Partial Differential Equations in Ecological Modeling

Such equations arise whenever the quantities of interest vary not only with time but also with another variable. For example, in age-structured models, this variable is the age of a given population subgroup. When there are spatial interactions in the model caused by diffusion and convection, the other variable is the spatial coordinate (or coordinates). In what follows we focus on those models where the effect of diffusion is on the order of, or greater than, the effect of convection. These models are described by so-called parabolic equations. Models where convection dominates diffusion (this case includes the age-structured models) pertain to a mathematically different type of problem called hyperbolic, or wave, equations; they will only be mentioned briefly at the end of this article. To further narrow the scope of this article, we will consider only finite-difference and elementary spectral numerical methods. The reader can find accessible expositions of finite-element methods in, for example, the books by Stanoyevitch and Chandra and Singh. The advantage of those methods over the finite-difference ones is that they allow one to use a nonuniform spatial discretization and also to account for complex shapes of two- and three-dimensional spatial domains. The price one pays for this advantage is a more involved programming.

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