The aim of this work is to study the influence of patch selection on the dynamics of a system describing the interactions between two populations, generically called `population N' and `population ...P'. Our model may be applied to prey–predator systems as well as to certain host–parasite or parasitoid systems. A situation in which population P affects the spatial distribution of population N is considered. We deal with a heterogeneous environment composed of two spatial patches: population P lives only in patch 1, while individuals belonging to population N migrate between patch 1 and patch 2, which may be a refuge. Therefore they are divided into two patch sub-populations and can migrate according to different migration laws. We make the assumption that the patch change is fast, whereas the growth and interaction processes are slower. We take advantage of the two time scales to perform aggregation methods in order to obtain a global model describing the time evolution of the total populations, at a slow time scale. At first, a migration law which is independent on population P density is considered. In this case the global model is equivalent to the local one, and under certain conditions, population P always gets extinct. Then, the same model, but in which individuals belonging to population N leave patch 1 proportionally to population P density, is studied. This particular behavioral choice leads to a dynamically richer global system, which favors stability and population coexistence. Finally, we study a third example corresponding to the addition of an aggregative behavior of population N on patch 1. This leads to a more complicated situation in which, according to initial conditions, the global system is described by two different aggregated models. Under certain conditions on parameters a stable limit cycle occurs, leading to periodic variations of the total population densities, as well as of the local densities on the spatial patches.
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.PUBLICATION ABSTRACT
The aim of this work is to study the effects of different individual behaviours on the overall growth of a spatially distributed population. The population can grow on two spatial patches, a source ...and a sink, that are connected by migrations. Two time scales are involved in the dynamics, a fast one corresponding to migrations and a slow one associated with the local growth on each patch. Different scenarios of densitydependent migration are proposed and their effects on the population growth are investigated. A general discussion on the use of aggregation methods for the study of integration of different ecological levels is proposed.
Le but de ce travail est l’étude des effets de différents comportements individuels sur la croissance à long terme d’une population spatialement distribuée. La population peut se développer sur deux sites, une source et un puits, connectés par des flux migratoires. Deux échelles de temps sont impliquées dans la dynamique, une échelle rapide correspondant à la migration et une échelle lente associée à la croissance de la population sur chaque site. Différents scénarios de migrations densité dépendantes sont proposés et leurs effets sur la dynamique de population sont étudiés. Une discussion générale sur l’utilité des méthodes d’agrégation des variables pour l’étude de l’intégration des différents niveaux d’organisation des systèmes écologiques conclut l’article.
Our aim is to model the
Salmo trutta population dynamics (three age-classes) in an arborescent river network (four levels, 15 patches), by considering both migrations (fast time scale) and demography ...(slow time scale). We study how the environmental management can influence the global population dynamics. We present a general model coupling both a linear discrete model for constant migrations and a non-linear density-dependent Leslie model for the demography, with (15 × 3) difference equations (15 patches, three age-classes). The variable aggregation method applied to discrete time models allows us to aggregate the previous model into a new one with only three equations. We assume fecundity and survival gradients with respect to the river network levels. The
Salmo trutta whole population tends towards an equilibrium state depending on the environmental structure, and we show that dams have a stronger influence than channelling on this equilibrium.
La croissance d'une population de truites (trois classes d'âge) dans un réseau de rivière arborescent (quatre niveaux, 15 sites) est modélisée en considérant simultanément les migrations (échelle de temps rapide) et la démographie (échelle de temps lente). Nous étudions comment la gestion de l'environnement peut influencer la dynamique globale de la population. Nous présentons un modèle général discret, avec (15 × 3) équations récurrentes (15 sites, trois classes d'âge), couplant une matrice de migration à coefficients constants et une matrice de Leslie densité-dépendante pour la démographie, La méthode d'agrégation des variables permet d'agréger le modèle précédent en un nouveau modèle ne contenant plus que trois équations. Nous supposons qu'il existe des gradients de fécondité et de survie, fonction des différents niveaux du réseau de rivières. La population totale de truites tend vers un équilibre dépendant de la structure de l'environnement, et nous montrons que les barrages ont une influence plus négative que les canaux sur cet équilibre.
