A random free-boundary diffusive logistic model: Analysis, computing and simulation

TL;DR

This paper extends the diffusive logistic model with a free boundary to incorporate randomness in parameters, using numerical methods to analyze the stochastic behavior of the moving front and solution.

Contribution

It introduces two numerical approaches for solving the random free boundary problem and compares their effectiveness in capturing stochastic dynamics.

Findings

Both approaches effectively model the stochastic free boundary problem.

Numerical schemes demonstrate stability and convergence.

The methods reveal the spreading-vanishing dichotomy in stochastic settings.

Abstract

A free boundary diffusive logistic model finds application in many different fields from biological invasion to wildfire propagation. However, many of these processes show a random nature and contain uncertainties in the parameters. In this paper we extend the diffusive logistic model with unknown moving front to the random scenario by assuming that the involved parameters have a finite degree of randomness. The resulting mathematical model becomes a random free boundary partial differential problem and it is addressed numerically combining the finite difference method with two approaches for the treatment of the moving front. Firstly, we propose a front-fixing transformation, reshaping the original random free boundary domain into a fixed deterministic one. A second approach is using the front-tracking method to capture the evolution of the moving front adapted to the random framework. Statistical moments of the approximating solution stochastic process and the stochastic moving boundary solution are calculated by the Monte Carlo technique. Qualitative numerical analysis establishes the stability and positivity conditions. Numerical examples are provided to compare both approaches, study the spreading-vanishing dichotomy, prove qualitative properties of the schemes and show the numerical convergence.