Fractional Order Plant-Herbivore Dynamics: From Stability to Chaos Control

This study investigates the dynamic behavior of a discrete-time plant-herbivore model incorporating conformable fractional-order derivatives and a toxin-dependent functional response. The model is discretized using a piecewise constant argument approach, enabling the analysis of memory effects and nonlocal interactions in ecological dynamics. By applying the Jury stability criterion, we derive necessary and sufficient conditions for the local asymptotic stability of the positive equilibrium. A comprehensive bifurcation analysis demonstrates that the system undergoes a supercritical Neimark-Sacker bifurcation as key parameters vary, leading to the emergence of quasiperiodic and chaotic dynamics. Notably, no evidence of flip (period-doubling) bifurcations is observed within the explored parameter space. To address the destabilizing effects of chaos, a hybrid control strategy tailored for the fractional discrete setting is implemented, successfully restoring stability in the system. Numerical simulations corroborate the theoretical results and highlight the novel dynamic regimes introduced by fractional-order memory. Our findings underscore the importance of incorporating both plant toxicity and memory effects for a realistic understanding and effective management of plant-herbivore interactions.