Population Growth and Competition Models with Decay and Competition Consistent Delay

TL;DR

This paper introduces a modified delayed logistic model incorporating intraspecific competition and provides a comprehensive global analysis showing no sustained oscillations, along with extensions to a two-species competition system and evolutionary delay dynamics.

Contribution

It presents a new delay logistic equation consistent with population decay, extends it to a competition model, and analyzes stability, bifurcations, and evolutionary trends.

Findings

No sustained oscillations in the model.

Existence of a threshold between extinction and survival.

Evolutionary delay approaches a critical value under certain conditions.

Abstract

We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of [Arino et al., J.~Theoret.~Biol.~241(1):109--119, 2006] by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition of two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, $\tau^*$, such that the evolutionary trend is for the delay to approach $\tau^*$.