| Popis: |
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 fromwhich the criticality of the system can be deduced. Finally, numerical simulations are performed with biologically plausible parameters to substantiate the analytical findings. Using numerical continuation methods we detect Generalized Hopf and Zero-Hopf bifurcation points. We discuss their normal form coefficients, compute their two-parameter unfoldings and relate these results to the mathematical theory of codimension two bifurcations. |
