Run-and-tumble particles on a line with a fertile site

TL;DR

This paper models run-and-tumble particles on a line with a fertile site, deriving equations for their density evolution, revealing exponential growth and a stationary state with a local minimum at the fertile site.

Contribution

It introduces a solvable model of RTPs with fertility at a site, providing analytical expressions for density growth and stationary states.

Findings

Density grows exponentially at large times depending on fertility parameters.

Stationary density profile has a local minimum at the fertile site.

The model's equations are solvable in the Laplace domain.

Abstract

We propose a model of run-and-tumble particles (RTPs) on a line with a fertile site at the origin. After going through the fertile site, a run-and-tumble particle gives rise to new particles until it flips direction. The process of creation of new particles is modelled by a fertility function (of the distance to the fertile site), multiplied by a fertility rate. If the initial conditions correspond to a single RTP with even probability density, the system is parity-invariant. The equations of motion can be solved in the Laplace domain, in terms of the density of right-movers at the origin. At large time, this density is shown to grow exponentially, at a rate that depends only on the fertility function and fertility rate. Moreover, the total density of RTPs (divided by the density of right-movers at the origin), reaches a stationary state that does not depend on the initial conditions, and presents a local minimum at the fertile site.