Bifurcation analysis and chaos control for fractional predator-prey model with Gompertz growth of prey population

This paper discusses a fractional-order prey–predator system with Gompertz growth of prey population in terms of the Caputo fractional derivative. The non-negativity and boundedness of the solutions of the considered model are successfully analyzed. We utilize the Mittag-Leffler function and the Laplace transform to prove the boundedness of the solutions of this model. We describe the topological categories of the fixed points of the model. It is theoretically demonstrated that under certain parametric conditions, the fractional-order prey–predator model can undergo both Neimark–Sacker and period-doubling bifurcations. The piecewise constant argument approach is invoked to discretize the considered model. We also formulate some necessary conditions under which the stability of the fixed points occurs. We find that there are two fixed points for the considered model which are semi-trivial and coexistence fixed points. These points are stable under some specific constraints. Using the bifurcation theory, we establish the Neimark–Sacker and period-doubling bifurcations under certain constraints. We also control the emergence of chaos using the OGY method. In order to guarantee the accuracy of the theoretical study, some numerical investigations are performed. In particular, we present some phase portraits for the stability and the emergence of the Neimark–Sacker and period-doubling bifurcations. The biological meaning of the given bifurcations is successfully discussed. The used techniques can be successfully employed for other models.