Stability of steady distributions of asocial populations dispersing in one dimension

Abstract

Asocial populations, for which the population growth function at small densities is negative (due either to paucity of reproductive opportunity or to other causes) are fundamentally different in character from logistic populations. This paper establishes the spatial distributions which are steady under the combined influence of birth, death, and one-dimensional dispersion, together with a general analysis of their stability. The distributions possible for a given growth function F( N) ( F is population density growth rate at normalized density N) depend on whether ∫ 0 1 F d N ⋛ 0 The principal ones of interest are spatially periodic (e.g. in a finite region with reflecting boundaries) or aperiodic (e.g. in a finite region with absorbing boundaries). The theory of stability to small disturbances is established with the aid of Sturm-Liouville comparison theorems. It is proved for arbitrary F( N) (not only for asocial populations) that all periodic and all “intersecting” aperiodic distributions are unstable, and that all “non-intersecting” aperiodic distributions are stable. For asocial populations with ∫ 0 1 F d N > 0 , the aperiodic distribution with central density N 0 = N 0† and minimum range is neutrally stable. Distributions with N 0 > N 0† are stable, and all others (apart from the trivial case N 0 = 0) are unstable. This result that the population distribution can be stable only if the habitat and the central density exceed certain critical sizes has important and timely ecological implications.