Abstract
Integrodifference (IDE) models can be used to determine the critical domain size required for persistence of populations with distinct dispersal and growth phases. Using this modelling framework, we develop a novel spatially implicit approximation to the proportion of individuals lost to unfavourable habitat outside of a finite domain of favourable habitat, which consistently outperforms the most common approximations. We explore how results using this approximation compare to the existing IDE results on the critical domain size for populations in a single patch of good habitat, in a network of patches, in the presence of advection, and in structured populations. We find that the approximation consistently provides results which are in close agreement with those of an IDE model except in the face of strong advective forces, with the advantage of requiring fewer numerical approximations while providing insights into the significance of disperser retention in determining the critical domain size of an IDE.
Highlights
Integrodifference equations (IDEs) are an established way to model populations in which growth does not occur simultaneously with dispersal
While IDEs provide us with a way to model a variety of dispersal patterns and growth functions, they are seldom analytically tractable and the critical domain size can often only be obtained via methods involving numerical integration
Approximations to an IDE framework are useful for those wishing to compare the effects of different dispersal patterns and demographic rates
Summary
Integrodifference equations (IDEs) are an established way to model populations in which growth does not occur simultaneously with dispersal. Used to model the propagation of alleles (Slatkin 1973, 1975), IDEs were first applied to population biology by Kot and Schaffer (1986) Since they have been analysed for travelling wave solutions (Hsu and Zhao 2008; Kot 1992; Kot et al 1996; Neubert and Parker 2004; Weinberger et al 2008) and dispersal-driven instability in predator–prey interactions (Neubert et al 1995), and the effects of different dispersal strategies have been considered in both spatially and temporally varying habitats (Hardin et al 1990). We first consider the growth of a non-spatial population according to a function f (Nt ) where Nt is the density of the female population at time t ≥ 0. Though for other formulations, see Lutscher (2008)
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