Mathematical analysis of toxicants and diseases impact on a prey-predator system with control interventions

Abstract

This study developed a generic prey-predator mathematical model incorporating logistic growth for prey and a Holling Type I functional response for predator consumption. Prey-predator interactions form the foundation of ecological balance, but in recent decades these systems have been increasingly threatened by toxicants and infectious diseases. The combined effect of these challenges can lead to severe population declines, biodiversity loss, and ecosystem instability, making their study important for conservation and environmental management. The model includes the effects of diseases and toxicants on prey and toxicants only on predators, and explores optimal control strategies to mitigate these threats. Without considering time delays, the model’s stability was examined using differential equation theory. Local stability of equilibrium points was analyzed through Jacobian matrix and eigenvalue methods, while Lyapunov functions were used for analysing global stability. The model was confirmed to be well-posed, meaning its solutions are biologically feasible. Initial simulations without control measures revealed that both diseases and toxicants significantly reduce prey and predator populations. When time-dependent controls were introduced, two strategies were tested: spatial isolation for disease control and bioremedial-antitoxic measures for toxicant control. Results showed that each control strategy independently improved population sizes. However, the best outcomes occurred when both strategies were applied together, leading to the greatest increase in prey and predator populations. The findings highlight that integrated control measures are essential to sustain threatened ecological systems. These awaraness provide a quantitative framework for policymakers and conservation biologists to design effective intervations for wildlife protection and habitat restoration.