更一般与年龄相关随机时滞种群方程的全局稳定性
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摘要
讨论更一般的与年龄相关随机时滞种群方程的全局稳定性.如果传统假设(Lipschitz条件)缺失,与年龄相关随机时滞种群方程可能有多于一个弱解.然而,大量文献研究结果是在此类方程有唯一强解前提下获得.因此,有必要对更一般的有多于一个弱解情况进行相关概念推广.对更一般的与年龄相关随机时滞种群方程,随机稳定性概念被提出,一般的Barbashin-Krasovskii定理和Lasalle定理被建立,涵盖了多于一个弱解的情况.显然,这两个定理给出随机时滞种群方程稳定性的判定标准,并且通过实例说明定理的有效性.
This paper considers the global stability of more general age-dependent stochastic delay population equations. Due to the lack of the traditional assumption(Lipschitz condition),age-dependent stochastic delay population equations may have more than one weak solution.However, the most associated results are only applicable to these equations having a unique strong solution. So it is necessary to extend the relevant concepts and methods to the general case. For more general age-dependent stochastic delay population equations, the concept of the stochastic stability is put forward, and the general Barbashin-Krasovskii theorem and Lasalle theorem are established, covering more than a weak solution of age-dependent stochastic delay population equations. Obviously, the two theorems present criterions of stochastic stability,and the example is given to illustrate the validity of the theorem.
This paper considers the global stability of more general age-dependent stochastic delay population equations. Due to the lack of the traditional assumption(Lipschitz condition),age-dependent stochastic delay population equations may have more than one weak solution.However, the most associated results are only applicable to these equations having a unique strong solution. So it is necessary to extend the relevant concepts and methods to the general case. For more general age-dependent stochastic delay population equations, the concept of the stochastic stability is put forward, and the general Barbashin-Krasovskii theorem and Lasalle theorem are established, covering more than a weak solution of age-dependent stochastic delay population equations. Obviously, the two theorems present criterions of stochastic stability,and the example is given to illustrate the validity of the theorem.
引文
[1] Zhang Q M, Liu W A and Nie Z K, Existence, Uniqueness and exponential stability for stocastic age-dependent population[J]. Applied Mathematics and compution, 2004, 154, 183-201.
[2]李荣华,戴永红,孟红兵.与年龄相关的随机时滞种群方程的指数稳定性[J].数学年刊,2006, 27A(1):39-52.
[3]申芳芳,张启敏,杨洪福,李西宁.与年龄相关的模糊随机种群扩散系统的指数稳定性[J].数学的实践与认识,2014, 44(7):187-195.
[4] Mao X R. A Note on the Lasalle-Type Theorems for stochastic differential Delay Equation[J]. J Math Anal Appl, 2002, 268(1):125-142.
[5] Mao X R, Yuan C G, Zou J Z. stochastic diferential delay equation of population dynamics[J]. J Math Anal Appl, 2005, 304(1):296-320.
[6] A A Levakov. stability analysis of stochastic differential equations with the use of Lyapunov functions of constant sign[J]. Differ Equ, 2011, 47(9):1271-1280.
[7] O Ignatyev, V Mandrekar, Barbashin-Krasovskii theorem for stochastic differential equation[J], Proc Amer Math Soc, 2010, 138(11):4123-4128.
[8] Li F Z, Liu Y G. Global stability and stabilization of more general stochastic nonlinear systems[J].J Math Anal Appl, 2014, 413:841-855.
[9] I Karatzas, S E Shreve. Brownian Motion and Stochastic Calculus[M], Springer-Verlag, New York,1991.
[10] N Ikdea, S Watanabe. Stochastic Differential Equations and Diffusion Processes[M], North-Holland Publishing, Amsterdam, 1989.
[2]李荣华,戴永红,孟红兵.与年龄相关的随机时滞种群方程的指数稳定性[J].数学年刊,2006, 27A(1):39-52.
[3]申芳芳,张启敏,杨洪福,李西宁.与年龄相关的模糊随机种群扩散系统的指数稳定性[J].数学的实践与认识,2014, 44(7):187-195.
[4] Mao X R. A Note on the Lasalle-Type Theorems for stochastic differential Delay Equation[J]. J Math Anal Appl, 2002, 268(1):125-142.
[5] Mao X R, Yuan C G, Zou J Z. stochastic diferential delay equation of population dynamics[J]. J Math Anal Appl, 2005, 304(1):296-320.
[6] A A Levakov. stability analysis of stochastic differential equations with the use of Lyapunov functions of constant sign[J]. Differ Equ, 2011, 47(9):1271-1280.
[7] O Ignatyev, V Mandrekar, Barbashin-Krasovskii theorem for stochastic differential equation[J], Proc Amer Math Soc, 2010, 138(11):4123-4128.
[8] Li F Z, Liu Y G. Global stability and stabilization of more general stochastic nonlinear systems[J].J Math Anal Appl, 2014, 413:841-855.
[9] I Karatzas, S E Shreve. Brownian Motion and Stochastic Calculus[M], Springer-Verlag, New York,1991.
[10] N Ikdea, S Watanabe. Stochastic Differential Equations and Diffusion Processes[M], North-Holland Publishing, Amsterdam, 1989.