Time-delayed models for the effects of toxicants on populations in contaminated aquatic ecosystems

Ecotoxicological models play a vital role in understanding the influence of toxicants on population dynamics in contaminated aquatic ecosystems. Traditional differential equation models describing interactions between populations and toxicants typically assume instantaneous population growth, overlooking potential time delays associated with reproductive and maturation processes. In this paper, we introduce two models with time delays to investigate the interaction between a population and a toxicant, where the population growth is governed by a delayed logistic equation. We mainly focus on the stability analysis of the steady states of the models. Our findings indicate that high toxicant concentrations result in population extinction, whereas moderate toxicant levels can potentially induce bistability, where the population's fate, whether persistence or extinction, depends on the initial densities of the population and toxicant. Furthermore, both our theoretical analysis and numerical simulations demonstrate that the time delay can lead to the destabilization of the coexistence steady states and the appearance of periodic solutions through Hopf bifurcation.