一类具有强Allee效应的捕食-食饵模型共存解的存在性

The existence of coexistence solutions of a predator-prey model with strong Allee effect

研究一类具有强Allee效应的捕食-食饵模型的共存解。首先,以捕食者增长率b为分歧参数,利用局部分歧定理证明发自半平凡解局部分支的存在性;其次,利用全局分歧定理将该局部分支延拓成全局分支,因此得到共存解存在性的充分条件;最后,刻画了全局分支的走向。结果表明:该模型的分歧图像形成一个Loop。由分歧图像可知,当Allee效应强度M∈(0,1/2),食饵增长率r>r*且b∈(λ_1(-(du_2*)/(1+mu_2*),λ_1(-)du_1*/(1+mu_1*))时,捕食者与食饵可以共存。

The coexistence solutions of a predator-prey model with strong Allee effect are studied.Firstly,by using local bifurcation theory and regarding the growth rate of the predator b as a bifurcation parameter,the existence of the local bifurcation branch from the semi-trivial solution branch is proved.Secondly,the local bifurcation branch can be extended to a global bifurcation branch by global bifurcation theory.The sufficient condition for the existence of coexistence solutions is got.Finally,the trend of the global bifurcation branch is depicted.It is shown that the bifurcation diagram of this model is a Loop,which indicates that coexistence solutions of the model are existed whenever b∈（λ1（-（du2＊）/（1＋mu2＊）,λ1（-）du1＊/（1＋mu1＊））,the parameter M ∈（0,1/2）（which reflects the intensity of strong Allee effect）,and the inherent growth rate of the prey r〉r＊.