带Poisson跳年龄相关随机时滞种群系统的均方稳定性
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摘要
本文研究了带Poisson跳年龄相关随机时滞种群系统均方稳定性的问题.在一定条件下,给出了数值解均方稳定的定义.利用补偿随机θ法讨论系统数值解的均方稳定性,给出数值解稳定的充分条件.获得了当1/2≤θ≤1时,对于任意的步长?τ/m,数值解是均方稳定的;当0≤θ<1,时,如果步长?t∈(0,?t0),数值解是指数均方稳定的的结果.最后通过数值算例推广并验证了结果的有效性和正确性.
This paper deals with the mean-square stability problem of stochastic agedependent delay population systems with Poisson jumps. Under the certain conditions, the definition of mean-square stability of the numerical solution is given. By utilizing the compensated stochastic θ methods, the mean-square stability of the numerical solution is investigated and a sufficient condition for mean-square stability of the numerical solution is presented. It is shown that the compensated stochastic θ methods are mean-square stable for any stepsize ?τ/m when1/2 ≤ θ ≤ 1, and they are exponentially mean-square stable if the stepsize ?t ∈(0, ?t0) when0 ≤ θ < 1. Finally, the theoretical results are also confirmed by a numerical experiment.
This paper deals with the mean-square stability problem of stochastic agedependent delay population systems with Poisson jumps. Under the certain conditions, the definition of mean-square stability of the numerical solution is given. By utilizing the compensated stochastic θ methods, the mean-square stability of the numerical solution is investigated and a sufficient condition for mean-square stability of the numerical solution is presented. It is shown that the compensated stochastic θ methods are mean-square stable for any stepsize ?τ/m when1/2 ≤ θ ≤ 1, and they are exponentially mean-square stable if the stepsize ?t ∈(0, ?t0) when0 ≤ θ < 1. Finally, the theoretical results are also confirmed by a numerical experiment.
引文
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[10] Wang W Q, Chen Y P. Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations[J]. Appl. Numer. Math., 2011, 61(5):696–701.
[11] Wang L S, Wang X J. Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps[J]. Appl. Math. Model., 2010, 34(8):2034–2043.
[12]辛志贤,张启敏,李强,哈金才.具有年龄结构随机种群系统数值解的渐近有界性[J].数学杂志, 2018,38(1):147–154.
[13]李强,张启敏.广义Khasminskii-type条件下与年龄相关随机时滞种群系统的数值解[J].数学的实践与认识, 2016, 46(18):159–169.
[14] Ma W J, Zhang Q M, Wang Z P. Asymptotic stability of stochastic age-dependent population equations with Markovian switching[J]. Appl. Math. Comput., 2014, 227(227):309–319.
[15] Ma W J, Ding B C, Zhang Q M. The existence and asymptotic behaviour of energy solutions to stochastic age-dependent population equations driven by Levy processes[J]. Appl. Math. Comput.,2015, 256:656–665.
[16] Mao X R. Stochastic differential equations and applications[M]. 2nd ed., Chichester:Horwood Publishing, 2007.
[17] Baker C T H, Buckwar E. On Halanay-type analysis of exponential stability for the theta-Maruyama method for stochastic delay differential equations[J]. Stoch. Dynam., 2005, 5(2):201–209.
[2] Jacod J, Shiryaev A N. Limit theorems for stochastic processes(2nd ed.)[M]. New York:Springer Verlag, 2002.
[3] Mao X R, Shah A. Exponential stability of stochastic differential equations[M]. New York:Marcel Dekker, 1994.
[4] Zhang Q M, Liu W A, Nie Z K. Existence, uniqueness and exponential stability for stochastic age-dependent population[J]. Appl. Math. Comput., 2004, 154(1):183–201.
[5] Li R H, Leung P K, Pang W K. Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching[J]. Appl. Math. Comput., 2009, 233(4):1046–1055.
[6] Ma W J, Zhang Q M, Han C Z. Numerical analysis for stochastic age-dependent population equations with fractional Brownian motion[J]. Commun. Nonl. Sci. Numer. Simul., 2012, 17(4):1884–1893.
[7] Pang W K, Li R H, Liu M. Convergence of the semi-implicit Euler method for stochastic agedependent population equations[J]. Appl. Math. Comput., 2008, 195(2):466–474.
[8] Sobczyk K. Stochastic differential equations with applications to physics and engineerin[M]. Netherlands:Springer 1991.
[9] Li Q Y, Gan S Q. Stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps[J]. Abst. Appl. Anal., 2012, 2012(2):183–194.
[10] Wang W Q, Chen Y P. Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations[J]. Appl. Numer. Math., 2011, 61(5):696–701.
[11] Wang L S, Wang X J. Convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps[J]. Appl. Math. Model., 2010, 34(8):2034–2043.
[12]辛志贤,张启敏,李强,哈金才.具有年龄结构随机种群系统数值解的渐近有界性[J].数学杂志, 2018,38(1):147–154.
[13]李强,张启敏.广义Khasminskii-type条件下与年龄相关随机时滞种群系统的数值解[J].数学的实践与认识, 2016, 46(18):159–169.
[14] Ma W J, Zhang Q M, Wang Z P. Asymptotic stability of stochastic age-dependent population equations with Markovian switching[J]. Appl. Math. Comput., 2014, 227(227):309–319.
[15] Ma W J, Ding B C, Zhang Q M. The existence and asymptotic behaviour of energy solutions to stochastic age-dependent population equations driven by Levy processes[J]. Appl. Math. Comput.,2015, 256:656–665.
[16] Mao X R. Stochastic differential equations and applications[M]. 2nd ed., Chichester:Horwood Publishing, 2007.
[17] Baker C T H, Buckwar E. On Halanay-type analysis of exponential stability for the theta-Maruyama method for stochastic delay differential equations[J]. Stoch. Dynam., 2005, 5(2):201–209.