具有常数收获的阶段结构种群模型的全局分支分析  

Global bifurcation analysis of a stage structured population model with constant harvesting

作　　者：张婧[1] 范丽[1] 郭艳阳 ZHANG Jing;FAN Li;GUO Yan-yang(School of Mathematics and Statistics,Shaanxi Normal University,Xi'an 710119,China)

机构地区：[1]陕西师范大学数学与统计学院,西安710119

出　　处：《兰州大学学报（自然科学版）》2022年第5期678-688,共11页Journal of Lanzhou University(Natural Sciences)

基　　金：国家自然科学基金面上项目(11771262)。

摘　　要：讨论具有常数收获率和年龄阶段结构的种群模型的稳定性及分支问题.应用分支分析的理论及解析方法,讨论该模型的鞍结分支、尖分支、Hopf分支和Bogdanov-Takens(BT)分支等多种分支,得到各种分支产生的条件、分支曲线计算公式和分支周期解的稳定性等.通过一系列近恒同变换推导了等价的余维二BT分支规范型和高余维退化BT分支的三参数开折规范型,证明BT分支点处平衡点为余维二尖点或二阶对称幂零尖点,分别讨论了BT和退化BT分支点附近系统的分支情况.给出了完整的分支图和各区域相轨线结构,分支分析结果表明收获项会引起种群数量的复杂变化,通过数值模拟验证了理论分析结果的准确性.The stability and bifurcation of a model with a constant harvesting rate and age stage structure were discussed. Via the analytical method of bifurcation analysis, the saddle node bifurcation, cusp bifurcation, Hopf bifurcation and Bogdanov-Takens(BT) bifurcation of the model were explored. The conditions for the generation of various bifurcations, the calculation formula of bifurcation curves and the stability of periodic solution to bifurcation were obtained. In particular, the equivalent codimensional two BT bifurcation norm forms and three parameters unfolding norm forms of higher codimension degenerate BT bifurcation were derived by means of a series of near-invariant transformations. It is proved that the equilibrium at BT bifurcation point was a cusp of codimension two or a nilpotent symmetric cusp of order two, and the bifurcation of systems near BT and degenerate BT bifurcation point was discussed respectively. The complete bifurcation diagram and phase portraits in each region were given. The results of bifurcation analysis also showed that the harvest term could cause complex changes in population quantity.The accuracy of the theoretical analysis results was verified by numerical simulation.

关 键 词：阶段结构种群模型 常数收获 Bogdanov-Takens分支 退化Bogdanov-Takens分支 

分 类 号：O175.14[理学—数学] Q141[理学—基础数学]