Steady-state analysis in a model for population diffusion in a multi-patch environment

We present a new fully spatially structured PDE metapopulation model for predator–prey dynamics in $d\le 3$ space dimensions. A nonlinear reaction–diffusion system of Rosenzweig–MacArthur form models predator–prey dynamics in two ‘high’ quality patches embedded in a ‘low’ quality subdomain, where species can diffuse, convect and die. Our model substantially generalizes and improves earlier fully structured metapopulation models. After a nondimensionalization procedure, in order to approximate the metapopulation model we present a fully discrete Galerkin finite element method in two space dimensions, which is a generalization of the finite element method analyzed in a previous single patch predator–prey model. The numerical solutions are illustrated for some test cases using MATLAB. Numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch. In an appendix we also give an outline for the proof of the well-posedness of the model.

A mathematical model is proposed to study the role of supplementary self-renewable resource on species population in a two-patch habitat. It is assumed that the density of forest resource biomass is governed by the logistic equation in both the regions but with the different intrinsic growth rate but the same carrying capacity in the entire habitat. It is further assumed that the densities of species population is also governed by the generalized logistic equations in both the regions but with different growth rates and carrying capacities. It is shown that the steady state solutions are positive, monotonic and continuous under both reservoir and no-flux boundary conditions. The linear and non-linear asymptotic stability conditions of non-uniform steady state are compared with the case of the model with and without diffusion in a homogeneous habitat.

We construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The models are based on random walks where there may be a bias in the direction an individual moves when it encounters an interface. This sort of dispersal process is called skew Brownian motion. Our models take the form of diffusion equations with matching conditions across the interface between regions for population densities and fluxes. We combine the dispersal models with linear population growth models which assume that the population growth rate differs between regions of different habitat types. We use those models to study issues of refuge design. We specifically consider how the effectiveness of buffer zones depends on their size, quality, and the population's response to the interface between the buffer zone and the refuge.

A spatially explicit model for competition with dispersal in a heterogeneous environment is used to study the effects of individual size and the spatial scale of the environment on the competitive interactions between species. The model is a Lotka-Volterra competition system with diffusion and with spatial variation in some coefficients. The coefficients in the model are taken to reflect a situation where the larger competitor typically disperses farther in unit time than the smaller and reproduces less rapidly, but has an advantage in contests or other forms of interference competition. The environment is assumed to be closed, i.e., it is assumed that individuals do not leave through the boundary. The environment is generally assumed to consist of a patch of favorable habitat surrounded by less favorable regions. The effects of spatial scale are studied by examining how the predictions of the model change as the size of the favorable patch is varied. The predictions turn out to be in qualitative agreement with the results of some empirical studies.