Dynamics and control of a plant-herbivore model incorporating Allee's effect

This research focuses on the interaction between the grape borer and grapevine using a discrete-time plant-herbivore model with Allee's effect. We specifically investigate a model that incorporates a strong predator functional response to better understand the system's qualitative behavior at positive equilibrium points. In the present study, we explore the topological classifications at fixed points, stability analysis, Neimark-Sacker, Transcritical bifurcation and State feedback control in the two-dimensional discrete-time plant-herbivore model. It is proved that for all involved parameters ${\varsigma }_1,{\varrho }_1,{\gamma }_1$ and ${\Upsilon }_1$, discrete-time plant-herbivore model has boundary and interior fixed points: $c^1=(0,0)$, $c^2=(\frac{{\varsigma }_1-1}{{\varrho }_1},0)$ and $c^3=(\frac{{\Upsilon }_1(1-{\gamma }_1)}{2{\gamma }_1-1},\sqrt{\frac{{\gamma }_1(2{\varsigma }_1+{\varrho }_1{\Upsilon }_1-2)-{\varrho }_1{\Upsilon }_1+1-{\varsigma }_1}{2{\gamma }_1-1}})$ respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: $c^1=(0,0)$, $c^2=(\frac{{\varsigma }_1-1}{{\varrho }_1},0)$ and $c^3=(\frac{{\Upsilon }_1(1-{\gamma }_1)}{2{\gamma }_1-1},\sqrt{\frac{{\gamma }_1(2{\varsigma }_1+{\varrho }_1{\Upsilon }_1-2)-{\varrho }_1{\Upsilon }_1+1-{\varsigma }_1}{2{\gamma }_1-1}})$. Our investigation uncovers that the boundary equilibrium $c^2=(\frac{{\varsigma }_1-1}{{\varrho }_1},0)$ experiences a transcritical bifurcation, whereas the unique positive steady-state $c^3=(\frac{{\Upsilon }_1(1-{\gamma }_1)}{2{\gamma }_1-1},\sqrt{\frac{{\gamma }_1(2{\varsigma }_1+{\varrho }_1{\Upsilon }_1-2)-{\varrho }_1{\Upsilon }_1+1-{\varsigma }_1}{2{\gamma }_1-1}})$ of the discrete-time plant-herbivore model undergoes a Neimark-Sacker bifurcation. To address the periodic fluctuations in grapevine population density and other unpredictable behaviors observed in the model, we propose implementing state feedback chaos control. To support our theoretical findings, we provide comprehensive numerical simulations, phase portraits, dynamics diagrams, and a graph of the maximum Lyapunov exponent. These visual representations enhance the clarity of our research outcomes and further validate the effectiveness of the chaos control approach.