Flat metrics for non-negarive Radon measures and structural stability of a nonlinear population model
by
Piotr Gwiazda
University of Warsaw
Coauthors: Anna Marciniak-Czochra Tommas Lorenz

Models describing time evolution of physiologically structured populations

have been extensively studied for many years, [2], [3]. Traditionally the

dynamics of such populations are described by partial differential equations

(PDE) of transport type.

The first general results on global existence and stability of the solutions

of structured population models were established for the states defined in

Banach space L^1 [2]. In this case it was possible to prove strong continuity

and structural stability of the solutions. However, it is often necessary to

describe populations, where all individuals have the same state or their distribution

is concentrated in respect to the structure, ie. initial distribution

of the individuals is not uniformly continuous with respect to the Lebesgue

measure. In such cases it is relevant to consider initial data in the space of

Radon measures as proposed in [4]. Analytical results on the existence of

the solutions are given in [4]. However, continuous dependence of solutions

on time and state is shown only in the weak-star topology.

The above motivated us to study the problem of structural stability of the

structured population dynamics. Our approach based on the theory of the

nonlinear semigroups in the metric spaces, instead of weak-star semigroups on

Banach spaces. Framework of the Wasserstein metric in spaces of probability

measures is a very important issue in the analysis of transport equations and

was recently developed, see for example [1]. To apply this framework to the

nonlinear structured population models, we replace the Wasserstein metric

by the flat metric to work with general nonnegative Radon measures. An additional problem comparing

to the case considered in [1], is that our system consists of the transport

equation with a nonlocal boundary condition.

References:

[1] Ambrosio, L., Gigli, N. & Savare, G. (2005): Gradient Flows in metric

spaces and in the space of probability measures, Birkhauser, ETH

Lecture Notes in Mathematics

[2] Webb G.F. (1985): Nonlinear Age-Dependent Population Dynamics,

Marcel Dekker, New York.

[3] Thieme, H. R. Mathematics in population biology.Woodstock Princeton

university press. Princeton (2003).

[4] Diekmann O. and Getto P. (2005): Boundedness, global existenceand

continuous dependence for nonlinear dynamical systems describing

physiologically structured populations. J. Differ. Equations, 215,

pp. 268--319.

Date received: July 16, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-44.