Periodic behavior and dynamical analysis of a prey–predator model incorporating the Allee effect and fear effect

This study analyzed the dynamics of a two-dimensional prey–predator model that incorporates the Allee and fear effects. We conducted stability analysis of the fixed points in discrete and continuous forms and focused on the periodic behavior of the discrete-time model. Our study used novel complex period graphs and the Lyapunov exponent to understand better the system’s complexities and the parameters’ interdependence. We found that periodic oscillations confirm the coexistence of populations, the sensitivity to initial conditions, and the complicated dynamics of the model. In addition, we looked into the bifurcation behavior of discrete and continuous models using bifurcation theory and presented numerical examples to validate our theoretical findings. We also identified the direction of bifurcation using attractive bifurcation plots and employed a simple control technique to avoid bifurcation. Our research contributes to a better understanding of the prey–predator system and has implications for other complex systems in various fields, including population dynamics, physical models, epidemiology, and economics. Overall, our work reveals additional illumination on the prey–predator model’s dynamics and increases our understanding of its dynamic behavior.