# Convergence of solutions in a mean-field model of go-or-grow type with reservation of sites for proliferation and cell cycle delay

## TL;DR

This paper analyzes a mean-field model of cell motility and proliferation incorporating volume exclusion, cell cycle delay, and go-or-grow behavior, proving convergence of solutions and exploring different growth dynamics.

## Contribution

It introduces a delay differential equation model capturing cell cycle delay and site reservation, and proves convergence of solutions in a biologically feasible setting.

## Key findings

- All space becomes filled by motile cells over time.
- Total cell population can follow logistic or step-function growth.
- Model behavior varies with parameters and initial conditions.

## Abstract

We consider the mean-field approximation of an individual-based model describing cell motility and proliferation, which incorporates the volume exclusion principle, the go-or-grow hypothesis and an explicit cell cycle delay. To utilise the framework of on-lattice agent-based models, we make the assumption that cells enter mitosis only if they can secure an additional site for the daughter cell, in which case they occupy two lattice sites until the completion of mitosis. The mean-field model is expressed by a system of delay differential equations and includes variables such as the number of motile cells, proliferating cells, reserved sites and empty sites. We prove the convergence of biologically feasible solutions: eventually all available space will be filled by mobile cells, after an initial phase when the proliferating cell population is increasing then diminishing. By comparing the behaviour of the mean-field model for different parameter values and initial cell distributions, we illustrate that the total cell population may follow a logistic-type growth curve, or may grow in a step-function-like fashion.