Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, People's Republic of China.
Department of Mathematics, Jiangxi University of Science and Technology, Ganzhou 341000, People's Republic of China.
Chaos. 2022 Apr;32(4):043125. doi: 10.1063/5.0082733.
In this article, a fractional-order prey-predator system with Beddington-DeAngelis functional response incorporating two significant factors, namely, dread of predators and prey shelter are proposed and studied. Because the life cycle of prey species is memory, the fractional calculus equation is considered to study the dynamic behavior of the proposed system. The sufficient conditions to ensure the existence and uniqueness of the system solution are found, and the legitimacy and well posedness in the biological sense of the system solution, such as nonnegativity and boundedness, are proved. The stability of all equilibrium points of the system is analyzed by an eigenvalue analysis method, and it is proved that the system generates Hopf bifurcation nearby the coexistence equilibrium with regard to three parameters: the fear coefficient k, the rate of prey shelters p, and the order of fractional derivative q. Compared with the integer derivative, the system dynamics in the situation of fractional derivative is more stable. We observe an interesting phenomenon through the simulation: with the increase in the level of the fear effect, the stability of the positive equilibrium point changes from stable to unstable and then to stable. At this time, there are two Hopf branches nearby the positive equilibrium point with respect to the fear coefficient k, and the system can be in a stable state at very low or high level of the fear effect. In addition, when the order of the fractional differential equation of the system decreases continuously, the stability of the system will change from unstable to stable, especially in the case of low-level fear caused by predators and low rate of prey shelters. Therefore, our findings support the view that the strong memory can promote the stable coexistence of two species in the prey-predator system, while fading memory of species will worsen the stable coexistence of two species in the proposed system.
本文提出并研究了一个具有 Beddington-DeAngelis 功能反应和两个重要因素的分数阶捕食-被捕食系统，即捕食者的恐惧和猎物庇护。由于猎物物种的生命周期具有记忆性，因此考虑分数阶微积分方程来研究所提出系统的动态行为。找到了确保系统解存在且唯一的充分条件，并证明了系统解在生物学意义上的合法性和适定性，如非负性和有界性。通过特征值分析方法分析了系统所有平衡点的稳定性，并证明了在三个参数：恐惧系数 k、猎物庇护率 p 和分数阶导数 q 的情况下，系统在共存平衡点附近产生 Hopf 分岔。与整数导数相比，分数导数情况下的系统动力学更稳定。通过仿真观察到一个有趣的现象：随着恐惧效应水平的增加，正平衡点的稳定性从稳定变为不稳定，然后再变为稳定。此时，正平衡点附近存在两个关于恐惧系数 k 的 Hopf 分支，并且系统可以在非常低或高水平的恐惧效应下处于稳定状态。此外，当系统分数微分方程的阶数连续降低时，系统的稳定性将从不稳定变为稳定，特别是在捕食者引起的恐惧水平低和猎物庇护率低的情况下。因此，我们的研究结果支持这样一种观点，即强记忆可以促进捕食-被捕食系统中两种物种的稳定共存，而物种记忆的衰退会恶化所提出系统中两种物种的稳定共存。
