食饵有非线性收获率的Holling-Ⅳ类功能性反应的捕食系统定性分析

The qualitative analysis of a predator-prey system with nonlinear-rate prey harvesting under type-Ⅳ functional response

研究一类食饵种群具有非线性收获率的Holling-Ⅳ类功能性反应的捕食系统:{dx/dt=x(a-bx-cx2)x+y/β+x2-hx2 dy/dt=y(-d+μx/β+x2)分析该系统平衡点的存在性与性态,并利用Poincare形式级数法计算正平衡点Q(x1,y1)的焦点量,得出了正平衡点Q(x1,y1)至少可达到二阶细焦点.还讨论了该系统极限环的不存在性的条件及在特定条件下的Hopf分支问题,并分析该系统正平衡点附近可能存在极限环的拓扑结构及其生态意义.

A class of a predator-prey system with nonlinear-rate prey harvesting under type-Ⅳ functional response is investigated：{dx/dt=x（a-bx-cx2）x＋y/β＋x2-hx2 dy/dt=y（-d＋μx/β＋x2） We analyse the existence of the equilibria of the above system.And we make use of the Poincare＇s method to calculate the focus value and obtain that the positive equilibria Qx1,y1 is a fine focus of 2 order at least.At the same time,we study the condition of the nonexistence of limit cycle of the above system,the problem about Hopf bifurcation under some given conditions,the probable existence of limit cycle portraits topological structure near the positive equilibrium and analyse their biological sense.