Existence of Ecological Prey Predator Mathematical Model in the Contaminated Domain

Abstract

We examine a predator–prey model that incorporates toxicant dynamics and a distributed delay representing gradual toxicant uptake. We prove positivity and boundedness of solutions and determine the system’s equilibria. Local stability of equilibria is examined via the Jacobian matrix and Routh–Hurwitz conditions, while global asymptotic stability of the coexistence equilibrium is established using a Lyapunov function and Sylvester’s criterion. Numerical simulations support the analytical results and compare the full model with a reduced model that omits toxicants and delay; these comparisons show that toxicants and delay affect transient dynamics but have limited impact on long-term stability. Our results highlight the importance of including toxicant dynamics and time delays when modeling population interactions in contaminated habitats.