Two-time scales in spatially structured models of population dynamics: A semigroup approach

Abstract

The aim of this work is to provide a unified approach to the treatment of a class of spatially structured population dynamics models whose evolution processes occur at two different time scales. In the setting of the C 0 -semigroup theory, we will consider a general formulation of some semilinear evolution problems defined on a Banach space in which the two-time scales are represented by a parameter ε > 0 small enough, that mathematically gives rise to a singular perturbation problem. Applying the so-called aggregation of variables method, a simplified model called the aggregated model is constructed. A nontrivial mathematical task consists of comparing the asymptotic behaviour of solutions to both problems when ε → 0 + , under the assumption that the aggregated model has a compact attractor. Applications of the method to a class of two-time reaction-diffusion models of spatially structured population dynamics and to models with discrete spatial structure are given.

An age-structured fishery model: Dynamics and optimal management with perfect elastic demand

“…Our approach consists on the following idea. We take advantage from existence of two time scales (see Sanchez et al 2011), a fast one for …sh di¤usion and a slow one for …sh growth and mortality to reduce our model to an ODE system. The major contributions of the present paper are: a) the persistence of the species depend on two factors:the …rst is biological and concerns the ability that a species reproduces itself, this is a function of birth rate, natural mortality of young population and the rate of maturity.…”

“…We will study the global existence and asymptotic behavior in time of solutions .The model is partially cooperative. This di¢ culty will be handled using the theory of fast -slow systems recently investigated by Sanchez et al (2011) , the approach used by the authors allows one to focus the study on an ordinary di¤erential equations governing the total population density, obtained by integration over the spatial domain and slow time scale. We will illustrate the results by giving an application to an optimal harvesting problem.…”

Extensions of the chemostat model with flocculation

“…Assuming that this slow manifold is asymptotically stable for the fast equation, the reduction of the system to the slow system gives an approximation of the solutions of (5) for arbitrarily small ε, in accordance with the theories of singular perturbations or variables aggregation [36][37][38]. The reduced model is obtained by replacing the fast variables u and v in the equations of S and x:…”

Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion

“…Those methods have been developed several classes of models including time continuous systems as well as discrete models [2]. A particular attention has been brought to the cases of Ordinary Differential Equations (ODE) [3,5], Partial Differential Equations (PDE) [4,20,21], and Delayed Differential Equations (DDE) [22].…”

“…Sanchez et al [20] introduced the aggregation of variables methods to define an aggregated model that simplifies a two-time scales semi-linear evolution equation defined on a Banach lattice, and applied it to two-time scales reaction-diffusion models with large diffusivity. Mathematical models of such processes are given by partial differential equations of reaction-diffusion type [12].…”