Stability and Bifurcation Analysis of Two-Species Prey–Predator Model Incorporating External Factors

Abstract

Constant prey refuge with immigration and harvesting in two species would result in significant diversity in the dynamics of a prey–predator population. The phrase “refuge” increases the likelihood of prey population survival in the face of a predator population. Based on these findings, we created and examined a two-species prey–predator system with immigration and harvesting factors, including refuge to only prey population. All ecologically possible equilibrium points are studied for the proposed system. Routh–Hurwitz stability criterion is used for local stability analysis. Global stability of the interior equilibrium point is examined with a suitable Lyapunov function. Local bifurcation of the proposed system, such as saddle-node bifurcation, is analyzed. The conditions for the emergence of this bifurcation at the critical threshold near the nonhyperbolic equilibrium point are established by utilizing Sotomayor’s theorem. The transversality condition is validated for the occurrence of Hopf-bifurcation. The first Lyapunov number is exploited for determining the nature of Hopf bifurcating periodic solution. Finally, numerical simulations are illustrated to validate our theoretical predictions.