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A random free-boundary diffusive logistic differential model: Numerical analysis, computing and simulation

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

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.

Suggested Citation

  • Casabán, M.-C. & Company, R. & Egorova, V.N. & Jódar, L., 2024. "A random free-boundary diffusive logistic differential model: Numerical analysis, computing and simulation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 55-78.
    • Handle: RePEc:eee:matcom:v:221:y:2024:i:c:p:55-78
    DOI: 10.1016/j.matcom.2024.02.016
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    References listed on IDEAS

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    1. Nepal, Surendra & Wondmagegne, Yosief & Muntean, Adrian, 2023. "Analysis of a fully discrete approximation to a moving-boundary problem describing rubber exposed to diffusants," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    2. María Consuelo Casabán & Rafael Company & Lucas Jódar, 2021. "Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    3. Vynnycky, M., 2023. "On boundary immobilization for one-dimensional Stefan-type problems with a moving boundary having initially parabolic-logarithmic behaviour," Applied Mathematics and Computation, Elsevier, vol. 444(C).
    4. Durrett, Rick & Foo, Jasmine & Leder, Kevin & Mayberry, John & Michor, Franziska, 2010. "Evolutionary dynamics of tumor progression with random fitness values," Theoretical Population Biology, Elsevier, vol. 78(1), pages 54-66.
    5. Casabán, M.-C. & Company, R. & Jódar, L., 2023. "Numerical difference solution of moving boundary random Stefan problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 878-901.
    6. María Consuelo Casabán & Rafael Company & Vera N. Egorova & Lucas Jódar, 2023. "Qualitative Numerical Analysis of a Free-Boundary Diffusive Logistic Model," Mathematics, MDPI, vol. 11(6), pages 1-19, March.
    7. Acevedo, Miguel A. & Marcano, Mariano & Fletcher, Robert J., 2012. "A diffusive logistic growth model to describe forest recovery," Ecological Modelling, Elsevier, vol. 244(C), pages 13-19.
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