R. Bravo de la Parra

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

Annual spawning migrations in modelling brown trout population dynamics inside an arborescent river network

In the present paper, the annual spawning migration of adults is introduced into a model, describing the Salmo trutta population dynamics in a hierarchically organized river network (four levels and 15 interconnected patches) model based on previous work. The model describes simultaneously demographic and migration processes taking place at different time scales: migrations of individuals between patches at a fast time scale (e.g. the week or the month), the annual spawning migration of adults and the demography at the slow time scale of the year. The S. trutta population is sub-divided into three age-classes (young of the year, juveniles, and adults). We used a Leslie-type model, coupled with a migration matrix associated with the annual spawning process, and a second migration matrix associated with fast movements of individuals between patches throughout the year. All demographic and migratory parameters are constant, leading to a linear model governing 45 state variables (15 patches three age-classes). By taking advantage of the two time scales and using aggregation techniques for the case of discrete time models, the complete model was approximated by a reduced one, with only three global variables (one per age-class) evolving at the slow time scale. Demographic indices were calculated for the population, and a sensibility analysis was performed to detect which parameters influence the most model predictions. We also quantified how modifications of the river network structure, by channels (change in connections between patches) or dams (patch deletion), influence the global population dynamics. We checked that the strategy of annual spawning migrations is actually beneficial for the population (the asymptotic population growth rate is increased), and that dams may have a more detrimental effect on the whole population dynamics than channelling.

Dynamics of a fishery on two fishing zones with fish stock dependent migrations: aggregation and control

This work presents a specific stock-effort dynamic model. The stock corresponds to two fish populations growing and moving between two fishing zones, on which they are harvested by two different fleets. The effort represents the number of fishing vessels of the two fleets which operate on the two fishing zones. The bioeconomical model is a set of four ordinary differential equations governing the stocks and the fishing efforts in the two fishing areas. Fish migration, as well as vessels displacements, between the two zones are assumed to take place at a faster time scale than the variation of the stocks and the changes of fleets sizes, respectively. The vessels movements between the two fishing areas are assumed to be stock dependent, i.e. the larger the stock density is in a zone the more vessels tends to remain in it. The global fish stock and the total number of vessels keep constant at the fast time scale. This property enables, via aggregation methods, the reduction of the system dimension in order to proceed to its qualitative analysis. Under some assumptions, we obtain either a stable equilibrium or a stable limit cycle which involves large cyclic variations of the total fish stock and fishing effort. Finally, we introduce a control parameter to maintain the system at a sustainable equilibrium and thus avoiding the important fluctuations founded otherwise.

Population dynamics modelling in an hierarchical arborescent river network: An attempt with salmo trutta

The balance between births and deaths in an age-structured population is strongly influenced by the spatial distribution of sub-populations. Our aim was to describe the demographic process of a fish population in an hierarchical dendritic river network, by taking into account the possible movements of individuals. We tried also to quantify the effect of river network changes (damming or channelling) on the global fish population dynamics. The Salmo trutta life pattern was taken as an example for.We proposed a model which includes the demographic and the migration processes, considering migration fast compared to demography. The population was divided into three age-classes and subdivided into fifteen spatial patches, thus having 45 state variables. Both processes were described by means of constant transfer coefficients, so we were dealing with a linear system of difference equations. The discrete case of the variable aggregation method allowed the study of the system through the dominant elements of a much simpler linear system with only three global variables: the total number of individuals in each age-class.From biological hypothesis on demographic and migratory parameters, we showed that the global population dynamics of fishes is well characterized in the reference river network, and that dams could have stronger effects on the global dynamics than channelling.

Aggregation methods in population dynamics discrete models

Aggregation methods try to approximate a large scale dynamical system, the general system, involving many coupled variables by a reduced system, the aggregated system, that describes the dynamics of a few global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the general system. Aggregation methods have been developed for general continuous time systems, systems of ordinary differential equations, and for linear discrete time models, with applications in population dynamics. In this contribution, we present aggregation methods for linear and nonlinear discrete time models. 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 in the nonlinear case. 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. In linear models, the asymptotic behaviours of the general and the aggregated systems are characterized by their dominant eigenelements, that are proved to coincide to a certain order. The general method is applied to aggregate a multiregional Leslie model in the constant rates case (linear) and also in the density dependent case (nonlinear).

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

In the modelisation 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: at the fast time scale, we have migration dynamics and at 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 @e (0 < @e @? 1). The main result decomposes the action of the semigroup associated to our problem into three parts: 1.(1) the semigroup associated to a demographic scalar problem times the vector of the equilibrium distribution of the migration process; 2.(2) the semigroup associated to the transitory process which leads to the first part; and 3.(3) an operator, bounded in norm, of order @e.

Approximate reduction of non-linear discrete models with two time scales

The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system.The aim of this work is to present a general class of nonlinear discrete time models with two time scales whose dynamics is susceptible of being approached by means of a reduced system. The reduction process is included in the so-called approximate aggregation of variables methods which consist of describing the dynamics of a complex system involving many coupled variables through the dynamics of a reduced system formulated in terms of a few global variables. For the time unit of the discrete system we use that of the slow dynamics and assume that fast dynamics acts a large number of times during it. After introducing a general two-time scales nonlinear discrete model we present its reduced accompanying model and the relationships between them. The main result proves that certain asymptotic behaviours, hyperbolic asymptotically stable (A.S.) periodic solutions, to the aggregated system entail that to the original system.

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process. Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models. An application of the results to the study of the dynamics of the Sole larvae is also provided.

Hawk-dove game and competition dynamics

In this article, we consider two populations subdivided into two categories of individuals (hawks and doves). Individuals fight to have access to a resource necessary for their growth. Conflicts occur between hawks of the same population and hawks of different populations. The aim of this work is to investigate the long term effects of these conflicts on coexistence and stability of the community of the two populations. This model involves four variables corresponding to the two tactics of individuals of the two populations. The model is composed of two parts, a fast part describing the encounters and fights, and the slow part describing the long term effects of encounters on the growth of the populations. We use aggregation methods allowing us to reduce this model into a system of two ODEs for the total densities of the two populations. This is found to be a classical Lotka-Volterra competition model. We study the effects of the different fast equilibrium proportions of hawks and doves in both populations on the global coexistence and the mutual exclusion of the two populations. We show that in some cases, mixed hawk and dove populations coexist. Aggressive populations of hawks exclude doves except in the case of interpopulation costs being smaller than intrapopulation ones.

A density-dependent model describing age-structured population dynamics using hawk–dove tactics

In this paper we deal with a nonlinear two-timescale discrete population model that couples age-structured demography with individual competition for resources. Individuals are divided into juvenile and adult classes, and demography is described by means of a density-dependent Leslie matrix. Adults compete to access resources; every time two adults meet, they choose either being aggressive (hawk) or non-aggressive (dove) to get the best pay-off. Individual encounters occur much more frequently than demographic events, what yields that the model takes the form of a two-timescale system. Approximate aggregation methods allow us to reduce the system while preserving at the same time crucial asymptotic information for the whole population. In this way, we are able to describe the total population size as function of individual aggressiveness level and environmental richness. Model analysis shows a general trend with species that look for richer environment having smaller proportions of hawk individuals with larger costs.