A convergence analysis for the approximation to the solution of an age-structured population model with infinite lifespan

Considering the numerical approximation of the density distribution for an age-structured population model with unbounded lifespan on a compact interval $[0,T]$, we prove second order of convergence for a discretization that adaptively selects its truncated age-interval according to the exponential rate of decay with age of the solution of the model. It appears that the adaptive capacity of the length in the truncated age-interval of the discretization to the infinity lifespan is a very convenient approach for a long-time integration of the model to establish the asymptotic behavior of its dynamics numerically. The analysis of convergence uses an appropriate weighted maximum norm with exponential weights to cope with the unbounded age lifespan. We report experiments to exhibit numerically the theoretical results and the asymptotic behavior of the dynamics for an age-structured squirrel population model introduced by Sulsky.