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<p>We consider the spreading dynamics of the Fisher-KPP equation in a shifting environment, by characterizing the limit of the rate function of the solution. For the environment with a weak monotone condition, it was demonstrated that the rate function converges to the unique viscosity solution of the underlying Hamilton-Jacobi equations. In case the environment does not satisfy the weak monotone condition, we show that the rate function is then characterized by the Hamilton-Jacobi equation with a dynamic junction condition, which depends additionally on the principal eigenvalue derived from the environmental function. This approach applies to the case when the environment has multiple shifting speeds, and clarifies the transition between nonlocally pulled fronts and forced traveling waves.</p>

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Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments II: flux-limited solutions

  • 期刊论文

  • 发表日期：

    2025年7月22日

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  • King-Yeung Lam,
  • Grégoire Nadin,
  • Xiao Yu

摘要

We consider the spreading dynamics of the Fisher-KPP equation in a shifting environment, by characterizing the limit of the rate function of the solution. For the environment with a weak monotone condition, it was demonstrated that the rate function converges to the unique viscosity solution of the underlying Hamilton-Jacobi equations. In case the environment does not satisfy the weak monotone condition, we show that the rate function is then characterized by the Hamilton-Jacobi equation with a dynamic junction condition, which depends additionally on the principal eigenvalue derived from the environmental function. This approach applies to the case when the environment has multiple shifting speeds, and clarifies the transition between nonlocally pulled fronts and forced traveling waves.