School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China.
Chaos. 2023 Jan;33(1):013131. doi: 10.1063/5.0130809.
In this paper, we study the dynamic behaviors of a fractional order predator-prey system, in which the prey population has three effects: Allee effect, fear effect, and shelter effect. First, we prove in detail the positivity, existence, uniqueness, and boundedness of the solutions of the model from the perspective of mathematical analysis. Second, the stability of the system is considered by analyzing the stability of all equilibria and possible bifurcations of the system. It is proved that the system undergoes Hopf bifurcation with respect to four important parameters at the positive equilibrium point. Third, through stability analysis of the system, we find that: (i) as long as the initial density of the prey population is small enough, it will enter the attraction region of an extinction equilibrium point, making the system population at risk of extinction; (ii) we can eliminate the limit-cycle to make the system achieve stable coexistence by appropriately increasing the fear level or refuge rate, or reducing the prey natality or the order of fractional order systems; (iii) fractional order system is more stable than integer order systems, when the system has periodic solution, the two species can coexist stably by increasing the fear level or refuge rate appropriately. The threshold of fear level and refuge rate in fractional order systems is smaller than that in integer order systems. Finally, the rationality of the research results is verified by numerical simulation.
本文研究了一类具有食饵的三种效应（艾里效应、恐惧效应和庇护所效应）的分数阶捕食-被捕食系统的动态行为。首先，从数学分析的角度详细证明了模型解的正性、存在性、唯一性和有界性。其次，通过分析系统所有平衡点的稳定性和可能的分岔，研究了系统的稳定性。结果表明，系统在正平衡点处关于四个重要参数发生 Hopf 分岔。第三，通过对系统的稳定性分析，发现：（i）只要食饵种群的初始密度足够小，它就会进入灭绝平衡点的吸引域，使系统种群面临灭绝的风险；（ii）通过适当增加恐惧水平或庇护率，可以消除极限环，使系统达到稳定共存；通过适当降低食饵的出生率或分数阶系统的阶数，也可以消除极限环，使系统达到稳定共存；（iii）当系统存在周期解时，通过适当增加恐惧水平或庇护率，两个物种可以稳定共存，且分数阶系统比整数阶系统更稳定。分数阶系统的恐惧水平和庇护率的阈值比整数阶系统的要小。最后，通过数值模拟验证了研究结果的合理性。
