Aggregated distributions in models for patchy populations

We investigate a model describing immigration, birth, and death of parasites on a dynamic host population. The model can also be interpreted as describing a herbivore population distributed on discrete patches of vegetation. We derive differential equations for the total number of hosts/patches and the mean number of parasites/herbivores per host/patch. The equations explicitly involve the variance-to-mean ratio of the distribution. It is shown that the positive equilibrium is stable if and only if the variance-to-mean ratio as a function of the mean increases with increasing mean. Thus aggregation of the parasites alone is not sufficient to stabilize the system; it is rather the density-dependent increase in parasite mortality due to a higher aggregation at higher mean parasite loads that causes stability. From this it follows that introducing a distribution with a constant clumping parameter into the model artificially stabilizes the steady state. We derive a three-dimensional model based on an assumption about the form of the distribution of the parasites on the hosts, but without introducing additional parameters into the model. We compare stability results for this model for different types of aggregated distributions and show that the underlying distribution determines the qualitative results about the stability of the equilibrium.