The stabilizing effect of small prey immigration on competitive predator-prey dynamics

In this paper, we propose competitive predator-prey models with a small immigration into the prey. The dynamics of these models are investigated by addressing the boundedness, coexistence, and extinction conditions, as well as the local and global stability of equilibria. Immigrants stabilize the systems and increase the probability of coexistence. A Hopf bifurcation analysis shows that the model with Holling type II exhibits a Hopf bifurcation with respect to immigration parameter, but there is no bifurcation of the model with Holling type I. The numerical results support the theoretical results. Additionally, incorporating a few immigrants into the prey has a high sensitivity when the dynamic is periodic, but it has a lower sensitivity when the dynamic is stable. The obtained results can be biologically interpreted to improve the survival of species in the environment by adding immigrants. The rescue effect is considered as one of the implications in the real world that interpret the obtained results in this study.