Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

TL;DR

This paper introduces a numerical algorithm for solving a complex nonlinear kinetic equation modeling spatial population dynamics with coalescence and repulsive jumps, validated through one-dimensional simulations and error analysis.

Contribution

It presents a novel numerical algorithm combining space-time discretization and advanced methods for a nonlinear nonlocal kinetic equation in population dynamics.

Findings

Solutions can exhibit unexpected time behavior for certain parameters.

The algorithm effectively handles nonlinear, nonlocal integro-differential equations.

Numerical error analysis confirms the method's reliability.

Abstract

An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point articles placed in $\mathbb{R}^d (d \geq 1)$. The particles perform random jumps with pair wise repulsion, in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The derivation of the algorithm is based on the use of space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, adjustable system-size schemes, etc. The algorithm is then applied to the one-dimensional version of the equation with various initial conditions. It is shown that for special choices of the model parameters, the solutions may have unexpectable time behaviour. A numerical error analysis of the obtained results is also carried out.