Convergence in a multi-layer population model with age-structure

In this paper we investigate the convergence of a multi-layer population model to a single-layer limit. In a previous paper Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223], we considered a Gurtin–MacCamy type model based on the fact that the population diffuses through a one dimensional habitat, partitioned into $n$ homogeneous layers. In each layer the population has its own age-dependent growth and diffusion parameters, so that within each layer the dynamics is not subject to environmental variations, while changes occur from a layer to another, according to different conditions. Such kind of a model may describe the growth of a population in a fragmented environment, but the same framework may be used to give an approximate view of the population growth and diffusion in a general spatially heterogeneous habitat, because the layer structure may arise by approximation of the original problem.In the present paper we show that this view is actually mathematically sound and justified. In fact, on the basis of the previous results (see Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223]) the approximating problem actually converges and the multi-layer solution may be considered a patch-wise picture of the original problem.