Dynamical behaviours of discrete amensalism system with fear effects on first species

Qianqian Li; Ankur Jyoti Kashyap; Qun Zhu; Fengde Chen; Qianqian Li; Ankur Jyoti Kashyap; Qun Zhu; Fengde Chen

doi:10.3934/mbe.2024035

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 1: 832-860. doi: 10.3934/mbe.2024035

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Research article

Dynamical behaviours of discrete amensalism system with fear effects on first species

1.
School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China
2.
Department of Mathematics, Girijananda Chowdhury University, Assam, India

Received: 10 October 2023 Revised: 03 December 2023 Accepted: 10 December 2023 Published: 20 December 2023

Amensalism, a rare yet impactful symbiotic relationship in ecological systems, is the focus of this study. We examine a discrete-time amensalism system by incorporating the fear effect on the first species. We identify the plausible equilibrium points and analyze their local stability conditions. The global attractivity of the positive equilibrium, $ E^* $, and the boundary equilibrium, $ E_1 $, are analyzed by exploring threshold conditions linked to the level of fear. Additionally, we analyze transcritical bifurcations and flip bifurcations exhibited by the boundary equilibrium points analytically. Considering some biologically feasible parameter values, we conduct extensive numerical simulations. From numerical simulations, it is observed that the level of fear has a stabilizing effect on the system dynamics when it increases. It eventually accelerates the extinction process for the first species as the level of fear continues to increase. These findings highlight the complex interplay between external factors and intrinsic system dynamics, enriching potential mechanisms for driving species changes and extinction events.
- fear effect,
- amensalism,
- global attractivity,
- chaos control,
- transcritical bifurcation,
- flip bifurcation
Citation: Qianqian Li, Ankur Jyoti Kashyap, Qun Zhu, Fengde Chen. Dynamical behaviours of discrete amensalism system with fear effects on first species[J]. Mathematical Biosciences and Engineering, 2024, 21(1): 832-860. doi: 10.3934/mbe.2024035

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Abstract

Amensalism, a rare yet impactful symbiotic relationship in ecological systems, is the focus of this study. We examine a discrete-time amensalism system by incorporating the fear effect on the first species. We identify the plausible equilibrium points and analyze their local stability conditions. The global attractivity of the positive equilibrium, $ E^* $, and the boundary equilibrium, $ E_1 $, are analyzed by exploring threshold conditions linked to the level of fear. Additionally, we analyze transcritical bifurcations and flip bifurcations exhibited by the boundary equilibrium points analytically. Considering some biologically feasible parameter values, we conduct extensive numerical simulations. From numerical simulations, it is observed that the level of fear has a stabilizing effect on the system dynamics when it increases. It eventually accelerates the extinction process for the first species as the level of fear continues to increase. These findings highlight the complex interplay between external factors and intrinsic system dynamics, enriching potential mechanisms for driving species changes and extinction events.

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