Aggregation of Variables and Applications to Population Dynamics
Ecological modelers produce models with more and more details, leading to dynamical systems involving lots of variables. This chapter presents a set of methods which aim to extract from these complex models some submodels containing the same information b
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IRD UR G´eodes, Centre IRD de l’Ile de France, 32, Av. Henri Varagnat, 93143 Bondy cedex, France [email protected] Departamento de Matem´ aticas, Universidad de Alcal´ a, 28871 Alcal´ a de Henares, Madrid, Spain [email protected] Laboratoire de Microbiologie, G´eochimie et d’Ecologie Marines, UMR 6117, Centre d’Oc´eanologie de Marseille (OSU), Universit´e de la M´editerran´ee, Case 901, Campus de Luminy, 13288 Marseille Cedex 9, France [email protected] Departamento de Matem´ aticas, E.T.S.I. Industriales, U.P.M., c/ Jos´e Guti´errez Abascal, 2, 28006 Madrid, Spain [email protected] IXXI, ENS Lyon, 46 all´ee d’Italie, 69364 Lyon cedex 07, France [email protected]
Summary. Ecological modelers produce models with more and more details, leading to dynamical systems involving lots of variables. This chapter presents a set of methods which aim to extract from these complex models some submodels containing the same information but which are more tractable from the mathematical point of view. This “aggregation” of variables is based on time scales separation methods. The first part of the chapter is devoted to the presentation of mathematical aggregation methods for ODE’s, discrete models, PDE’s and DDE’s. The second part presents several applications in population and community dynamics.
5.1 Introduction Ecology aims to understand the relations between living organisms and their environment. This environment constitutes a set of physical, chemical and biological constraints acting at the individual level. In order to deal with the complexity of an ecosystem, ecology has been developed on the basis of a wide range of knowledge starting from the molecular level (molecular ecology) to the ecosystem level. One of the current aims of ecological modelling is to use the mathematical formalism for integrating all this knowledge.
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On the other hand, mathematical ecology provided a large amount of rather simple models involving a small number of state variables and parameters. The time continuous Lotka–Volterra models, published in the beginning of the twentieth century [59, 60, 94] as well as the discrete host–parasite Nicholson– Bailey models [69, 70] are classic examples and can be found in many biomathematical textbooks as the book by Edelstein-Keshet [48] and the book by Murray [66] in which many other examples and references are given. In such population dynamics models, the state variables are often chosen as the population densities and the model is a set of nonlinear coupled ordinary differential equations (ODE’s) or discrete equations. The models describe the time variation of the interacting populations. Of course, mathematical ecologists proposed also more realistic models taking account of some populations structures (space, age, physiology, etc.). Mathematical methods have been developed to deal with these structured population models, but which may fail to get robust results for high dimensional systems. During the last decades, supported by
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