On a model for interference between searching insect parasites

Abstract

The purpose of this paper is to study a stochastic model which assesses the effect of mutual interference on the searching efficiency in populations of insect parasites. By looking carefully at the assumptions which govern the model, I shall explain why the searching efficiency is of the same order as the total number, N, in the population, a conclusion which is consistent with the predictions of population biologists; previous studies have reached the conclusion that the efficiency is of order \/~N . The major results of the paper establish normal approximations for the distribution of the numbers of active parasites. These are valid at all stages of the process, in particular the non-equilibrium phase, where explicit analytic formulae for the state-probabilities are unavailable.

A model for a spatially structured metapopulation accounting for within patch dynamics

“…We will apply Theorem 3.1 of Pollett [24] which allows us to approximate the path of our process by the solution to a system of differential equations. To do this we first need to establish that our model is density dependent in the sense of Kurtz [23], or at least asymptotically density dependent [24].…”

“…To do this we first need to establish that our model is density dependent in the sense of Kurtz [23], or at least asymptotically density dependent [24].…”

“…, J. Therefore, the family of processes indexed by the population ceiling N is asymptotically density dependent according to Definition 3.1 of [24]. Next we apply Theorem 3.1 of [24], the analogue of Theorem 3.1 of Kurtz [23] for asymptotically density dependent families of processes.…”

“…Therefore, the family of processes indexed by the population ceiling N is asymptotically density dependent according to Definition 3.1 of [24]. Next we apply Theorem 3.1 of [24], the analogue of Theorem 3.1 of Kurtz [23] for asymptotically density dependent families of processes. The conditions of this theorem are fulfilled as f N (x, l) is bounded on E for all N and l and is nonzero for only finitely many l. Recall that λ ij is the proportion of individuals emanating from patch i who are destined for patch j. Thus,…”

“…In this case, the stationary distribution would necessarily assign all its probability mass to the extinction state, and thus would not provide useful information about any quasi-stationary regime (being a common feature of metapopulation models [22]). Instead, we analyse this model by determining a simpler approximating differential equation based on the work of Kurtz [23] and Pollett [24].…”