Existence and uniqueness of solutions for single-population McKendrick-von Foerster models with renewal

TL;DR

This paper proves the existence and uniqueness of solutions for a generalized single-species McKendrick-von Foerster model with renewal, incorporating history-dependent nonlocal interactions, using a fixed-point approach.

Contribution

It introduces a generalized model with history-dependent renewal, extending previous size-structured models, and provides rigorous proof of solution existence and uniqueness.

Findings

Existence and uniqueness of solutions established.

Model generalizes earlier delay and integral models.

Applicable to size-structured population dynamics.

Abstract

We study a McKendrick-von Foerster type equation with renewal. This model is represented by a single equation which describes one species which produces young individuals. The renewal condition is linear, but takes into account some history of the population. This model addresses nonlocal interactions between individuals structured by age. The vast majority of size-structured models are also treatable. Our model generalizes a number of earlier models with delays and integrals. The existence and uniqueness is proved through a fixed-point approach to an equivalent integral problem in $L^{\infty}\cap L^1.$