Dynamical and Phase Analyses on Chaos Control of Discrete Predator–Prey Systems of Mixed Holling Types with Density-Dependent Birth Rates

This paper discusses the effect of density-dependent birth rate in a discrete predator–prey system with mixed functional responses of Holling types I and III. We use the Beverton–Holt function to modify the birth rate parameter of the prey species. The steady states and their stability are established. The criteria for flip bifurcation (FB) and Neimark–Sacker bifurcation (NSB) are proposed analytically using the center manifold theorem and normal bifurcation theory. Using the state feedback control method, it is shown that chaotic orbit can be stabilized at an unstable steady-state. This study deduces that the density of predator species is negatively affected by the density-dependent birth rate of prey species. Moreover, density-dependent birth rate can increase the possibility of extinction in prey species, which means that this birth rate may cause Allee effect. Some numerical simulations are given to verify these analytical results.