Global stabilization and regulation of prey immigration for predator–prey models

Observations of predator–prey dynamics in nature often reveal varying predation rates at different prey densities. However, the stabilizing or regulatory influence of constant prey immigration in predator–prey models incorporating a Holling type III functional response remains unclear. This study aims to address this issue by examining prey immigration as a stabilizing and regulating mechanism in ecosystems. This is achieved by employing a predator–prey model, with the predation process represented by a functional response involving Holling type III. The model admits at most one interior equilibrium point. A mathematical formulation is used to establish the interplay between the destabilization of the interior equilibrium point owing to increased carrying capacity and stabilization arising from constant prey immigration. Under certain conditions, an increase in carrying capacity disrupts the previously stable interior equilibrium point, resulting in the emergence of a stable limit cycle. The amplitude of this limit cycles grows as the carrying capacity increases, heightening the risk of ecosystem extinction. Conversely, constant prey immigration attenuates the destabilization of the interior equilibrium point and mitigates the amplification of the limit cycle. Necessary and sufficient conditions for global asymptotic stability of the interior equilibrium point and uniqueness of limit cycles are derived by transforming the model into a Liénard-type system with multiple parameters to validate these findings. The study findings enable the formulation of the effects of constant prey immigration on the stabilization and regulation in predator–prey models.