Dubey Balram, Kumar Sasmal Sourav, Sudarshan Anand
Department of Mathematics, BITS Pilani, Pilani Campus, Pilani 333031, Rajasthan, India.
Chaos. 2022 Dec;32(12):123132. doi: 10.1063/5.0126782.
This study presents a qualitative analysis of a modified Leslie-Gower prey-predator model with fear effect and prey refuge in the presence of diffusion and time delay. For the non-delayed temporal system, we examined the dissipativeness and persistence of the solutions. The existence of equilibria and stability analysis is performed to comprehend the complex behavior of the proposed model. Bifurcation of codimension-1, such as Hopf-bifurcation and saddle-node, is investigated. In addition, it is observed that increasing the strength of fear may induce periodic oscillations, and a higher value of fear may lead to the extinction of prey species. The system shows a bistability attribute involving two stable equilibria. The impact of providing spatial refuge to the prey population is also examined. We noticed that prey refuge benefits both species up to a specific threshold value beyond which it turns detrimental to predator species. For the non-spatial delayed system, the direction and stability of Hopf-bifurcation are investigated with the help of the center manifold theorem and normal form theory. We noticed that increasing the delay parameter may destabilize the system by producing periodic oscillations. For the spatiotemporal system, we derived the analytical conditions for Turing instability. We investigated the pattern dynamics driven by self-diffusion. The biological significance of various Turing patterns, such as cold spots, stripes, hot spots, and organic labyrinth, is examined. We analyzed the criterion for Hopf-bifurcation for the delayed spatiotemporal system. The impact of fear response delay on spatial patterns is investigated. Numerical simulations are illustrated to corroborate the analytical findings.
本研究对一个具有恐惧效应和猎物避难所的改进型莱斯利 - 高尔捕食者 - 猎物模型进行了定性分析，该模型存在扩散和时间延迟。对于无延迟的时间系统，我们研究了解的耗散性和持久性。进行了平衡点的存在性和稳定性分析，以理解所提出模型的复杂行为。研究了一维余维分岔，如霍普夫分岔和鞍结分岔。此外，观察到恐惧强度的增加可能会引发周期性振荡，而较高的恐惧值可能导致猎物种群灭绝。该系统表现出涉及两个稳定平衡点的双稳属性。还研究了为猎物种群提供空间避难所的影响。我们注意到，猎物避难所在达到特定阈值之前对两个物种都有益，超过该阈值则对捕食者物种有害。对于非空间延迟系统，借助中心流形定理和范式理论研究了霍普夫分岔的方向和稳定性。我们注意到，增加延迟参数可能会通过产生周期性振荡使系统不稳定。对于时空系统，我们推导了图灵不稳定性的解析条件。我们研究了由自扩散驱动的模式动力学。研究了各种图灵模式，如冷斑、条纹、热点和有机迷宫的生物学意义。我们分析了延迟时空系统的霍普夫分岔准则。研究了恐惧反应延迟对空间模式的影响。通过数值模拟来证实分析结果。
