Stability Analysis of a Predator-Prey Model with Disease in Prey, Predator Harvesting, and Internal Migration of Susceptible Prey in a Closed Ecosystem

Abstract

This study develops and analyzes a mathematical model of a predator-prey system incorporating three simultaneous ecological factors: disease in the prey population, harvesting of the predator, and internal migration of healthy prey within a closed ecosystem. The prey population is divided into two compartments healthy prey and infected prey where disease transmission follows the law of mass action. The predator population is subject to a constant harvesting rate representing external pressure such as hunting or capture. Internal migration of healthy prey is modeled as a density-dependent flux governed by a migration parameter m, which drives the redistribution of healthy prey across spatial patches without allowing permanent emigration from the system. The model is formulated as a system of three autonomous ordinary differential equations. Six equilibrium points are identified, representing a range of ecological scenarios from total extinction to full coexistence of all populations. The local stability of each equilibrium point is analyzed by linearization via the Jacobian matrix, and stability conditions are derived in terms of the model parameters, including the application of the Routh–Hurwitz criterion for the coexistence equilibrium. A key finding is that the migration parameter m does not alter the location of equilibrium points but directly influences the eigenvalues of the Jacobian, thereby affecting the rate at which the system recovers from perturbations. Numerical simulations are conducted to verify and illustrate the analytical results. This work extends the model of Mansur et al. [1] by introducing an internal spatial dimension, offering a more ecologically realistic framework for understanding population dynamics in ecosystems where disease, harvesting, and migration co-occur