一类具年龄结构的中立型种群模型的分支研究
发布时间：2019-05-23 22:20
【摘要】：随着生物数学理论的不断发展,中立型泛函微分方程已经被越来越广泛地用于描述生物种群模型的演化规律。中立型泛函微分方程一般被用来描绘当前时刻状态变化率依赖于历史时刻状态变化率的发展系统,在种群模型中这意味着当前时刻的种群数量增长率依赖于历史某时刻的增长率。首先,对具年龄结构的双曲模型进行约化,得到了一类具有年龄结构的种群增长的中立型方程。通过选择不同的出生函数,得到了两类要研究的模型,第一类模型的增长率按照logistic形式进行,第二类模型的增长率按照指数形式进行。其次,针对第一类模型,讨论了模型平衡解的稳定性,通过分析特征方程,得到了关于零解和正平衡解的稳定性结果。接下来分成两种情形研究了方程的Hopf分支性质。第一种是幼年个体死亡率被忽略的情形;第二种是幼年个体死亡率没有被忽略的情形。应用中心流形定理与规范型理论,研究了正平衡解处的Hopf分支方向与分支周期解的稳定性。针对第二类模型,研究了按照指数形式增长的微分方程平衡解的稳定性,并分两种情形研究了方程的Hopf分支性质。此外,得到了正平衡解处的Hopf分支方向与分支周期解的稳定性。最后,以第二类模型为例研究了方程的全局Hopf分支。利用全局Hopf分支定理给出了方程周期解的大范围存在性条件。同时,对理论分析结果给予了数值算例支撑。
[Abstract]:With the continuous development of biological mathematics theory, neutral functional differential equations have been more and more widely used to describe the evolution of biological population models. Neutral functional differential equations are generally used to describe the development system in which the state change rate of the current time depends on the state change rate of the historical moment. In the population model, this means that the population growth rate at the current time depends on the growth rate at a certain time in history. Firstly, the hyperbolic model with age structure is reduced, and a class of neutral equation of population growth with age structure is obtained. By selecting different birth functions, two kinds of models to be studied are obtained. the growth rate of the first model is carried out in the form of logistic, and the growth rate of the second model is carried out in the exponential form. Secondly, for the first kind of model, the stability of the equilibrium solution of the model is discussed, and the stability results of the zero solution and the positive equilibrium solution are obtained by analyzing the characteristic equation. Next, the Hopf bifurcation properties of the equation are studied in two cases. The first is that the infant mortality rate is ignored; the second is the case where the juvenile individual mortality rate is not ignored. In this paper, the Hopf bifurcation direction and the stability of bifurcation periodic solutions at the positive equilibrium solution are studied by using the central manifolds theorem and the canonical form theory. For the second kind of model, the stability of equilibrium solutions of differential equations growing in exponential form is studied, and the Hopf bifurcation properties of the equations are studied in two cases. In addition, the stability of the Hopf bifurcation direction and the bifurcation periodic solution at the positive equilibrium solution is obtained. Finally, the global Hopf bifurcation of the equation is studied by taking the second kind of model as an example. By using the global Hopf bifurcation theorem, the existence conditions of periodic solutions for the equation are given. At the same time, numerical examples are given to support the theoretical analysis results.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2484280
[Abstract]:With the continuous development of biological mathematics theory, neutral functional differential equations have been more and more widely used to describe the evolution of biological population models. Neutral functional differential equations are generally used to describe the development system in which the state change rate of the current time depends on the state change rate of the historical moment. In the population model, this means that the population growth rate at the current time depends on the growth rate at a certain time in history. Firstly, the hyperbolic model with age structure is reduced, and a class of neutral equation of population growth with age structure is obtained. By selecting different birth functions, two kinds of models to be studied are obtained. the growth rate of the first model is carried out in the form of logistic, and the growth rate of the second model is carried out in the exponential form. Secondly, for the first kind of model, the stability of the equilibrium solution of the model is discussed, and the stability results of the zero solution and the positive equilibrium solution are obtained by analyzing the characteristic equation. Next, the Hopf bifurcation properties of the equation are studied in two cases. The first is that the infant mortality rate is ignored; the second is the case where the juvenile individual mortality rate is not ignored. In this paper, the Hopf bifurcation direction and the stability of bifurcation periodic solutions at the positive equilibrium solution are studied by using the central manifolds theorem and the canonical form theory. For the second kind of model, the stability of equilibrium solutions of differential equations growing in exponential form is studied, and the Hopf bifurcation properties of the equations are studied in two cases. In addition, the stability of the Hopf bifurcation direction and the bifurcation periodic solution at the positive equilibrium solution is obtained. Finally, the global Hopf bifurcation of the equation is studied by taking the second kind of model as an example. By using the global Hopf bifurcation theorem, the existence conditions of periodic solutions for the equation are given. At the same time, numerical examples are given to support the theoretical analysis results.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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1 苏颖;单种群模型的分支问题[D];哈尔滨工业大学;2011年
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