Bifurcation and chaotic dynamics in a fractional-order modified Richards growth model with varying carrying capacity and Allee effect

We consider a fractional-order Richards growth model incorporating environmental feedback on the carrying capacity and the Allee effect. To reveal the rich dynamical behavior of the system, a discretization procedure based on piecewise constant arguments is applied, leading to a system of difference equations. By employing the Schur–Cohn criterion, stability conditions for the two equilibrium points E1 and E2 are derived with respect to the growth parameter r. It is shown theoretically that the discrete system exhibits a flip bifurcation around the equilibrium point E1, while a Neimark–Sacker bifurcation occurs at the equilibrium point E2. When the corresponding critical thresholds are exceeded, the presence of chaotic dynamics in both cases is confirmed through the computation of Lyapunov exponents. To regulate these chaotic behaviors, two effective control strategies, namely state feedback control and a hybrid control approach, are implemented. In addition, we deal with the basin of attraction, demonstrating that some initial states evolve toward a stable long term regime, whereas others escape the feasible domain, emphasizing the strong dependence of the system’s behavior on initial conditions. All theoretical findings are supported by numerical simulations and interpreted from a biological perspective.