Coexistence in the Competition Model with Diffusion

Eilbeck, J. C.; Furter, Jacques E.; López-Gómez, Julián

doi:10.1006/jdeq.1994.1005

Cited by 50 publications

(49 citation statements)

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“…Proof: we consider the set Ω ′ ⊂⊂ Ω compactly enclosed in Ω and the function φ(x, r) defined as in (19), restricted to the set D := {x ∈ Ω ′ , r ∈ R + : 2r < dist(∂Ω ′ , ∂Ω)}. We have to prove that φ is bounded on D. We argue by contradiction, assuming that …”

Section: Is Lipschitz Continuous In the Interior Of ωmentioning

confidence: 99%

A variational problem for the spatial segregation of reaction-diffusion systems

Conti¹,

Terracini²,

Verzini³

2005

Indiana Univ. Math. J.

108

166

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In this paper we study a class of stationary states for reaction-diffusion systems of k ≥ 3 densities having disjoint supports. For a class of segregation states governed by a variational principle we prove existence and provide conditions for uniqueness. Some qualitative properties and the local regularity both of the densities and of their free boundaries are established in the more general context of a functional class characterized by differential inequalities.

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Section: Is Lipschitz Continuous In the Interior Of ωmentioning

confidence: 99%

A variational problem for the spatial segregation of reaction-diffusion systems

Conti¹,

Terracini²,

Verzini³

2005

Indiana Univ. Math. J.

108

166

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“…A wide literature is devoted to this topic, mainly for the case of competition models of Lotka-Volterra type (see e.g. [1,7,8,9,10,11,12,13,14,15,16]). As a prototype for the study of this phenomenon, in [4] we consider the competition-diffusion system of k differential equations:…”

Section: Introductionmentioning

confidence: 99%

Uniqueness and least energy property for solutions to strongly competing systems

Conti¹,

Terracini²,

Verzini³

2006

Interfaces Free Bound.

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“…Fixed points of T are equilibria of (6). Indeed, it follows from (b) and σ 12 = 0 that ρ(DT (E 1 )) = 1, that one is a simple eigenvalue strictly larger in modulus than any other and with eigenspace spanned by z.…”

Section: Proofsmentioning

confidence: 99%

“…For related results on the existence, uniqueness and stability of positive solutions of competitive systems with diffusion see [4,1,6,5].…”

Section: Theorem 1 Let F Be Analytic and Assume That There Ismentioning

confidence: 99%

Asymptotically stable equilibria for monotone semiflows

Hirsch

Smith

2006

Discrete &Amp; Continuous Dynamical Systems - A

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Abstract. Conditions for the existence of a stable equilibrium and for the existence of an asymptotically stable equilibrium for a strongly order preserving semiflow are presented. Analyticity of the semiflow and the compactness of certain subsets of the set of equilibria are required for the latter and yield finiteness of the equilibrium set. Our results are applied to semilinear parabolic partial differential equations and to the classical Kolmogorov competition system with diffusion.

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Coexistence in the Competition Model with Diffusion

Cited by 50 publications

References 0 publications

A variational problem for the spatial segregation of reaction-diffusion systems

A variational problem for the spatial segregation of reaction-diffusion systems

Uniqueness and least energy property for solutions to strongly competing systems

Asymptotically stable equilibria for monotone semiflows

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