Temporal Consequences of Spatial Density Dependence

(1) It has been suggested (e.g. by Hassell, Southwood & Reader 1987) that spatial density dependence can be an important cause of temporal density dependence and stability, while at the same time making these effects difficult to detect in the field. In particular, these authors claim that spatially density-dependent mortality stabilizes a discrete-time single-species model, and that field populations failing to show temporal density dependence may be stabilized by such spatial density dependence. (2) We show that the single-species model of spatial density dependence due to De Jong (1979), and a variant of it used by Hassell, Southwood & Reader (1987), are stable because they contain explicit temporal, as well as spatial, density dependence. Because of this, the models are stable even without spatial heterogeneity. (3) To obtain spatial density dependence without temporal density dependence, we derive a model differing from De Jong's only in making the fraction dying in a patch a function of the relative density, i.e. (density in the patch)/(average patch density for that generation), rather than the absolute density. Thus, mortality on a patch with the average density is the same in each generation. (4) Our model shows that spatial heterogeneity and density dependence can induce temporal density dependence and greater stability. When there is no heterogeneity, mortality is constant and our model becomes the null model of geometric growth or decline (or stasis, if fertility and mortality are exactly matched). When heterogeneity is added, our model yields a region of values of fertility and mortality (on a patch of average density) giving stability. (5) However, our model suggests that spatial heterogeneity is more likely to be destabilizing than stabilizing. Spatial density dependence introduces inverse temporal density dependence at low (and often all) densities. For the negative binomial distribution of patch densities and any given level of fertility, the mortality on a patch of average density must lie in a very small range if stability is to be possible, and this range shrinks steadily as clumping is increased. The range is negligible or non-existent for commonly observed values of the clumping parameter.