Physiologically structured populations with diffusion and dynamic boundary conditions

TL;DR

This paper develops a mathematical model for size-structured populations incorporating diffusion and dynamic boundary conditions, proving existence, positivity, and long-term growth behaviors of solutions.

Contribution

It introduces a novel size-structured population model with generalized boundary conditions and analyzes its long-term dynamics and solution properties.

Findings

Solutions are governed by a positive quasicontractive semigroup.

The model admits a finite dimensional global attractor.

Solutions exhibit asynchronous exponential growth under positive fertility.

Abstract

We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. The model is equipped with generalized Wentzell-Robin (or dynamic) boundary conditions. This allows modelling of "adhesion" at extremely small or large sizes. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. This is carried out via establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is our model admits a finite dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.