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Metapopulation models have been used to better understand the conditions necessary for the persistence of the metapopulation. In this paper, we study a stochastic patch occupancy model that incorporates variation in quality and connectivity of the habitat patches. Two important assumptions are imposed in our analysis. Firstly, the distance between patches has a special form. This amounts to assuming that migrating individuals follow certain pathways. Secondly, the area of the habitat patches is assumed to scale with the number of patches in the metapopulation. Under these assumptions, a deterministic limit is obtained as the number of patches goes to infinity. Using the deterministic limiting process, a condition for persistence of the metapopulation is derived. [ABSTRACT FROM AUTHOR]

In this paper we study the well-posedness of different models of population of leaky integrate-and-fire neurons with a population density approach. The synaptic interaction between neurons is modeled by a potential jump at the reception of a spike. We study populations that are self excitatory or self inhibitory. We distinguish the cases where this interaction is instantaneous from the one where there is a repartition of conduction delays. In the case of a bounded density of delays both excitatory and inhibitory population models are shown to be well-posed. But without conduction delay the solution of the model of self excitatory neurons may blow up. We analyze the different behaviours of the model with jumps compared to its diffusion approximation. [ABSTRACT FROM AUTHOR]