霍普夫分叉
中央歧管
同宿轨道
捕食
捕食者
分叉
数学
鞍结分岔
平衡点
控制理论(社会学)
跨临界分岔
理论(学习稳定性)
应用数学
同宿分支
倍周期分岔
统计物理学
计算机科学
生态学
数学分析
物理
生物
非线性系统
人工智能
机器学习
微分方程
量子力学
控制(管理)
作者
Ashvini Gupta,Balram Dubey
出处
期刊:Chaos
[American Institute of Physics]
日期:2023-09-01
卷期号:33 (9)
被引量:8
摘要
The present work highlights the reverse side of the same ecological coin by considering the counter-attack of prey on immature predators. We assume that the birth rate of prey is affected by the fear of adult predators and its carry-over effects (COEs). Next, we introduce two discrete delays to show time lag due to COEs and fear-response. We observe that the existence of a positive equilibrium point and the stability of the prey-only state is independent of fear and COEs. Furthermore, the necessary condition for the co-existence of all three species is determined. Our system experiences several local and global bifurcations, like, Hopf, saddle-node, transcritical, and homoclinic bifurcation. The simultaneous variation in the attack rate of prey and predator results in the Bogdanov–Takens bifurcation. Our numerical results explain the paradox of enrichment, chaos, and bi-stability of node-focus and node-cycle types. The system, with and without delay, is analyzed theoretically and numerically. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also derived. The cascade of predator attacks, prey counter-attacks, and predator defense exhibit intricate dynamics, which sheds light on ecological harmony.
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