Discrete-time versus continuous-time toxic predation models

Abstract

In this paper, we establish and analyze a discrete analogue of the toxin-dependent predator–prey model mechanistically derived from element flows. We systematically compare the discrete- and continuous-time models to study the robustness of the model dynamics to time discretization. A theoretical study yields the conclusion that the stability conditions of equilibria in the discrete model are either the same or stronger than those in the continuous model. A comparison of the dynamical behaviours in the two models is conducted by numerical analysis. When the toxin level is low, the dynamical behaviours of the continuous and discrete models are quite different. The populations in the continuous-time model tend to a steady-state, whereas the discrete-time model produces periodic oscillations. When the toxin level is high, the dynamical behaviours of the continuous-time model are robust to time discretization. Some important features can be observed in both models, such as backward bifurcation, bistability, and the deterministic extinction of populations as a result of high toxin concentrations.