# Nonlocal birth-death competitive dynamics with volume exclusion

## TL;DR

This paper introduces a stochastic lattice model for particles with volume exclusion, deriving macroscopic equations that reveal how nonlocal competition influences spatial patterns and dynamics.

## Contribution

It develops a novel nonlocal birth-death model with volume exclusion, deriving macroscopic equations and analyzing pattern formation and stability.

## Key findings

- Steady-state solutions and their stability are characterized.
- Spatial pattern formation is demonstrated through stability analysis.
- Numerical simulations confirm theoretical predictions.

## Abstract

A stochastic birth-death competition model for particles with excluded volume is proposed. The particles move, reproduce, and die on a regular lattice. While the death rate is constant, the birth rate is spatially nonlocal and implements inter-particle competition by a dependence on the number of particles within a finite distance. The finite volume of particles is accounted for by fixing an upper value to the number of particles that can occupy a lattice node, compromising births and movements. We derive closed macroscopic equations for the density of particles and spatial correlation at two adjacent sites. Under different conditions, the description is further reduced to a single equation for the particle density that contains three terms: diffusion, a linear death, and a highly nonlinear and nonlocal birth term. Steady-state homogeneous solutions, their stability which reveals spatial pattern formation, and the dynamics of time-dependent homogeneous solutions are discussed and compared, in the one-dimensional case, with numerical simulations of the particle system.