Asymptotic stability of delayed consumer age-structured population models with an Allee effect

Abstract

In this article we study a nonlinear age-structured consumer population model with density-dependent death and fertility rates, and time delays that model incubation/gestation period. Density dependence we consider combines both positive effects at low population numbers (i.e., the Allee effect) and negative effects at high population numbers due to intra-specific competition of consumers. The positive density-dependence is either due to an increase in the birth rate, or due to a decrease in the mortality rate at low population numbers. We prove that similarly to unstructured models, the Allee effect leads to model multi-stability where, besides the locally stable extinction equilibrium, there are up to two positive equilibria. Calculating derivatives of the basic reproduction number at the equilibria we prove that the upper of the two non-trivial equilibria (when it exists) is locally asymptotically stable independently of the time delay. The smaller of the two equilibria is always unstable. Using numerical simulations we analyze topologically nonequivalent phase portraits of the model.