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<p>This paper studies the dynamic behavior of a novel fractional-order three-dimensional predator-prey system with competition and fear effects. Considering that two competing predator populations prey on the same prey population, the prey population exhibits fear towards both predator populations, and the competitive effect of these two predators is manifested as mutual interference during foraging activities. In order to incorporate this competitive effect into the functional response function, we introduce it based on classical Holling time budget parameter. Additionally, to account for the memory effect of the population, a fractional differential equation is established using Caputo fractional derivatives. First, we mathematically prove the non-negativity, uniqueness, and boundedness of the system’s solution. Then, we analyze the existence and stability of all possible biological equilibrium points of the system, and give the conditions for the occurrence of Hopf bifurcations near the equilibrium points with respect to the parameters of fear level and memory level. Finally, the theoretical analysis results are verified through detailed numerical simulations. Through the analysis of the system’s dynamic behavior, we find that two boundary equilibrium points exhibit three types of bistability under the influence of fear and memory effects. The initial densities of the three populations will determine the final state of the corresponding solutions. The fear effect and competition effect will affect the existence of positive equilibrium points, but will not change their stability. When there is a positive equilibrium point, it is always unstable. The competition between two predator populations for the same food source makes them unable to coexist for a long time, which is completely consistent with the Gauss competition exclusion principle in ecology. The memory effect has no effect on the existence of system equilibrium points and the stability of the positive equilibrium, but it affects the stability of equilibrium points at two boundary interfaces, resulting in rich dynamic behavior in the system. These three effects collaborate to exert profound influences on environmental protection and population ecology. By conducting in-depth exploration of these effects, we can develop more effective conservation strategies to sustain the health and stability of ecosystems.</p>

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Dynamic analysis of a fractional order three-dimensional predator-prey system with competitive and fear effects

  • 期刊论文

  • 发表日期：

    2025年6月24日

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  • Binfeng Xie,
  • Zhengce Zhang,
  • Na Zhang

摘要

This paper studies the dynamic behavior of a novel fractional-order three-dimensional predator-prey system with competition and fear effects. Considering that two competing predator populations prey on the same prey population, the prey population exhibits fear towards both predator populations, and the competitive effect of these two predators is manifested as mutual interference during foraging activities. In order to incorporate this competitive effect into the functional response function, we introduce it based on classical Holling time budget parameter. Additionally, to account for the memory effect of the population, a fractional differential equation is established using Caputo fractional derivatives. First, we mathematically prove the non-negativity, uniqueness, and boundedness of the system’s solution. Then, we analyze the existence and stability of all possible biological equilibrium points of the system, and give the conditions for the occurrence of Hopf bifurcations near the equilibrium points with respect to the parameters of fear level and memory level. Finally, the theoretical analysis results are verified through detailed numerical simulations. Through the analysis of the system’s dynamic behavior, we find that two boundary equilibrium points exhibit three types of bistability under the influence of fear and memory effects. The initial densities of the three populations will determine the final state of the corresponding solutions. The fear effect and competition effect will affect the existence of positive equilibrium points, but will not change their stability. When there is a positive equilibrium point, it is always unstable. The competition between two predator populations for the same food source makes them unable to coexist for a long time, which is completely consistent with the Gauss competition exclusion principle in ecology. The memory effect has no effect on the existence of system equilibrium points and the stability of the positive equilibrium, but it affects the stability of equilibrium points at two boundary interfaces, resulting in rich dynamic behavior in the system. These three effects collaborate to exert profound influences on environmental protection and population ecology. By conducting in-depth exploration of these effects, we can develop more effective conservation strategies to sustain the health and stability of ecosystems.