Stability, bifurcation analysis of a predator-prey system with density dependent disease recovery using center manifold theory

Abstract

The center manifold is an invariant manifold that plays a crucial role in the bifurcation analysis of dynamical systems. The center manifold existence theorem assures the local existence of an invariant submanifold of the state space of a dynamical system around a non-hyperbolic equilibrium point. Center manifold theory is essential in the reduction of different bifurcation scenarios to their normal forms. Our study focuses on a predator-prey interactive system with density-dependent growth in predators subject to a contagious disease. The disease is assumed to be horizontally transmitted, and the rate of recovery of the infected predator is assumed to be density-dependent. At the trivial (zero) equilibrium, the center manifold is calculated whose dynamical behaviour is similar to that of the original system. Further, using the center manifolds, the normal form of a
Hopf bifurcation point is determined from which the criticality of the system
can be deduced. Finally, numerical simulations are performed with biologically plausible parameters to substantiate the analytical findings.