zbMATH — the first resource for mathematics

A model for an age-structured population with two time scales. (English) Zbl 1043.92519
Summary: In the modelling of the dynamics of a sole population, an interesting issue is the influence of daily vertical migrations of the larvae on the whole dynamical process. As a first step towards getting some insight on that issue, we propose a model that describes the dynamics of an age-structured population living in an environment divided into \(N\) different spatial patches. We distinguish two time scales: on the fast time scale, we have migration dynamics and on the slow time scale, the demographic dynamics. The demographic process is described using the classical McKendrick model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that the migration process is conservative 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 and that the characteristic elements of that asymptotic behaviour can be approximated by those of a scalar classical McKendrick model. In the present work, 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 \(\epsilon\) \((0<\epsilon\ll1)\). The main result decomposes the action of the semigroup associated to our problem into three parts: (1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process; (2) the semigroup associated to the transitory process which leads to the first part; and (3) an operator, bounded in norm, of order \(\epsilon\).

92D25 Population dynamics (general)
Full Text: DOI
[1] Koutsikopoulos, C.; Fortier, L.; Gagne, J.A., Cross-shelf dispersion of dover sole (solea solea) eggs and larvae in biscay bay and recruitment to inshore nurseries, Jour. plankton res., 13, 923-945, (1991)
[2] Champalbert, G.; Koutsikopoulos, C., Behaviour, transport and recruitment of biscay bay sole (solea solea): laboratory and field studies, J. mar. biol. ass. U.K., 75, 93-108, (1995)
[3] Arino, O.; Koutsikopoulos, C.; Ramzi, A., Elements of a model of the evolution of the density of a sole population, J. biol. systems, 4, 445-458, (1996)
[4] Arino, O.; Sánchez, E.; de la Parra, R.Bravo, A model of an age-structured population in a multipatch environment, Mathl. comput. modelling, 27, 4, 137-150, (1998) · Zbl 1185.35115
[5] de la Parra, R.Bravo; Auger, P.; Sánchez, E., Aggregation methods in discrete models, J. biol. systems, 3, 603-612, (1995)
[6] Sánchez, E.; de la Parra, R.Bravo; Auger, P., Linear discrete models with different time scales, Acta biotheoretica, 43, 465-479, (1995)
[7] Metz, J.A.J.; Diekmann, O., The dynamics of physiologically structured populations, () · Zbl 0614.92014
[8] Seneta, E., Nonnegative matrices and Markov chains, (1981), Springer-Verlag · Zbl 0471.60001
[9] Webb, G.F., Theory of nonlinear age-dependent population dynamics, (1985), Marcel Dekker New York · Zbl 0555.92014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.