Bifurcation Analysis and Chaos Control for Prey-Predator Model With Allee Effect

Abstract

The main purpose of this work is to discuss the dynamics of a predator-prey dynamical system with Allee effect. The conformable fractional derivative is applied to convert the fractional derivatives which appear in the governing model into ordinary derivatives. We use the piecewise-constant approximation method to discritize the considered model. We also investigate the occurrence of positive equilibrium points. Moreover, we analyse the stability of the equilibrium point using some stability theorems. This work also explores a Neimark-Sacker bifurcation and a period-doubling bifurcation using the theory of bifurcations. The distance between the obtained equilibrium point and some closed curves is examined for various values for the considered bifurcation parameter. The chaos control is nicely analysed using the hybrid control approach. Furthermore, we present the maximum Lyapunov exponents for different values for the bifurcation parameter. Numerical simulations are utilized to ensure that the obtained theoretical results are correct. The used techniques can be applied to deal with predator-prey models in various versions.