具有合作狩猎的食物链模型的Hopf分支

Hopf bifurcation of a food chain model with cooperative hunting

为研究捕食者间合作行为对食物链系统动力学行为的影响,在原有合作捕食模型基础上引入顶层捕食者(仅以捕食者种群为食),建立了具有合作狩猎的食物链模型。运用线性化方法讨论了平衡点的局部稳定性,并通过构造合适的Lyapunov函数,给出了系统全局稳定的充分条件。利用中心流形约简定理导出了分支周期解稳定性的显式公式,并通过数值模拟验证了理论分析结果。结果显示,当捕食者间无合作时,正平衡点为稳定焦点,随着合作捕食参数α的增大,系统出现稳定的极限环且随α的增大不断胀大。说明如果捕食者种群密度过大,系统将产生持续的周期振荡,即食饵、中级捕食者、顶层捕食者要么以周期振荡的形式共存,要么种群数最终趋于稳定。因此,合作狩猎更有利于维护生态平衡。

In order to study the impact of cooperative behaviors between predators on the dynamic behavior of the food chain system,a top-level predator(which only feeds on the predator population)on the basis of the original cooperative hunting model is introduced,and a food chain model with cooperative hunting is established.The local stability of the equilibrium point is discussed by using the linearization method.By establishing an appropriate Lyapunov function,a sufficient condition for the global stability of the system at the equilibrium point is given.In addition,by using the central manifold reduction theorem,the explicit formulas of the branch periodic solution is studied.Numerical simulations are conducted to verify our theoretical analysis.The results show that when there is no cooperative relationship between predators,the positive equilibrium point is stable.With the increase of cooperative predation parameterα,the system will have a stable limit cycle,and the limit cycle will continue to expand following the increase of cooperative predation parameterα.Moreover,if the predator population density is too large,the system will produce continuous periodic oscillation,that is,the three species i.e.diet,mid-level predator,top-level predator,either coexist in the form of periodic oscillation,or the number of species eventually tends to be stable.Therefore,cooperative hunting is more conducive to maintaining ecological balance.