Numerical analysis of linear θ-methods with two-layer boundary conditions for age-structured population models

In this paper, we consider a fully discretization scheme for infinite age-structured population models with time-variable fertility rate and mortality rate. Based on the characteristics, the classical linear θ-methods with a kind of two-layer boundary condition are constructed for preserving an invariance of total populations. We are interested in the finite-time convergence and the stability for a long time. With the classical approach, some conjecture on the first order convergence is proved. For the time-independent model the numerical stability is studied by an embedded infinite dimensional dynamical system, which provides a numerical basic reproduction number by the infinite Leslie operator. Furthermore, it is shown that the numerical solutions replicate the un-stability and stability of the analytical solutions for small stepsize. Finally, three examples are given to verify the feasibility of our methods.