The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator

Weili Kong; Yuanfu Shao; Weili Kong; Yuanfu Shao

doi:10.3934/math.20231498

AIMS Mathematics

2023, Volume 8, Issue 12: 29260-29289. doi: 10.3934/math.20231498

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The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator

Weili Kong ¹,
Yuanfu Shao ^{2
,
,}

1.
College of Teacher Education, Qujing Normal University, Qujing, Yunnan 655011, China
2.
College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

Received: 12 September 2023 Revised: 16 October 2023 Accepted: 23 October 2023 Published: 27 October 2023
MSC : 34C23, 92D25

Taking into account the delayed fear induced by predators on the birth rate of prey, the counter-predation sensitiveness of prey, and the direct consumption by predators with stage structure and interference impacts, we proposed a prey-predator model with fear, Crowley-Martin functional response, stage structure and time delays. By use of the functional differential equation theory and Sotomayor's bifurcation theorem, we established some criteria of the local asymptotical stability and bifurcations of the system equilibrium points. Numerically, we validated the theoretical findings and explored the effects of fear, counter-predation sensitivity, direct predation rate and the transversion rate of the immature predator. We found that the functional response as well as the stage structure of predators affected the system stability. The fear and anti-predation sensitivity have positive and negative impacts to the system stability. Low fear level and high anti-predation sensitivity are beneficial to the system stability and the survival of prey. Meanwhile, low anti-predation sensitivity can make the system jump from one equilibrium point to another or make it oscillate between stability and instability frequently, leading to such phenomena as the bubble, or bistability. The fear and mature delays can make the system change from unstable to stable and cause chaos if they are too large. Finally, some ecological suggestions were given to overcome the negative effect induced by fear on the system stability.
- fear,
- delay,
- stage structure,
- counter-predator sensitivity,
- bifurcation analysis
Citation: Weili Kong, Yuanfu Shao. The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator[J]. AIMS Mathematics, 2023, 8(12): 29260-29289. doi: 10.3934/math.20231498

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Abstract

Taking into account the delayed fear induced by predators on the birth rate of prey, the counter-predation sensitiveness of prey, and the direct consumption by predators with stage structure and interference impacts, we proposed a prey-predator model with fear, Crowley-Martin functional response, stage structure and time delays. By use of the functional differential equation theory and Sotomayor's bifurcation theorem, we established some criteria of the local asymptotical stability and bifurcations of the system equilibrium points. Numerically, we validated the theoretical findings and explored the effects of fear, counter-predation sensitivity, direct predation rate and the transversion rate of the immature predator. We found that the functional response as well as the stage structure of predators affected the system stability. The fear and anti-predation sensitivity have positive and negative impacts to the system stability. Low fear level and high anti-predation sensitivity are beneficial to the system stability and the survival of prey. Meanwhile, low anti-predation sensitivity can make the system jump from one equilibrium point to another or make it oscillate between stability and instability frequently, leading to such phenomena as the bubble, or bistability. The fear and mature delays can make the system change from unstable to stable and cause chaos if they are too large. Finally, some ecological suggestions were given to overcome the negative effect induced by fear on the system stability.

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