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On the evolution of slow dispersal in multi-species communities

Robert Stephen Cantrell, King-Yeung Lam

Abstract

For any $N \geq 2$, we show that there are choices of diffusion rates $\{d_i\}_{i=1}^N$ such that for $N$ competing species which are ecologically identical and having distinct diffusion rates, the slowest disperser is able to competitive exclude the remainder of the species. In fact, the choices of such diffusion rates is open in the Hausdorff topology. Our result provides some evidence in the affirmative direction regarding the conjecture by Dockery et al. in 1998. The main tools include Morse decomposition of the semiflow, as well as the theory of normalized principal Floquet bundle for linear parabolic equations. A critical step in the proof is to establish the smooth dependence of the Floquet bundle on diffusion rate and other coefficients, which may be of independent interest.

On the evolution of slow dispersal in multi-species communities

Abstract

For any , we show that there are choices of diffusion rates such that for competing species which are ecologically identical and having distinct diffusion rates, the slowest disperser is able to competitive exclude the remainder of the species. In fact, the choices of such diffusion rates is open in the Hausdorff topology. Our result provides some evidence in the affirmative direction regarding the conjecture by Dockery et al. in 1998. The main tools include Morse decomposition of the semiflow, as well as the theory of normalized principal Floquet bundle for linear parabolic equations. A critical step in the proof is to establish the smooth dependence of the Floquet bundle on diffusion rate and other coefficients, which may be of independent interest.

Paper Structure

This paper contains 17 sections, 22 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Two conjectures of Dockery et al.
  3. Definitions
  4. Proofs of Propositions \ref{['prop:1.1']} and \ref{['prop:1.2']}
  5. Setting up the proof of Theorem \ref{['thm:main']}
  6. Outline of the proof
  7. The complete Lyapunov function for the unperturbed semiflow $\Phi$
  8. Uniform continuity of the intermediate semiflow $\varphi$ and perturbed semiflow $\varphi_\varepsilon$
  9. Rough estimates for the perturbed semiflow
  10. The normalized principal Floquet bundle
  11. The normalized principal Floquet bundle
  12. Completion of the proof of main theorem
  13. Conclusion
  14. The normalized principal Floquet bundle
  15. Existence and uniqueness results
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Suppose $N=2$ and $d_1<d_2$. Then every positive solution of eq:1.1 converges to the equilibrium $(\theta_{d_1},0)$. Furthermore, a Morse decomposition for $\textup{Inv}\,K^+$ is given by Here $\textup{Inv}\,K^+$ denotes the maximal bounded invariant set of the dynamical system generated in $K^+=\{(u_i)_{i=1}^N \in [C(\overline\Omega)]^N:\, u_i \geq 0\}$ under eq:1.1.

Theorems & Definitions (53)

  • Theorem \oldthetheorem: Dockery1998
  • Conjecture 1
  • Conjecture 2
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof : Proof of Proposition \ref{['prop:1.1']}
  • ...and 43 more