A singular perturbation in an age-structured population model

The aim of this work is to study a model of age-structured population with two time scales: the first one is slow and corresponds to the demographic p rocess and the second one is comparatively fast and describes the migration process between different spatial patches. From a mathematical point of vi ew the model is a linear system of partial differential equations, where th e state variables are the population densities in each spatial patch, toget her with a boundary condition of integral type, the birth equation. Due to the two different time scales, the system depends on a small parameter epsi lon and can be thought of as a singular perturbation problem. The main resu lts of the work are that, for epsilon > 0 small enough, the solutions of th e system can be approximated by means of the solutions of a scalar problem, where the fast process has been avoided by supposing it has attained an eq uilibrium. The state variable of the scalar system represents the global de nsity of the population. The birth equation causes a singularity for ages c lose to 0 to appear, which produces a boundary layer type phenomenon. This work originated from the study of some fisheries of the West Coast of the Atlantic Ocean, namely, small pelagic fish (anchovy and sardine) and fl atfish (sole) of the Bay of Biscay. The general model of fish population dy namics considered throughout the paper was elaborated as part of this study .