Behavioral choices based on patch selection: a model using aggregation methods

Adaptation of prey and predators between patches

Mathematical models are proposed to simulate migrations of prey and predators between patches. In the absence of predators, it is shown that the adaptation of prey leads to an ideal spatial distribution in the sense that the maximal capacity of each patch is achieved. With the introduction of co-adaptation of predators, it is proved that both prey and predators achieve ideal spatial distributions when the adaptations are weak. Further, it is shown that the adaptation of prey and predators increases the survival probability of predators from the extinction in both patches to the persistence in one patch. It is also demonstrated that there exists a pattern that prey and predators cooperate well through adaptations such that predators are permanent in every patch in the case that predators become extinct in each patch in the absence of adaptations. For strong adaptations, it is proved that the model admits periodic cycles and multiple stability transitions.

Effects of density dependent migrations on the dynamics of a predator prey model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.

Integrative biology: linking levels of organization

Biological systems are composed of different levels of organization. Usually, one considers the atomic, molecular, cellular, individual, population, community and ecosystem levels. These levels of organization also correspond to different levels of observation of the system, from microscopic to macroscopic, i.e., to different time and space scales. The more microscopic the level is, the faster the time scale and the smaller the space scale are. The dynamics of the complete system is the result of the coupled dynamical processes that take place in each of its levels of organization at different time scales. Variables aggregation methods take advantage of these different time scales to reduce the dimension of mathematical models such as a system of ordinary differential equations. We are going to study the dynamics of a system which is hierarchically organized in the sense that it is composed of groups of elements that can be themselves divided into further smaller sub-groups and so on. The hierarchical structure of the system results from the fact that the intra-group interactions are assumed to be larger than inter-group ones. We present aggregation methods that allow one to build a reduced model that governs a few global variables at the slow time scale.

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.

Emergence of individual behaviour at the population level. Effects of density-dependent migration on population dynamics

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 density-dependent 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.