School of Mathematics Science, Huaiyin Normal University, Huaian 223300, People's Republic of China.
School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, People's Republic of China.
Chaos. 2019 Aug;29(8):083109. doi: 10.1063/1.5111121.
In this paper, we investigate the dynamics of an improved Leslie-Gower predator-prey model which is characterized by the reduction of the prey growth rate due to fear of the predator (i.e., antipredator behavior). The value of this study lies in two aspects: mathematically, (i) it provides the existence and the stability of the positive equilibrium; (ii) it gives the existence of the Hopf bifurcation and limit cycle; and (iii) it shows the mechanisms of the fear factor and the prey refuge on the level of the positive equilibrium. Biologically, we find that the influence of the fear factor is complex: (i) increasing the level of fear can cause the level of the population density to decrease and the prey to become extinct; (ii) the effect of the cost of fear on the stability of the positive equilibrium is rich and complex: it can either destabilize the stability and benefit the emergency of the periodic behavior or stabilize the system by excluding the existence of periodic solutions; (iii) with a fixed level of fear, the prey refuge is beneficial to the coexistence of the prey and the predator, and with the increase of the level of the prey refuge, the positive equilibrium may change from stable spiral sink to unstable spiral source to stable spiral sink. These results may enrich the dynamics of the predator-prey systems.
本文研究了一种改进的 Leslie-Gower 捕食者-猎物模型的动态，该模型的特点是由于对捕食者的恐惧（即反捕食行为）而降低了猎物的增长率。这项研究的价值在于两个方面：从数学角度来看，（i）它提供了正平衡点的存在性和稳定性；（ii）它给出了 Hopf 分岔和极限环的存在性；（iii）它展示了恐惧因素和猎物避难所在正平衡点水平上的作用机制。从生物学角度来看，我们发现恐惧因素的影响是复杂的：（i）增加恐惧水平会导致种群密度降低，猎物灭绝；（ii）恐惧成本对正平衡点稳定性的影响是丰富而复杂的：它可以使系统不稳定，有利于周期性行为的出现，也可以通过排除周期性解的存在来稳定系统；（iii）在固定的恐惧水平下，猎物避难所有利于猎物和捕食者的共存，并且随着猎物避难所水平的增加，正平衡点可能从稳定的螺旋汇点变为不稳定的螺旋源点，再变为稳定的螺旋汇点。这些结果可能丰富捕食者-猎物系统的动力学。
