In this paper, we deal with the existence and stability of an equilibrium age distribution of the linearly implicit Euler-Riemann method for nonlinear age-structured population model with density dependence, i.e., the Gurtin-MacCamy models. It is shown that a dynamical invariance is replicated by numerical solutions for a long time. With the help of infinite-dimensional Leslie operators, the numerical processes are embedded into a nonlinear dynamical process in an infinite dimensional space, which provides a numerical basic reproduction function and numerical endemic equilibrium distributions. As an application to the Logistic model, a numerical reproduction number $R_0^h$ ensures the global stability of disease-free equilibrium whenever $R_0^h<1$ and the existence of the numerical endemic equilibrium for $R_0^h>1$. Moreover, instead of the convergence of numerical solutions, it is much more interesting that the numerical solutions preserve the existence of endemic equilibrium for small stepsize, since the numerical reproduction numbers, numerical endemic equilibrium and distribution converge to the exact ones with accuracy of order 1. Finally, some numerical experiments illustrate the verification and the efficiency of our results.

在本文中，我们处理具有密度依赖性的非线性年龄结构人口模型（即Gurtin-MacCamy模型）的线性隐式Euler-Riemann方法的平衡年龄分布的存在性和稳定性。结果表明，动力学不变性在很长一段时间内都可以通过数值解得到重复。借助无穷维Leslie算子，将数值过程嵌入到无穷维空间中的非线性动力学过程中，该过程提供了数值基本的再生函数和数值局部均衡分布。作为Logistic模型的应用，数值复制数${\mathrm{[R}}_0^H$ 确保无病平衡的全局稳定性 ${\mathrm{[R}}_0^H<\mathrm{1个}$ 和存在的数值地方均衡的存在 ${\mathrm{[R}}_0^H>\mathrm{1个}$。此外，代替数值解的收敛，更有趣的是，数值解保留了小步长的地方均衡的存在，因为数值复制数，数值地方均衡和分布收敛到精确的阶次为1的精度。最后，一些数值实验说明了我们的结果的有效性和验证性。