A model for an age-structured population with two time scales

In the modelisation of the dynamics of a sole population, an interesting is sue is the influence of daily vertical migrations of the larvae on the whol e dynamical process. As a first step towards getting some insight on that i ssue, we propose a model that describes the dynamics of an age-structured p opulation living in an environment divided into N different spatial patches . We distinguish two time scales: at the fast time scale, we have migration dynamics and at the slow time scale, the demographic dynamics. The demogra phic process is described using the classical McKendrick model for each pat ch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conse rvative with respect to the total population and some additional technical assumptions, we proved in a previous work that the semigroup associated to our problem has the property of positive asynchronous exponential growth an d that the characteristic elements of that asymptotic behaviour can be appr oximated by those of a scalar classical McKendrick model. In the present wo rk, we develop the study of the nature of the convergence of the solutions of our problem to the solutions of the associated scalar one when the ratio between the time scales is epsilon (0 < epsilon much less than 1). The mai n result decomposes the action of the semigroup associated to our problem i nto three parts: (1) the semigroup associated to a demographic scalar problem times the vect or of the equilibrium distribution of the migration process; (2) the semigroup associated to the transitory process which leads to the f irst part; and (3) an operator, bounded in norm, of order epsilon. (C) 2000 Elsevier Scien ce Ltd. All rights reserved.