-127]. See also Gyllenberg and Hanski [Theoretical Population Biology 52, 198-215] who study a differential equation metapopulation model that incorporates variation in habitat quality and use this model to examine the effect of habitat degradation and destruction. In addition to habitat degradation, the persistence of a metapopulation is affected by its dynamical properties. One such property is called an Allee-like effect. This term is borrowed from population biology where the Allee effect refers to populations exhibiting a critical threshold below which the population goes extinct. For metapopulations, an Allee-like effect refers to a metapopulation exhibiting a similar threshold behaviour. Amarasekare [The American Naturalist 152, 298-302] summarises some of the evidence supporting the operation of an Allee-like effect in real metapopulations and proposes a modification of the Levins's model which exhibits this phenomenon. Note that Amarasekare's model does not allow variation in habitat characteristics. In this paper, we examine the effect of habitat degradation on a metapopulation exhibiting an Allee-like effect using the metapopulation model introduced by McVinish and Pollett [Advances in Applied Probability 42, 1172-1186]. This model incorporates variation in habitat quality between patches by allowing the local survival probabilities to vary between patches (but not in time). The model can also incorporate an Allee-like effect by imposing certain conditions on the colonisation process. The resulting discrete time Markov chain is difficult to analyse directly, but can be approximated by a deterministic process when the number of habitat patches is large. This approximation is used to study the effect habitat degradation has on the persistence the metapopulation. We show that even a small amount of habitat degradation in metapopulations exhibiting an Allee-like effect can cause a metapopulation with a high level of persistence to go extinct. This can be contrasted with a metapopulation that does not exhibit an Allee-like effect where a small change to the habitat quality will result in only a small change to the level of persistence of the metapopulation. We conclude that metapopulations exhibiting an Allee-like effect are in much greater need of protection from habitat degradation and destruction