Bifurcation and chaos in a discrete-time fractional-order logistic model with Allee effect and proportional harvesting

The Allee effect and harvesting always get a pivotal role in studying the preservation of a population. In this context, we consider a Caputo fractional-order logistic model with the Allee effect and proportional harvesting. In particular, we implement the piecewise constant arguments (PWCA) method to discretize the fractional model. The dynamics of the obtained discrete-time model are then analyzed. Fixed points and their stability conditions are established. We also show the existence of saddle-node and period-doubling bifurcations in the discrete-time model. These analytical results are then confirmed by some numerical simulations via bifurcation, Cobweb, and maximal Lyapunov exponent diagrams. The occurrence of period-doubling bifurcation route to chaos is also observed numerically. Finally, the occurrence of period-doubling bifurcation is successfully controlled using a hybrid control strategy.