Qualitative analysis of plant-herbivore interaction with Allee effect: stability, bifurcation and chaos control

In this study, the dynamics of plant-herbivore interactions is explored while incorporating the Allee effect on prey population. This study offers valuable insights into the dynamics of plant-herbivore systems by examining the role of the Allee effect and its influence on population dynamics. The Allee effect, which occurs when reduced population size negatively impacts individual fitness and population growth rate, is considered a critical factor in understanding these interactions. This investigation is focused on analyzing the stability of the system through the examination of fixed points, including a trivial steady-state, a predator-free steady-state, and a coexisting steady-state. Also their local stability criteria is examined with the help of the method of linearization. Bifurcation analysis is utilized to explore conditions leading to qualitative changes in the system’s behavior, including the emergence of complex dynamics. The system undergoes a transcritical bifurcation and Hopf bifurcation at positive steady-state. To address chaotic behavior, chaos control techniques is implemented to stabilize the system at the desired state including the emergence of chaotic attractors. Additionally, numerical simulations is conducted to validate the theoretical and mathematical analyses, offering a visual and qualitative comprehension of the system’s behavior across different parameter settings. For numerical simulations, MATHEMATICA software is used to illustrate the study.