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

M. -C. Casabán, R. Company, V. N. Egorova, L. Jódar

TL;DR

The paper extends a radially symmetric diffusive logistic model to a random Stefan-type problem with a moving front by introducing finite randomness in model parameters. It develops two numerical strategies, a random Front-Fixing (FF) method and a random Front-Tracking (FT) method, and derives stable, positive finite-difference schemes (RFDS-FF and RFDS-FT) to simulate the stochastic front dynamics. Uncertainty is handled via Monte Carlo sampling to estimate the mean and variance of the population density $u$ and front position $H$, enabling analysis of the spreading-vanishing dichotomy under randomness. Numerical experiments compare FF and FT, study convergence of the Monte Carlo approach, and reveal a trade-off: FF is computationally efficient for short horizons with fixed grids, while FT maintains accuracy for longer simulations by adapting the grid. These findings inform method selection for RPDEs with free boundaries in applications like ecological invasion and wildfire spread.

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.

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

TL;DR

The paper extends a radially symmetric diffusive logistic model to a random Stefan-type problem with a moving front by introducing finite randomness in model parameters. It develops two numerical strategies, a random Front-Fixing (FF) method and a random Front-Tracking (FT) method, and derives stable, positive finite-difference schemes (RFDS-FF and RFDS-FT) to simulate the stochastic front dynamics. Uncertainty is handled via Monte Carlo sampling to estimate the mean and variance of the population density and front position , enabling analysis of the spreading-vanishing dichotomy under randomness. Numerical experiments compare FF and FT, study convergence of the Monte Carlo approach, and reveal a trade-off: FF is computationally efficient for short horizons with fixed grids, while FT maintains accuracy for longer simulations by adapting the grid. These findings inform method selection for RPDEs with free boundaries in applications like ecological invasion and wildfire spread.

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.
Paper Structure (14 sections, 3 theorems, 84 equations, 19 figures, 7 tables, 3 algorithms)

This paper contains 14 sections, 3 theorems, 84 equations, 19 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction and motivation
  2. Preliminaries
  3. Random operational calculus
  4. Spreading-Vanishing dichotomy of the model
  5. A random Front-Fixing (FF) method
  6. Construction of the random finite difference scheme for FF method (RFDS-FF)
  7. Stability and qualitative properties of the numerical solutions s.p.'s
  8. Algorithm for computing the numerical solution s.p.'s for FF method
  9. A random Front-Tracking (FT) method
  10. Algorithm for computing the numerical solution s.p.'s for FT method
  11. Monte Carlo technique
  12. Numerical examples
  13. Comparison between the random FF and FT methods
  14. Conclusions

Key Result

Theorem 1

Let $\left\{ u(r,t;\omega_{\ell}), \, H(t;\omega_{\ell}) \right\}$ be the solution of the sample problem eq:PDE--eq:BC for a given realization $\omega_{\ell}\in{\Omega}$, then there is a positive constant $R^*(\omega_{\ell})$, such that

Figures (19)

  • Figure 1: Evolution of the position of the moving boundary with FT method for a fixed realization $\omega_\ell\in{\Omega}$. The moving front, $H^n(\omega_\ell)$, in a time level $n$ is highlighted in red and the fractional step forward in the space, $p^n(\omega_\ell)$, in a time level $n$ is marked in blue.
  • Figure 2: The two possible scenarios where a sample of the front, $H^n(\omega_\ell)$, can be placed at time $t^n$. The sample fractional spatial distance, $\Delta^{n}(\omega_\ell)$, is computed by \ref{['eq:p_FT']}.
  • Figure 3: Numerical solution for the deterministic case of the problem \ref{['eq:PDE']}--\ref{['eq:ConditionsCI']} derived from the FF method (blue solid line) and the FT method (red dashed line) across various values of $T=\{1, 5,10,50,75,100 \}$.
  • Figure 4: Numerical solution for the deterministic version of problem \ref{['eq:PDE']}--\ref{['eq:ConditionsCI']}, comparing the FF method with different numbers of grid points $M$, and the FT method using $M=50$ grid points.The plot on the right shows a zoomed-in view for $r$ close to $H(T)$ at $T=50$.
  • Figure 5: Absolute deviations of the mean (left) and the standard deviation (right) of the random free boundary $H(T;\omega_{\ell})$, $1 \leq \ell \leq K$, computed between random FF and FT method up to time $T=10$ by \ref{['AbsErr-FFvsFT_H']} when parameters $\alpha$ and $\beta$ are constant (see Table \ref{['table:Comparison_parameters']}) and the sample $K$ realizations varies.
  • ...and 14 more figures

Theorems & Definitions (7)

  • Theorem 1: Spreading-Vanishing dichotomy Du2011Spreading-vanishingII
  • Lemma 1
  • Theorem 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4