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Persistence, extinction and spreading properties of non-cooperative Fisher--KPP systems in space-time periodic media

Léo Girardin

TL;DR

This work advances the theory of non-cooperative Fisher-KPP reaction-diffusion systems in space-time periodic media by linking long-time dynamics to a family of generalized principal eigenvalues $\lambda_{1,z}$. The authors develop a robust framework using cooperative-system controls, Harnack inequalities, and Gaussian bounds to establish extinction criteria, hair-trigger persistence, and precise spreading speeds via a Freidlin--Gärtner-type formula. Key contributions include sharp dichotomies determined by $\lambda_1'$, $\lambda_1$, and $\lambda_{1,z}$, existence of a uniformly positive space-time periodic entire solution when $\lambda_1'<0$, and a rigorous spreading description for compactly supported initial data. The methods extend scalar KPP results to arbitrary system size $N$, with implications for multi-type population dynamics under spatially and temporally heterogeneous environments.

Abstract

This paper is concerned with asymptotic persistence, extinction and spreading properties for non-cooperative Fisher-KPP systems with space-time periodic coefficients. Results are formulated in terms of a family of generalized principal eigenvalues associated with the linearized problem. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time, at least one space-time periodic uniformly positive entire solution exists, and solutions with compactly supported initial condition asymptotically spread in space at a speed given by a Freidlin-G{ä}rtner-type formula. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the twogeneralized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial condition. The proofs rely upon double-sided controls by solutions of cooperative systems. The control from below is new for such systems and makes it possible to shorten the proofs and extend the generality of the system simultaneously.

Persistence, extinction and spreading properties of non-cooperative Fisher--KPP systems in space-time periodic media

TL;DR

This work advances the theory of non-cooperative Fisher-KPP reaction-diffusion systems in space-time periodic media by linking long-time dynamics to a family of generalized principal eigenvalues . The authors develop a robust framework using cooperative-system controls, Harnack inequalities, and Gaussian bounds to establish extinction criteria, hair-trigger persistence, and precise spreading speeds via a Freidlin--Gärtner-type formula. Key contributions include sharp dichotomies determined by , , and , existence of a uniformly positive space-time periodic entire solution when , and a rigorous spreading description for compactly supported initial data. The methods extend scalar KPP results to arbitrary system size , with implications for multi-type population dynamics under spatially and temporally heterogeneous environments.

Abstract

This paper is concerned with asymptotic persistence, extinction and spreading properties for non-cooperative Fisher-KPP systems with space-time periodic coefficients. Results are formulated in terms of a family of generalized principal eigenvalues associated with the linearized problem. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time, at least one space-time periodic uniformly positive entire solution exists, and solutions with compactly supported initial condition asymptotically spread in space at a speed given by a Freidlin-G{ä}rtner-type formula. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the twogeneralized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial condition. The proofs rely upon double-sided controls by solutions of cooperative systems. The control from below is new for such systems and makes it possible to shorten the proofs and extend the generality of the system simultaneously.
Paper Structure (19 sections, 20 theorems, 134 equations)

This paper contains 19 sections, 20 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Motivations
  4. Notations
  5. Results
  6. Comments
  7. Preliminaries
  8. Global boundedness estimates and absorbing set
  9. Harnack inequality for linear cooperative systems
  10. Comparison principle for semilinear cooperative systems
  11. Comparison between the components in present-time
  12. Proofs
  13. Global extinction (Theorem \ref{['thm:extinction']})
  14. Conditional extinction of small solutions (Theorem \ref{['thm:extinction_small_solutions']})
  15. Existence of a nonnegative nonzero space-time periodic entire solution (Theorem \ref{['thm:existence_persistent_entire_solution']})
  16. ...and 4 more sections

Key Result

Theorem 1.1

Assume $\lambda_1'\geq 0$. Then all solutions $\bm{\mathbf{u}}$ of the Cauchy problem sys:KPP--IC satisfy

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Proposition 2.3: Proposition 2.4 in Girardin_Mazari_2022
  • ...and 26 more