Loops and branches of coexistence states in a Lotka-Volterra competition model

Yuan Lou, Salomé Martínez, Peter Poláčik

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36 Scopus citations

Abstract

A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.

Original languageEnglish (US)
Pages (from-to)720-742
Number of pages23
JournalJournal of Differential Equations
Volume230
Issue number2
DOIs
StatePublished - Nov 15 2006

Bibliographical note

Funding Information:
S.M. was partially supported by Fondecyt Grant #1050754, FONDAP de Matemáticas Apli-cadas and Nucleus Millennium P04-069-F Information and Randomness; P.P. was supported by NSF grant DMS-0400702. The visits of S.M. to Ohio State University was partially supported by Fondecyt Grant #1050754, MRI and MBI at Ohio State University. This work is supported in part by NSF grant upon Agreement No. 0112050. The authors thank the anonymous referee for his/her careful reading of the manuscript and helpful comments.

Keywords

  • Bifurcation
  • Competing species
  • Reaction-diffusion
  • Spatial heterogeneity
  • Stability

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