Transcritical Bifurcation and Flip Bifurcation of a New Discrete Ratio-Dependent Predator-Prey System

Abstract

After a discrete two-species predator-prey system with ratio-dependent functional response is topologically and equivalently reduced, some new dynamical properties for the new discrete system are formulated. The one is for the existence and local stability for all equilibria of this new system.Although the corresponding results for the equilibrium E 3 have been presented in a known literature, our results are more complete. The other is, what's more important and difficult, to derive some sufficient conditions for the transcritical bifurcation and period-doubling bifurcation of this system at the equilibria E 1 , E 2 and E 3 to occur, which are completely new. Numerical simulations are performed to not only illustrate the theoretical results obtained but also find new dynamics -chaos occuring. Our results sufficiently display that this system is very sensitive to its parameters. Namely, the perturbations of different parameters in this system will produce different bifurcations.

Dynamics of a Discrete Leslie–Gower Model with Harvesting and Holling-II Functional Response

“…Generally speaking, it is impossible to obtain an exact solution for a complex differential equation system. So, many researchers derive the corresponding approximate solution by computer [19][20][21][22][23][24]. What the computer considers is the discrete points in the calculation process; so, the discrete model has practical significance in deriving the numerical solution.…”