A Singular Perturbation in an Age-Structured Population Model

Abstract

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 process and the second one is comparatively fast and describes the migration process between different spatial patches. From a mathematical point of view the model is a linear system of partial differential equations, where the state variables are the population densities in each spatial patch, together with a boundary condition of integral type, the birth equation. Due to the two different time scales, the system depends on a small parameter $\varepsilon$ and can be thought of as a singular perturbation problem. The main results of the work are that, for $\varepsilon>0$ small enough, the solutions of the 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 equilibrium. The state variable of the scalar system represents the global density of the population. The birth equation causes a singularity for ages close 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 flatfish (sole) of the Bay of Biscay. The general model of fish population dynamics considered throughout the paper was elaborated as part of this study.