Aggregation methods in dynamical systems and applications in population and community dynamics (original) (raw)
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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 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.
Aggregation and emergence in ecological modelling: integration of ecological levels
Ecological Modelling, 2000
Modelling ecological systems implies to take into account different ecological levels: the individual, population, community and ecosystem levels. Two large families of models can be distinguished among different approaches: (i) completely detailed models involving many variables and parameters; (ii) more simple models involving only few state variables. The first class of models are usually more realistic including many details as for example the internal structure of the population. Nevertheless, the mathematical analysis is not always possible and only computer simulations can be performed. The second class of models can mathematically be analysed, but they sometimes neglect some details and remain unrealistic. We present here a review of aggregation methods, which can be seen as a compromise between these two previous modelling approaches. They are applicable for models involving two levels of organisation and the corresponding time scales. The most detailed level of description is usually associated to a fast time scale, while the coarser one rather corresponds to a slow time scale. A detailed model is thus considered at the individual level, containing many micro-variables and consisting of two parts: a fast and a slow one. Aggregation methods allow then to reduce the dimension of the initial dynamical system to an aggregated one governing few global variables evolving at the slow time scale. We focus our attention on the emerging properties of individual behaviours at the population and community levels.
Aggregation methods in population dynamics discrete models
Mathematical and Computer Modelling, 1998
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).
Aggregation in model ecosystems. I. Perfect aggregation
Ecological Modelling, 1987
Determining appropriate levels of aggregation or complexity constitutes a major problem in describing ecological systems. We give necessary and sufficient conditions for perfect aggregation in non-linear dynamical systems. This condition has a form explicitly related to that for aggregation of linear dynamics in automatic control theory and to that for aggregation of linear functions in economics. Application of the theory to several ecological examples supports the general principle of aggregating variables that functionally are either similar or complementary.
5Aggregation of Variables and Applications to Population Dynamics
2016
Summary. Ecological modelers produce models with more and more details, lead-ing to dynamical systems involving lots of variables. This chapter presents a set of methods which aim to extract from these complex models some submodels contain-ing 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 aggre-gation methods for ODE’s, discrete models, PDE’s and DDE’s. The second part presents several applications in population and community dynamics. 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
Aggregation and emergence in systems of ordinary differential equations
Mathematical and Computer Modelling, 1998
The aim of this article is to present aggregation methods for a system of ordinary differential equations (ODE's) involving two time scales. The system of ODE's is composed of the sum of fast parts and a perturbation. The fast dynamics are assumed to be conservative. The corresponding first integrals define a few global variables. Aggregation corresponds to the reduction of the dimension of the dynamical system which is replaced by an aggregated system governing the global variables at the slow time scale. The centre manifold theorem is used in order to get the slow reduced model as a Taylor expansion of a small parameter. We particularly look for the conditions necessary to get emerging properties in the aggregated model with respect to the nonaggregated one. We define two different types of emergences, functional and dynamical. Functional emergence corresponds to different functions for the two dynamics, aggregated and nonaggregated. Dynamical emergence means that both dynamics are qualitatively different. We also present averaging methods for aggregation when the fast system converges towards a stable limit cycle.
Discrete & Continuous Dynamical Systems - S, 2021
The main goal of this paper is to adapt a class of complexity reduction methods called aggregation of variables methods to the construction of reduced models of two-time reaction-diffusion-chemotaxis models of spatially structured populations and to provide an error bound of the approximate dynamics. Aggregation of variables methods are general techniques that allow reducing the dimension of a mathematical dynamical system. Here we reduce a system of Partial Differential Equations to a simpler Ordinary Differential Equation system, provided that the evolution processes occur at two different time scales: a slow one for the demography and a fast one for migrations and chemotaxis, with a ratio ε > 0 small enough. We give an approximation of the error between solutions of both original and reduced model for a generic function representing the demography. Finally, we provide an optimization of the error bound and validate numerically this result for a spatial inter-specific model with constant diffusion and population growth given by a logistic law in population dynamics.
Aggregation in Biological Systems: Computational Aspects
2007
Many biologically relevant dynamical systems are aggregable, in the sense that one can divide their microvariables x 1 ,. .. , x n into several (k) nonintersecting groups and find functions y1,. .. , y k (k < n) from these groups (macrovariables) whose dynamics only depend on the initial state of the macrovariable. For example, the state of a population genetic system can be described by listing the frequencies x i of different genotypes, so that the corresponding dynamical system describe the effects of mutation, recombination, and natural selection. The goal of aggregation approaches in population genetics is to find macrovariables y a ,. .. , y k to which aggregated mutation, recombination, and selection functions could be applied. Population genetic models are formally equivalent to genetic algorithms, and are therefore of wide interest in the computational sciences. Another example of a multi-variable biological system of interest arises in ecology. Ecosystems contain many interacting species, and because of the complexity of multi-variable nonlinear systems, it would be of value to derive a formal description that reduces the number of variables to some macrostates that are weighted sums of the densities of several species. In this chapter, we explore different computational aspects of aggregability for linear and non-linear systems. Specifically, we investigate the problem of conditional aggregability (i.e., aggregability restricted to modular states) and aggregation of variables in biologically relevant quadratic dynamical systems.
Modelling of ecosystem with different types of components aggregation
2012
The paper presents several models of the Bering Sea ecosystem. The models differ in the degree and method of grouping. The use of the multimodal approach to research of ecosystems makes better use of available information about the object. The system is considered in two ways: as a closed object and given the influence of the environment. Equilibriums of the system were examined for stability. A comparison of the solutions of the aggregated model with the dynamics of the original model is performed.