# A cell-cell repulsion model on a hyperbolic Keller-Segel equation

## TL;DR

This paper introduces a hyperbolic Keller-Segel model for two-cell populations, proving solution existence and uniqueness, and explores how initial distributions influence short-term and long-term cell population dynamics through numerical simulations.

## Contribution

The work develops a novel hyperbolic Keller-Segel model for cell populations, proving mathematical properties and analyzing the effects of initial conditions on population dynamics.

## Key findings

- Model exhibits competitive exclusion differing from ODE models.
- Initial total cell number influences population ratio.
- Fast dispersion provides short-term advantage.

## Abstract

In this work, we discuss a cell-cell repulsion population dynamic model based on a hyperbolic Keller-Segel equation with two populations. This model can well describe the cell growth and dispersion in the cell co-culture experiment in the work of Pasquier et al. \cite{Pasquier2011}. With the notion of solutions integrated along the characteristics, we prove the existence and uniqueness of the solution and the segregation property of the two species. From a numerical perspective, we can also observe that the model admits a competitive exclusion (the results are different from the corresponding ODE model). More importantly, our model shows the complexity of the short term (6 days) co-cultured cell distribution depending on the initial distribution of each species. Through numerical simulations, the impact of the initial distribution on the population ratio lies in the initial total cell number and our study shows that the population ratio is not impacted by the law of initial distribution. We also find that a fast dispersion rate gives a short-term advantage while the vital dynamic contributes to a long-term population advantage.