与年龄相关的随机种群模型数值解的散逸性
发布时间：2018-11-18 20:47
【摘要】：目前,在金融、生物、化学、通讯等多个研究领域中随机微分方程理论都已被普遍地应用.但是在实际生活中,任何领域中都将会出现各种各样随机因素的影响.因此,借助随机扰动参数对微分方程的研究更具有说服力,更符合真实反映.本文在Brown运动及Poisson过程产生扰动情况下对随机微分系统的散逸性进行了研究.另一方面,由于随机系统自身的复杂性,通常情况下随机微分方程大都无精确解或精确解难以解出,带Poisson跳的方程更是这般.因而,借助数值方法对随机微分方程的解以及其性质的分析就显得更为重要.本文的主要工作是探究了与年龄相关的随机种群模型数值解的散逸性问题.内容主要包括下面三方面:(1)讨论了一类基于倒向Euler法的随机种群模型数值解的均方散逸性.利用倒向Euler法以及根据其步长h受限制和无限制的两种条件下,对该随机种群模型数值解的均方散逸性进行研究并加以证明.最后通过数值例子以及结合MATLAB软件包演示了结果的有效性.(2)利用 Ito 公式、Cauchy-Schwarz 不等式和 Bellman-Gronwall-Type 估计式,在满足假设条件的情况下讨论了随机种群模型数值解的均方散逸性.并利用分步倒向Euler法和补偿的分步倒向Euler法证明了此系统数值解的均方散逸性,最后借助数值实例对本章重要的结论加以验证.(3)借助Lyapunov函数、Barbashin-Krasovskii定理及Ito公式讨论了基于年龄结构的随机种群模型数值解的全局稳定性问题.并且对该模型强解的存在性加以分析验证,从而获得该模型零解依概率全局稳定的充分条件;最后利用数值例子结合MATLAB软件包对结论的有效性进行演示.
[Abstract]:At present, stochastic differential equation theory has been widely used in finance, biology, chemistry, communication and so on. However, in real life, there will be a variety of random factors in any field. Therefore, the study of differential equations with stochastic perturbation parameters is more persuasive and more consistent with the real reflection. In this paper, the escapes of stochastic differential systems are studied in the case of Brown motion and perturbation of Poisson processes. On the other hand, due to the complexity of the stochastic system itself, most stochastic differential equations have no exact solution or exact solution, especially the equation with Poisson jump. Therefore, it is more important to analyze the solutions and properties of stochastic differential equations by numerical method. The main work of this paper is to investigate the problem of the numerical solution of the Age-related stochastic population model. The main contents are as follows: (1) the mean-square escape of numerical solutions for a class of stochastic population models based on backward Euler method is discussed. By using the backward Euler method and under the condition that the step size h is restricted and unrestricted, the mean square escape of the numerical solution of the stochastic population model is studied and proved. Finally, numerical examples and MATLAB software package are used to demonstrate the validity of the results. (2) using Ito formula, Cauchy-Schwarz inequality and Bellman-Gronwall-Type estimator, The mean square escape of the numerical solution of the stochastic population model is discussed under the condition that the assumption is satisfied. The mean-square escape of the numerical solution of the system is proved by the stepwise backward Euler method and the compensated stepwise backward Euler method. Finally, the important conclusions of this chapter are verified by numerical examples. (3) with the help of the Lyapunov function, Barbashin-Krasovskii theorem and Ito formula discuss the global stability of numerical solution of stochastic population model based on age structure. The existence of strong solutions of the model is analyzed and verified, and a sufficient condition for the global stability of the model with probability of zero solution is obtained. Finally, the validity of the conclusion is demonstrated by a numerical example combined with MATLAB software package.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2341177
[Abstract]:At present, stochastic differential equation theory has been widely used in finance, biology, chemistry, communication and so on. However, in real life, there will be a variety of random factors in any field. Therefore, the study of differential equations with stochastic perturbation parameters is more persuasive and more consistent with the real reflection. In this paper, the escapes of stochastic differential systems are studied in the case of Brown motion and perturbation of Poisson processes. On the other hand, due to the complexity of the stochastic system itself, most stochastic differential equations have no exact solution or exact solution, especially the equation with Poisson jump. Therefore, it is more important to analyze the solutions and properties of stochastic differential equations by numerical method. The main work of this paper is to investigate the problem of the numerical solution of the Age-related stochastic population model. The main contents are as follows: (1) the mean-square escape of numerical solutions for a class of stochastic population models based on backward Euler method is discussed. By using the backward Euler method and under the condition that the step size h is restricted and unrestricted, the mean square escape of the numerical solution of the stochastic population model is studied and proved. Finally, numerical examples and MATLAB software package are used to demonstrate the validity of the results. (2) using Ito formula, Cauchy-Schwarz inequality and Bellman-Gronwall-Type estimator, The mean square escape of the numerical solution of the stochastic population model is discussed under the condition that the assumption is satisfied. The mean-square escape of the numerical solution of the system is proved by the stepwise backward Euler method and the compensated stepwise backward Euler method. Finally, the important conclusions of this chapter are verified by numerical examples. (3) with the help of the Lyapunov function, Barbashin-Krasovskii theorem and Ito formula discuss the global stability of numerical solution of stochastic population model based on age structure. The existence of strong solutions of the model is analyzed and verified, and a sufficient condition for the global stability of the model with probability of zero solution is obtained. Finally, the validity of the conclusion is demonstrated by a numerical example combined with MATLAB software package.
【学位授予单位】：宁夏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
【参考文献】
相关硕士学位论文 前1条
1 王青;一类具有时滞非自治的Lotka-Volterra种群模型的动力学分析[D];北京交通大学;2016年
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