A discrete model with density dependent fast migration (original) (raw)
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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).
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 methods in dynamical systems and applications in population and community dynamics
Physics of Life Reviews, 2008
Approximate aggregation techniques allow one to transform a complex system involving many coupled variables into a simpler reduced model with a lesser number of global variables in such a way that the dynamics of the former can be approximated by that of the latter. In ecology, as a paradigmatic example, we are faced with modelling complex systems involving many variables corresponding to various interacting organization levels. This review is devoted to approximate aggregation methods that are based on the existence of different time scales, which is the case in many real systems as ecological ones where the different organization levels (individual, population, community and ecosystem) possess a different characteristic time scale. Two main goals of variables aggregation are dealt with in this work. The first one is to reduce the dimension of the mathematical model to be handled analytically and the second one is to understand how different organization levels interact and which properties of a given level emerge at other levels. The review is organized in three sections devoted to aggregation methods associated to different mathematical formalisms: ordinary differential equations, infinite-dimensional evolution equations and difference equations.
Emergence of Population Growth Models: Fast Migration and Slow Growth
Journal of Theoretical Biology, 1996
We present aggregation and emergence methods in large-scale dynamical systems with different timescales. Aggregation corresponds to the reduction of the dimension of a dynamical system which is replaced by a smaller model for a small number of global variables at a slow timescale. We study the couplings between fast and slow dynamics leading to the emergence of global properties in
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.
Approximate reduction of multiregional birth-death models with fast migration
2002
In this work, we deal with the reduction of a time discrete model for a population distributed among N spatial patches and whose dynamics is controlled both by reproduction and by migration. These processes take place at different time scales in the sense of the latter being much faster than the former. We incorporate the effect, of demographic stochasticity into the population, which results in both dynamics being modelled by multitype Galton-Watson branching processes. We present a multitype global model that incorporates the effect, of the two processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining a unitype "aggregated" process that approximates the evolution of the total size of the population. We show that, given the separation of time scales between the birth-death process and the migration process is sufficiently high, we can obtain both qualitative and quantitative information about the behavior of the multitype global model through the study of this simple aggregated model.
Fast Migration and Emergent Population Dynamics
Physical Review Letters, 2012
We consider population dynamics on a network of patches, each of which has a the same local dynamics, with different population scales (carrying capacities). It is reasonable to assume that if the patches are coupled by very fast migration the whole system will look like an individual patch with a large effective carrying capacity. This is called a "well-mixed" system. We show that, in general, it is not true that the well-mixed system has the same dynamics as each local patch. Different global dynamics can emerge from coupling, and usually must be figured out for each individual case. We give a general condition which must be satisfied for well-mixed systems to have the same dynamics as the constituent patches.
On an aggregation model with long and short range interactions
Nonlinear Analysis-real World Applications, 2007
In recent papers the authors had proposed a stochastic model for swarm aggregation, based on individuals subject to long range attraction and short range repulsion, in addition to a classical Brownian random dispersal. Under suitable laws of large numbers they showed that, for a large number of individuals, the evolution of the empirical distribution of the population can be expressed in terms of an approximating nonlinear degenerate and nonlocal parabolic equation, which describes the limit.In this paper the well-posedness of such evolution equation is investigated, which invokes a notion of entropy solutions extended to the nonlocal case. We motivate entropy solutions from the discrete particle system and use them to prove uniqueness. Moreover, we provide existence results and discuss some basic properties of solutions. Finally, we apply a Lagrangian numerical scheme to perform numerical simulations in spatial dimension one.