具有种内合作的种群动力系统的分叉

种群的种内关系包括种内干扰和种内合作两方面，其中关于种内干扰的种群动力系统的动力学研究已有很多，但是关于种内合作的研究却较少，尤其是合作捕食。本项目应用微分方程与动力系统的定性理论和分叉理论，对三类具有不同合作捕食功能反应函数的捕食者-食饵系统的定性性质和分叉现象进行研究。由于合作捕食因素使系统具有更多的参数和更复杂的非线性功能反应函数，我们在系统的动力学分析过程中需要克服更多的困难。对正平衡点的存在性和定性性质的确定，我们将避开对正平衡点坐标的解析表达式的依赖。对细焦点的判定，我们将利用结式消去法对Lyapunov量所生成的理想在非代数闭域中的代数簇进行不可约分解。对分叉的非横截条件和非退化条件的验证，我们将利用多项式完全判别系统以及实根分离方法判定含参多项式在区间内零点的存在性。另外，将系统的定性分析和分叉分析的理论结果给出合理的生态解释，从而对种群数量变化进行有效调控。

The intraspecific relationships of population include intraspecific interference and intraspecific cooperation. There have been a lot of studies on the dynamics of population dynamic systems with intraspecific interference, but there are few on intraspecific cooperation, especially cooperative predation. In this project, we study the qualitative properties and bifurcation phenomena of three types of predator-prey systems with cooperative predation and different functional response functions by the qualitative and bifurcation theory. Due to the cooperative predation, the systems contain more parameters and more complex nonlinear functional response functions, so that we need to overcome more difficulties in the analysis of dynamics. We will avoid the dependence on the analytic expressions of coordinates of positive equilibria, when discussing their existence and qualitative properties. In order to determine whether an center type equilibrium is a focus, we use resultants to decompose the algebraic varieties of the ideal of Lyapunov quantities over algebraically non-closed fields. To verify the non-transversal and non-degenerate conditions of bifurcations, we utilize the complete discrimination system of polynomial and the real-root isolation technique to determine the existence of real roots of parametric polynomial in some intervals. Furthermore, the reasonable ecological explanations are given on the theoretical results of qualitative and bifurcation analysis, so that the change of population amount can be effectively regulated.