Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Planar pulsating traveling wave solutions of non-cooperative Fisher--KPP systems in space-time periodic media

Léo Girardin, Grégoire Nadin

TL;DR

The article analyzes non-cooperative Fisher--KPP reaction--diffusion systems with space-time periodic coefficients and characterizes invading, constant-speed entire solutions along any direction. It develops a fixed-point framework that leverages the cooperative linear part and carefully constructed sub- and super-solutions in truncated cylinders, followed by compactness to obtain global planar pulsating waves. A sharp dichotomy is established: below a critical speed $c_e^\star$ no GTWs exist, while for $c>c_e^\star$ planar pulsating traveling waves with time and transverse-space periodicity exist for rational directions, with space-homogeneous coefficients allowing time-periodic waves for all speeds $c\ge c_e^\star$; a corresponding corollary applies to space-homogeneous media. The results connect to long-time spreading behavior in the Cauchy problem and relate to Freidlin--Gärtner-type spreading speeds, while outlining a natural conjecture for full time-space periodicity beyond rational directions.

Abstract

Non-cooperative Fisher-KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the invasion of open space by a persistent population at constant speed. These solutions are important in the understanding of long-time behaviors for the Cauchy problem. Adapting methods developed for scalar equations satisfying the comparison principle as well as methods developed for systems with homogeneous coefficients, we prove, in each spatial direction, the existence of a critical speed such that: there exists no almost planar generalized transition waves with a smaller speed; if the direction is rational, each rational speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time and transverse space periodicity; if the coefficients are homogeneous in space, each speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time periodicity.

Planar pulsating traveling wave solutions of non-cooperative Fisher--KPP systems in space-time periodic media

TL;DR

The article analyzes non-cooperative Fisher--KPP reaction--diffusion systems with space-time periodic coefficients and characterizes invading, constant-speed entire solutions along any direction. It develops a fixed-point framework that leverages the cooperative linear part and carefully constructed sub- and super-solutions in truncated cylinders, followed by compactness to obtain global planar pulsating waves. A sharp dichotomy is established: below a critical speed no GTWs exist, while for planar pulsating traveling waves with time and transverse-space periodicity exist for rational directions, with space-homogeneous coefficients allowing time-periodic waves for all speeds ; a corresponding corollary applies to space-homogeneous media. The results connect to long-time spreading behavior in the Cauchy problem and relate to Freidlin--Gärtner-type spreading speeds, while outlining a natural conjecture for full time-space periodicity beyond rational directions.

Abstract

Non-cooperative Fisher-KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the invasion of open space by a persistent population at constant speed. These solutions are important in the understanding of long-time behaviors for the Cauchy problem. Adapting methods developed for scalar equations satisfying the comparison principle as well as methods developed for systems with homogeneous coefficients, we prove, in each spatial direction, the existence of a critical speed such that: there exists no almost planar generalized transition waves with a smaller speed; if the direction is rational, each rational speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time and transverse space periodicity; if the coefficients are homogeneous in space, each speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time periodicity.

Paper Structure

This paper contains 25 sections, 18 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. General framework
  3. Organization of the paper
  4. State of the art
  5. Motivations
  6. Nondimensionalization
  7. Entire solutions and invasions of open space
  8. Almost planar generalized transition waves
  9. Planar pulsating traveling waves
  10. The linear part and the nonlinear part
  11. Notations
  12. Main result
  13. Comments
  14. Preliminaries
  15. Harnack inequality for linear cooperative systems
  16. ...and 10 more sections

Key Result

Theorem 1.1

Assume $\lambda_{1,\textup{per}}<0$. Then, for any $e\in\mathbb{S}^{n-1}$ and $c\in(-\infty,c_e^\star)$, there exists no almost planar generalized transition wave solution in direction $-e$ with global mean speed $c$.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Conjecture 1.5
  • Proposition 2.1: Proposition 2.4 in Girardin_Mazari_2022
  • Lemma 2.2: Lemma 3.5 in Girardin_2023
  • ...and 23 more