Asymptotic behaviour of a structured population model on a space of measures

Abstract

In this paper we consider a physiologically structured population model with distributed states at birth, formulated on the space of non-negative Radon measures. Using a characterisation of the pre-dual space of bounded Lipschitz functions, we show how to apply the theory of strongly continuous positive semigroups to such a model. In particular, we establish the exponential convergence of solutions to a one-dimensional global attractor.

Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation

“…It allowed us to obtain the weak-* convergence of the rescaled measure solutions to a periodic behavior. The question whether it can be strengthened into a strong convergence, for the weighted total variation norm or a weaker one such as the bounded Lipschitz norm [34], is a challenging natural fit for a continuation of the present work.…”

Spaces of Measures and their Applications to Structured Population Models

“…Analytical setting and current challenges. A promising framework for analysis of structured population systems is offered by the recently developed mathematical theory of PDE models defined in the space of Radon measures [1,2,3,6,15,16,17,18,23,24,25,45,46,47] and a related approach of measure differential equations [13,5,37,39]. To account for both discrete and continuous cell distributions as well as a possible non-Euclidean structure of the state space such as graphs, one needs to go beyond R d and define models on a reasonably generic metric space.…”