Rafael Bravo de la Parra

Methods of aggregation of variables in population dynamics

In ecology, we are faced with modelling complex systems involving many variables corresponding to interacting populations structured in different compartmental classes, ages and spatial patches. Models that incorporate such a variety of aspects would lead to systems of equations with many variables and parameters. Mathematical analysis of these models would, in general, be impossible. In many real cases, the dynamics of the system corresponds to two or more time scales. For example, individual decisions can be rapid in comparison to growth of the populations. In that case, it is possible to perform aggregation methods that allow one to build a reduced model that governs the dynamics of a lower dimensional system, at a slow time scale. In this article, we present a review of aggregation methods for time continuous systems as well as for discrete models. We also present applications in population dynamics. A first example concerns a continuous time model of a single population distributed on a system of two connected patches (a logistic source and a sink), by fast migration. It is shown that under a certain condition, the total equilibrium population can be larger than the carrying capacity of the logistic source. A second example concerns a discrete model of a population distributed on two patches, still a source and a sink, connected by fast migration. The use of aggregation methods permits us to conclude that density-dependent migration can stabilize the total population.

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 mathematical model of growth of population of fish in the larval stage: Density-dependence effects

A mathematical model for the growth of a population of fish in the larval stage is proposed. The emphasis is put on the first part of the larval stage, when the larvae are still passive. It is assumed that during this stage, the larvae move with the phytoplankton on which they feed and share their food equally, leading to ratio-dependence. The other stages of the life cycle are modeled using simple demographic mechanisms. A distinguishing feature of the model is that the exit from the early larval stage as well as from the active one is determined in terms of a threshold to be reached by the larvae. Simplifying the model further on, the whole dynamics is reduced to a two dimensional system of state-dependent delay equations. The model is put in perspective with some of the main hypotheses proposed in the literature as an explanation to the massive destruction which occurs between the egg stage and the adult stage.

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

Variables aggregation in a time discrete linear model.

\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 ...

A density dependent model describing Salmo trutta population dynamics in an arborescent river network. Effects of dams and channelling

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.