Existence and Spatial Decay of Forced Waves for the Fisher-KPP Equation with a Degenerate Shifting Environment

TL;DR

The paper addresses forced traveling waves in the Fisher-KPP equation with a moving degenerate environment, u_t = u_xx + u(a(x-ct) - u), by performing ODE-based asymptotics in the moving frame. It classifies local positive solutions near z=+∞, linking exponential decay to the finite-integral case and uncovering infinite families of non-exponential decays under a critical integrability condition of e^{-(1/c)∫ a}. Using this classification, the authors establish a complete existence, multiplicity, and spatial-decay theory: a unique exponentially decaying forced wave for c∈(0,2√α); in the super-critical weight case, infinitely many non-exponential waves with a non-L^1 maximal wave; and, in many degenerate scenarios, precise decay rates and ordering of forced waves, addressing open problems on forced waves in degenerate moving habitats. These results illuminate how degenerate shifting environments shape propagation and persistence in Fisher-KPP dynamics and provide a framework for analyzing climate-change-inspired habitat shifts.

Abstract

This paper studies forced waves for the heterogeneous Fisher-KPP equation $u_t = u_{xx} + u(a(x-ct)-u)$, where $c>0$ and $a(z)>0$ satisfies $a(-\infty)=α>0=a(+\infty)$, $a'(z)\le0$ ($z\gg1$). Using ODE asymptotic analysis, we classify all local positive solutions near $z=+\infty$. Exponential decay solutions always exist; non-exponential decay solutions exist if and only if $\mathrm{e}^{-\frac{1}{c}\int_{z_0}^z a(s)ds}\in L^1$ (or equivalently, when $a(z)$ decays slower than a critical algebraic rate). We establish a complete existence, multiplicity and spatial decay theory for forced waves. For each $c\in(0,2\sqrtα)$, there exists a unique exponentially decaying forced wave. This wave is either the unique forced wave or the minimal forced wave, depending on the integrability condition. In the super-critical case $\mathrm{e}^{-\frac{1}{c}\int_{z_0}^z a(s)ds}\in L^1$, for any $c>0$ there exist infinitely many non-exponentially decaying forced waves. The maximal wave is not in $L^1$, and for nearly all such $a(z)$ we establish the existence, multiplicity and precise decay of these waves. These results provide nearly complete answers to open problems concerning the existence, uniqueness, multiplicity and spatial decay rates of forced waves in Fisher-KPP models with degenerate moving environments.

Figures (1)

• Figure 1: The graph of $\lambda_1(c)$ and $\lambda'_1(c)$ defined by \ref{['S']} for the case $f_u(-\infty,0)=\alpha>0=f_u(+\infty,0)$ and $\frac{d}{dz}f_u(z,0)<0$.

Theorems & Definitions (19)

• Lemma 2.1
• Remark 2.1
• Theorem 2.1
• Remark 2.2
• Theorem 2.2
• Theorem 2.3
• Remark 2.3
• Theorem 2.4
• Lemma 3.1
• Proposition 3.1