La représentation de l'espace dans les modèles de dynamique des populations. L’exemple des modèles dynamiques déterministes à temps et espace continus

First analytic approaches in population dynamics modelling did not really begin until the 1920s and 1930s. By the 1950s, models that described how fish populations respond to fishing were developed, and provided useful tools in the management of living resources. These early studies, as well as the bulk of existing theoretical ecology, are usually in spatially homogeneous settings: they deal with temporal processes, and ask specifically about steady-state solutions to nonspatial systems. However, several lines of inquiry have stressed the critical role of spatial complexity when thinking about populations and communities as dynamical systems varying in response to natural or human-created disturbance. The introduction of this report is designed to highlight the fundamental effects of space on the dynamics of populations of individual organisms, each of these discrete entities interacting only with its immediate neighborhood, i.e., the rather confined region through which it moves. Various approaches are first briefly examined, before focusing throughout the rest of the report on spatially explicit models for continuous space and time, couched in terms of deterministic partial differential equations (although stochastic or individual-based models are important, this choice aims at keeping the report within reasonable bounds). Chapter 1 relies upon the classical Mc Kendrick-von Foerster equation, and introduces the integrodifferential formulation convoluting the local population density with a “dispersal kernel”: the instantaneous rate of change of the former thus depends on the influence of the spatial distribution of the population, the latter summarizing macroscopically the dynamics resulting from individual movements. Chapter 2 puts the emphasis on the case of spatially symmetric kernel functions whose range of influence is restricted to a small neighborhood, and recalls how the integrodifferential model reduces to a familiar reaction-diffusion system; the concept of “biological diffusion” is also explained. Typical features of the dynamical behaviour of reaction-diffusion models are presented, with examples of generalizations such as correlated random walk and nonlinear diffusion. Some interacting populations systems are then considered, the main concern being with models for two populations; the governing equations for diffusion driven instability mechanisms of pattern formation (Turing structures) are derived. Examples of spatial pattern generation in predator-prey systems are also given. The purpose of chapter 3 is to spark interest among field- and experimental ecologists in wedding a pertinent theoretical frame to the management of the variety of spatio-temporal scales characterizing the biological cycles of marine organisms. The conclusion of the report explores some challenges facing the definition of a relevant strategy of model building in “spatial population dynamics”.