Eva Sánchez

Aggregation of Variables and Applications to Population Dynamics

1 IRD UR Geodes, Centre IRD de l’Ile de France, 32, Av. Henri Varagnat, 93143 Bondy cedex, France pierre.auger@bondy.ird.fr 2 Departamento de Matematicas, Universidad de Alcala, 28871 Alcala de Henares, Madrid, Spain rafael.bravo@uah.es 3 Laboratoire de Microbiologie, Geochimie et d’Ecologie Marines, UMR 6117, Centre d’Oceanologie de Marseille (OSU), Universite de la Mediterranee, Case 901, Campus de Luminy, 13288 Marseille Cedex 9, France Jean-Christophe.Poggiale@univmed.fr 4 Departamento de Matematicas, E.T.S.I. Industriales, U.P.M., c/ Jose Gutierrez Abascal, 2, 28006 Madrid, Spain esanchez@etsii.upm.es 5 IXXI, ENS Lyon, 46 allee d’Italie, 69364 Lyon cedex 07, France tri.nguyen-huu@ens-lyon.fr

AGGREGATION METHODS IN DISCRETE MODELS

The aim of this work is to extend approximate aggregation methods for multi-time scale systems of ordinary differential equations to time discrete models. We give general methods in order to reduce a large scale time discrete model into an aggregated model for a few number of slow macro-variables. We study the case of linear systems. We demonstrate that the elements defining the asymptotic behaviours of the initial and aggregate models are similar to first order. We apply this method to the case of an agestructured population with sub-populations in each age classes associated to different spacial patches or different individual activities. A fast time scale is assumed for patch or activity dynamics with respect to aging and reproduction processes. Our method allows us to aggregate the system into a classical Leslie model in which the fecundity and aging parameters of the aggregated model are expressed in terms of the equilibrium proportions of individuals in the different activities or patches.

Linear discrete models with different time scales

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order.

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 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

A discrete model with density dependent fast migration.

\varepsilon

Time Scales in Density Dependent Discrete Models

and can be thought of as a singular perturbation problem. The main results of the work are that, for

Aggregation methods in population dynamics discrete models

\varepsilon>0

A predator–prey model with predators using hawk and dove tactics

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 ...

The impact of behavioral plasticity at individual level on domestic cat population dynamics

The aim of this work is to develop an approximate aggregation method for certain non-linear discrete models. Approximate aggregation consists in describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete models with two different time scales, the slow one considered to be linear and the fast one non-linear because of its transition matrix depends on the global variables. In our discrete model the time unit is chosen to be the one associated to the slow dynamics, and then we approximate the effect of fast dynamics by using a sufficiently large power of its corresponding transition matrix. In a previous work the same system is treated in the case of fast dynamics considered to be linear, conservative in the global variables and inducing a stable frequency distribution of the state variables. A similar non-linear model has also been studied which uses as time unit the one associated to the fast dynamics and has the non-linearity in the slow part of the system. In the present work we transform the system to make the global variables explicit, and we justify the quick derivation of the aggregated system. The local asymptotic behaviour of the aggregated system entails that of the general system under certain conditions, for instance, if the aggregated system has a stable hyperbolic fixed point then the general system has one too. The method is applied to aggregate a multiregional Leslie model with density dependent migration rates.

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

The aim of this work is to extend approximate aggregation methods for multi-time scale systems of nonlinear ordinary differential equations to time discrete models. Approximate aggregation consist on describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete time models with two different time scales, the fast one considered linear and the slow one generally nonlinear. We transform the system to make the global variables appear, and use a version of center manifold theory to build up the aggregated system. Simple forms of the aggregated system are enough for the local study of the asymptotic behaviour of the general system provided that it has certain stability under perturbations. The general method is applied to aggregate a multiregional density dependent Leslie model into a density dependent Leslie model in which the demographic rates are expressed in terms of the equilibrium proportions of individuals in the different patches.