一类延迟Gompertz方程的数值解的振动性分析

An Oscillation Analysis of Numerical Solution of A Class of Delayed Gompertz Equations

Gompertz方程常用于描述种群动态和肿瘤生长,本文研究了一类延迟Gompertz方程的振动性。首先利用泰勒公式线性化该方程,再对线性方程应用线性θ方法得到其差分格式。其次,运用振动理论分别分析线性化后的方程和所得差分格式。在研究方程数值解的振动性时,把差分方程中θ的取值范围分为2部分,通过分析差分方程的特征方程的解的性质,得到延迟Gompertz方程的解析解和数值解振动的充分条件,最后进行数值实验验证。

Oscillation of numerical solutions is studied with regard to a class of delayed Gompertz equations,which have been widely used in description of the population dynamics and tumour growth.Firstly,the linearized equations are obtained by Taylor formula and its corresponding difference equations by linearθmethod.Secondly,the oscillation theory is applied to analyze those obtained equations.In the process,the oscillation is primarily discussed through studying the properties of roots for the corresponding characteristic equations.For requirement,the oscillation of numerical solutions is discussed while the variableθbelongs in different scopes.Accordingly,the sufficient conditions under which numerical solutions oscillate are acquired.To verify the results,some numerical experiments are given.The first three experiments validate the conditions of the delayed Gompertz equation which has three delay terms.And the rest of the experiments check on another which has two delay terms.