Some Results Based on Maximal Regularity Regarding Population Models with Age and Spatial Structure

TL;DR

This paper reviews how maximal regularity theory helps analyze population models with age and spatial structure, enabling characterization of semigroup generators and existence proofs for equilibria.

Contribution

It provides new insights into the use of maximal regularity for linear and nonlinear population models with age and spatial components.

Findings

Characterization of semigroup generators using maximal regularity

Proof of asynchronous exponential growth of the semigroup

Existence of positive equilibrium solutions via fixed point and bifurcation methods

Abstract

We review some results on abstract linear and nonlinear population models with age and spatial structure. The results are mainly based on the assumption of maximal $L_p$-regularity of the spatial dispersion term. In particular, this property allows us to characterize completely the generator of the underlying linear semigroup and to give a simple proof of asynchronous exponential growth of the semigroup. Moreover, maximal regularity is also a powerful tool in order to establish the existence of nontrivial positive equilibrium solutions to nonlinear equations by fixed point arguments or bifurcation techniques. We illustrate the results with examples.