一类具有Holling III-II功能函数的三种群模型的余维二Bogdanov-Takens分岔研究

Research on Bogdanov-Takens Bifurcation of a Tritrophic Model with Holling III-II Functional Responses

摘要: 针对一类Leslie型三种群食物链模型，分析了该模型的余维二Bogdanov-Takens分岔行为。假设猎物满足Logistic增长速率，捕食者种群通过Holling III功能反应吃掉猎物，而超级捕食者种群则通过Holling II功能反应吃掉捕食者。首先，给出了保证系统具有正平衡点的条件，及在该正平衡点发生Bogdanov-Takens分岔的参数条件；其次，选取了满足上述条件的适当参数，对Bogdanov-Takens分岔的非退化条件及正则性条件分别进行了验证；最后，通过数值仿真，对Bogdanov-Takens平衡点进行了参数扰动，获得了分岔后的不同动力学行为。

Abstract: For a class of Leslie type tritrophic food chain models, the codimension-2 Bogdanov-Takens bifurcation behavior of the model was analyzed. Suppose the prey follows the Logistic growth rate, the predator population feeds on the prey through the Holling III functional response, while the superpredator population feeds on the predator through the Holling II functional response. Firstly, the conditions for ensuring that the system has a positive equilibrium point are given, as well as the parameter conditions for the Bogdanov-Takens bifurcation to occur at this positive equilibrium point. Secondly, appropriate parameters that meet the above conditions were selected, and the non-degenerate condition and regularity condition of the Bogdanov-Takens bifurcation were verified respectively. Finally, through numerical simulation, parameter perturbation was studied on the Bogdanov-Takens equilibrium point, and different dynamic behaviors were bifurcated.