具有年龄结构种群系统的数值解研究
发布时间：2019-02-19 20:52
【摘要】：近几年来,具有年龄结构种群系统被广泛关注.然而,因此数值解的研究就显得尤为重要.本文讨论具有年龄结构种群系统的数值解.该论文的研究内容主要有以下几个方面:(1)讨论了具有年龄结构种群扩散系统反问题的数值解.对原系统变形后建立了具有高精度的四阶Pad(?)差分格式来计算种群的密度和扩散系数,该格式的截断误差为O(Δt~2+h~4)并且无条件稳定,所得结果能更准确的描述种群密度和扩散系数.且数值算例验证了方法的精确性和可靠性.(2)讨论了具有年龄结构随机种群系统数值解问题.在线性增长条件下,利用Euler-Maruyama(EM)方法讨论了数值解的阶矩渐近有界性,并获得了渐近有界性准则.通过数值算例对所得的结论进行了验证.(3)将分裂倒向Euler法(SSBE)应用于具有年龄结构随机种群系统,减弱了上一问题中的假设条件.首先讨论了解的存在唯一性,其次利用离散半鞅收敛定理,建立了分裂倒向Euler法对应数值解的几乎必然指数稳定性的判定准则.最后,用数值算例验证了所得的结果.
[Abstract]:In recent years, the population system with age structure has been widely concerned. However, the study of numerical solutions is particularly important. In this paper, the numerical solution of population system with age structure is discussed. The main contents of this paper are as follows: (1) numerical solution of inverse problem of population diffusion system with age structure is discussed. The fourth order Pad (?) with high accuracy was established after the original system was deformed. The difference scheme is used to calculate the population density and diffusion coefficient. The truncation error of the scheme is O (螖 t ~ 2 h ~ (4) and is unconditionally stable. The obtained results can more accurately describe the population density and diffusion coefficient. Numerical examples verify the accuracy and reliability of the method. (2) the numerical solution of stochastic population system with age structure is discussed. Under the condition of linear growth, the asymptotic boundedness of the moment of numerical solutions is discussed by using Euler-Maruyama (EM) method, and the asymptotic boundedness criterion is obtained. The results obtained are verified by numerical examples. (3) the split backward Euler method (SSBE) is applied to the stochastic population system with age structure, which weakens the hypothesis in the previous problem. First, the existence and uniqueness of the solution are discussed. Then, by using the convergence theorem of discrete semimartingale, the criterion of almost inevitable exponential stability of the corresponding numerical solution of the split backward Euler method is established. Finally, numerical examples are used to verify the results obtained.
【学位授予单位】：北方民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2426867
[Abstract]:In recent years, the population system with age structure has been widely concerned. However, the study of numerical solutions is particularly important. In this paper, the numerical solution of population system with age structure is discussed. The main contents of this paper are as follows: (1) numerical solution of inverse problem of population diffusion system with age structure is discussed. The fourth order Pad (?) with high accuracy was established after the original system was deformed. The difference scheme is used to calculate the population density and diffusion coefficient. The truncation error of the scheme is O (螖 t ~ 2 h ~ (4) and is unconditionally stable. The obtained results can more accurately describe the population density and diffusion coefficient. Numerical examples verify the accuracy and reliability of the method. (2) the numerical solution of stochastic population system with age structure is discussed. Under the condition of linear growth, the asymptotic boundedness of the moment of numerical solutions is discussed by using Euler-Maruyama (EM) method, and the asymptotic boundedness criterion is obtained. The results obtained are verified by numerical examples. (3) the split backward Euler method (SSBE) is applied to the stochastic population system with age structure, which weakens the hypothesis in the previous problem. First, the existence and uniqueness of the solution are discussed. Then, by using the convergence theorem of discrete semimartingale, the criterion of almost inevitable exponential stability of the corresponding numerical solution of the split backward Euler method is established. Finally, numerical examples are used to verify the results obtained.
【学位授予单位】：北方民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
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