The Effect of Dispersal on Population Stability in One-Species, Discrete-Space Population Growth Models

Here I propose a general formulation of population growth with dispersal in one-species systems. The species range is viewed as a collection of spatially separate habitats, and population growth with dispersal is described as a discrete parameter process. I examine the effect of dispersal on system stability using a series of special cases of this general formulation. A system is considered relatively stable if, for a fixed pattern of external perturbation, it exhibits a low degree of variability. Measures of variability are proposed that quantify this idea. Dispersal increases the degree of stability of many of the systems examined. This effect arises if juveniles disperse as a fixed feature of the life history and population density does not severely reduce birth rate. It also arises if adults disperse facultatively in response to overcrowding. Habitat selection augments dispersal's stabilizing tendency. Explicit inclusion of habitat locations complicates the mathematics but does not alter this qualitative dispersal effect. Dispersal does not always increase population stability. If population density strongly suppresses birth rate, obligate juvenile dispersal can actually reduce stability. In this case, more juveniles are produced by sparse than by dense populations, and dispersal causes growing populations to lose more potential recruits than they gain. Adult dispersal strongly stimulated by population growth rate can also reduce population stability for the same reason. In discrete time, dispersal can actually destroy asymptotic stability. The conclusion that dispersal can sometimes stabilize and sometimes destabilize populations also emerges from more thoroughly studied continuous-space diffusion models. The discrete-space systems studied here are generally mathematically simpler than diffusion equations, and this approach may expedite the task of combining dispersal with other population phenomena into more inclusive models.