具有捕获项的Beddington-DeAnglis型捕食-食饵扩散模型的动力学分析

Dynamics Analysis of a Diffusive Predator-Prey Model with Beddington-DeAngelis Function Response and Harvesting

研究了一类具有捕获项的Beddington-DeAnglis型捕食-食饵扩散模型.首先利用不动点指数理论得到了正解存在的充分条件,并结合分歧理论讨论了正解的多重性;其次通过线性算子的扰动理论、度理论和稳定性理论考察了捕食者间的相互作用充分大时正解的唯一性和稳定性;接着运用抛物系统的比较原理分析了两物种的灭绝性和持久性;最后通过数值模拟对理论结果进行了验证和补充.研究结果表明,只要食饵的最大增长率较大且捕食者的低密度死亡率较小,当捕食者间相互作用的影响充分大且捕食者的捕获率在某一范围内时系统存在唯一稳定的正解;当捕食者的捕获率在另一范围内时系统至少存在两个正解.

A diffusive predator-prey model with Beddington-DeAngelis function response and harvesting is studied.Firstly,the sufficient conditions for the existence of positive solutions are obtained by the fixed point index theory,and the multiplicity of positive solutions is discussed by the bifurcation theorem.Then,by virtue of the combination of the perturbation theory for linear operators,degree theory and stability theory,the uniqueness and stability of positive solution are investigated when the interaction among predators is large.In addition,we analyze the extinction and permanence of the two species by means of the comparison principle of parabolic systems.Finally,we make some numerical simulations to validate and complement the theoretical results.If the maximal growth rate of the prey is large and low density mortality of the predator is small,the findings suggest that the model has only one unique asymptotically stable positive solution provided that the effect of the interaction among predators is sufficiently large and the harvesting rate of the predator lies in a certain range;and has at least two positive solutions when the harvesting rate of the predator belongs to another range.