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Spreading dynamics for the Lotka-Volterra system with general initial supports: the strong competition

Hongjun Guo

Abstract

This paper studies the spreading dynamics of a high-dimensional strong competition Lotka-Volterra system where two species initially occupy disjoint measurable (possibly unbounded) subsets in $\mathbb{R}^N$, which are called initial support. Recently, Hamel and Rossi [14] introduced some new geometric notions, such as bounded or unbounded directions and positive-distance interior, for single-species equations with general initial supports. Under these notions and appropriate assumptions, we characterize directional spreading behavior for the two-species system: precise spreading speeds and sets for both species are derived.

Spreading dynamics for the Lotka-Volterra system with general initial supports: the strong competition

Abstract

This paper studies the spreading dynamics of a high-dimensional strong competition Lotka-Volterra system where two species initially occupy disjoint measurable (possibly unbounded) subsets in , which are called initial support. Recently, Hamel and Rossi [14] introduced some new geometric notions, such as bounded or unbounded directions and positive-distance interior, for single-species equations with general initial supports. Under these notions and appropriate assumptions, we characterize directional spreading behavior for the two-species system: precise spreading speeds and sets for both species are derived.
Paper Structure (9 sections, 20 theorems, 241 equations, 1 figure)

This paper contains 9 sections, 20 theorems, 241 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Known results for a single species
  3. Interaction of two species
  4. Main results
  5. Comments on Theorems \ref{['Th1']}-\ref{['Th2']}
  6. Preliminaries
  7. Proofs of Theorems \ref{['Th1']}-\ref{['Th2']}
  8. Proof of Theorem \ref{['Th1']}
  9. Proof of Theorem \ref{['Th2']}

Key Result

Proposition 1.4

Assume that (A1)-(A2) hold. There is $\rho>0$ such that the solution $(u,v)(t,x)$ of pro1 associated with $U=B_\rho$ and $V$ being any set with $U\cap V=\emptyset$ satisfies

Figures (1)

  • Figure 1: Sketch of $\mathcal{S}_u$ and $\mathcal{S}_v$ in 2 dimension.

Theorems & Definitions (40)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: HR
  • Proposition 1.4: BG
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 30 more