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Propagation phenomena in periodic patchy landscapes with interface conditions

François Hamel, Frithjof Lutscher, Mingmin Zhang

Abstract

This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis. We first establish the well-posedness for the Cauchy problem. Next, we give existence and uniqueness results for the positive steady state and we analyze the long-time behavior of the solutions to the evolution problem. Afterwards, based on dynamical systems methods, we investigate the spreading properties and the existence of pulsating traveling waves in the positive and negative directions. It is shown that the asymptotic spreading speed, c * , exists and coincides with the minimal wave speed of pulsating traveling waves in positive and negative directions. In particular, we give a variational formula for c * by using the principal eigenvalues of certain linear periodic eigenvalue problems.

Propagation phenomena in periodic patchy landscapes with interface conditions

Abstract

This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis. We first establish the well-posedness for the Cauchy problem. Next, we give existence and uniqueness results for the positive steady state and we analyze the long-time behavior of the solutions to the evolution problem. Afterwards, based on dynamical systems methods, we investigate the spreading properties and the existence of pulsating traveling waves in the positive and negative directions. It is shown that the asymptotic spreading speed, c * , exists and coincides with the minimal wave speed of pulsating traveling waves in positive and negative directions. In particular, we give a variational formula for c * by using the principal eigenvalues of certain linear periodic eigenvalue problems.

Paper Structure

This paper contains 15 sections, 20 theorems, 227 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model presentation and statement of the main results
  3. The model and some equivalent formulations
  4. Well-posedness of the Cauchy problem \ref{['m-pb']}--\ref{['d-f']}
  5. Existence, uniqueness and attractiveness of a positive periodic steady state
  6. Spreading speeds and pulsating traveling waves
  7. Well-posedness of the Cauchy problem \ref{['m-pb']}--\ref{['d-f']}: proof of Theorem \ref{['thm-m-wellposedness']}
  8. Truncated problem
  9. Proof of Theorem \ref{['thm-m-wellposedness']}
  10. Existence and uniqueness of a stationary solution
  11. Existence of solutions: proof of Theorem \ref{['thm-2.1-existence']}
  12. Uniqueness of solutions: proof of Theorem \ref{['thm-2.4-uniqueness']}
  13. Long-time behavior: proof of Theorem \ref{['thm-long time behavior']}
  14. Spreading speeds and periodic traveling waves: proofs of Theo-rems \ref{['thm-spreading result']} and \ref{['thm-existence of PTW']}
  15. Comparison principles

Key Result

Theorem 2.2

Under assumption 2.5, for any nonnegative bounded continuous initial condition $u_0$, there is a unique nonnegative bounded classical solution $u$ in $[0,+\infty)\times\mathbb{R}$ of the Cauchy problem m-pb--d-f. Furthermore, for any $\tau>0$ and any patch $I\subset \mathbb{R}$, with a positive constant $C$ depending on $\tau$, $l_{1,2}$, $d_{1,2}$, $f_{1,2}$, $\sigma$ and $\Vert u_0 \Vert_{L^\in

Figures (2)

  • Figure 1: Schematic figure of the one-dimensional periodic patchy environment.
  • Figure 2: Truncated interval $[-nl, nl]$.

Theorems & Definitions (42)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 32 more