一类具有恐惧效应和避难所的捕食者-食饵模型的动力学分析

Dynamic Analysis of a Predator-Prey Model with Fear Effect and Refuge

为探究恐惧效应和避难所对生物种群的影响,建立一类食饵具有恐惧效应和避难所的服从Gompertz增长规律的捕食者-食饵模型。首先,证明了模型解的非负性;其次,给出了正平衡点的存在条件以及平衡点局部稳定的条件,利用Dulac准则证明模型在解的有界区域Ω上不存在极限环,进一步表明其全局渐近稳定性;最后,分析了恐惧强度及避难所的规模对种群密度的影响。研究发现食饵种群密度不受恐惧效应的影响,但是食饵避难所有利于食饵种群密度的增加;随着恐惧强度的增加,捕食者的种群密度会减少;较大的食饵避难所规模不利于捕食者种群的增长。

To explore the impact of fear effect and refuge on biological populations a predator-prey model with fear effect and refuge subject to Gompertz growth rule is established.Firstly,the nonnegativity of the solution in the model is proved.Secondly,the existence condition of the positive equilibrium and the conditions of the local stability of the equilibria are obtained,the Dulac criterion is used to show that there is no limit cycle on the bounded regionΩof the solution of such model,and the global asymptotic stability of the equilibriua are further demonstrated.Finally,the effects of fear intensity and refuge size on population density are analyzed.The results show that the prey population density is not affected by fear effect,but the prey refuge is conducive to the increase of prey population density;as the intensity of fear effect increases,the predator population density decreases;the large size of the prey refuge is not conducive to the growth of the predator population.