Analyzing multi-parameter bifurcation on a prey–predator model with the Allee effect and fear effect

• Published in: Chaos, solitons and fractals Vol. 180; p. 114498
• Main Authors: Abbasi, Muhammad Aqib, Samreen, Maria
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-03-2024
• Subjects: Bifurcation analysis; Fixed points stability; Numerical simulation; Fixed points stability; Numerical simulation; Bifurcation analysis
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2024.114498

In this research article, we present the dynamic analysis of the prey–predator model, adding the fear and the Allee effects. The main objective of this study is to analyze the periodicity and multi-parameter bifurcation of the model graphically. In particular, we studied the bifurcation between the two parameters. We critically analyzed the dynamics in the model with the help of detailed graphs of period ten and the Lyapunov exponent. Our study provides novel insights into the bifurcation behavior of the model, emphasizing the significance of Lyapunov exponents and the period-10 oscillation in understanding the dependencies among parameters. Through period ten oscillations, we confirm the complex dynamics, the coexistence of populations, and sensitivity to the initial conditions. Our analysis of the continuous-time model reveals that only Hopf bifurcation occurs at the positive fixed point, and we offer mathematical proof that no Hopf bifurcation occurs at other fixed points except the positive fixed point. From the numerical examples, we concluded that the crowding effect should be minimized for the stability of the model. Also, in the interior fixed point, when fear and Allee effects are taken as bifurcation parameters, backward bifurcations occur, which shows that in the presence of the crowding effect, the increase of the fear effect stabilizes the model.
Similarly, the more significant Allee effect stabilizes the model. While the decrease of these two effects causes an increase in growth rate, which causes bifurcation in the system due to overcrowding, the addition of the Allee effect and the fear effect should be, to a certain extent, so that the excess on both impacts controls the crowding effect. To prevent bifurcation, we employed a simple control method. This study improves our comprehension of the prey–predator system while potentially having implications for other complex systems in various fields, including population dynamics, epidemiology, and economics. Overall, our study contributes to the understanding of the dynamic behavior of the prey–predator model and provides new insights into its dynamics.
•The article discusses the dynamics of the prey–predator model, incorporating the Allee effect and fear effect.•It analyzes multi-parameter bifurcations.•The article discusses the periodicity and Lyapunov exponent of different parameters.•It obtains biological conclusions regarding the ecological stability of the model.