Delayed Density Dependence and the Stability of Interacting Populations and Subpopulations

Abstract Theoretical investigations of the dynamics of populations with discrete generations have traditionally been based on simple models of the formNt+1=f[Nt]. However, recent studies of the dynamics of natural populations indicate that density-dependent population regulation probably takes place over many generations (Nt+1=f[Nt,Nt−1, …]). In this paper, I explore the stability properties of interacting populations and contrast the predictions of discrete-generation models of population growth which do and do not include delayed density dependence. Relative to non-delayed models, inclusion of delayed density dependence changes the shape of population cycles (flip vs Hopf bifurcations) and decreases the range of parameters which predict stable equilibria. I also explore extensions of these models that include interspecific competition and coupling of spatially isolated patches. In both cases, delayed density dependence significantly changes the way in which demographic parameters scale to overall dynamics. For example, when delayed density dependence does not differ between two species, the asymptotic stability of both species is determined by a weighted average of the population growth rates of the two species. However, when species differ in time delay, some pairs of species that would both exhibit cyclical or chaotic dynamics in isolation can stably coexist. Analogous conclusions hold for the effects of deterministic spatial environmental variation among coupled patches. This implies that inclusion of delayed density dependence in investigations of population dynamics can dramatically change the inferences we draw from mathematical models and that further investigations of the effects of deterministic differences in demographic parameters and of delayed density dependence are warranted.

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