Singular limit of a competition–diffusion system with large interspecific interaction

Hilhorst, Danielle; Martin, Sébastien; Mimura, Mamoru

doi:10.1016/j.jmaa.2012.02.001

Cited by 12 publications

(14 citation statements)

References 32 publications

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“…Since initial data are uniformly bounded in L 2 (R d ) thanks to (11) and Remark 2, the first part of Proposition 1 is proved. For all T > 0, M and p are uniformly bounded in H 1 T for small enough.…”

Section: Lemmamentioning

confidence: 97%

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Reduction to a Single Closed Equation for 2-by-2 Reaction-Diffusion Systems of Lotka--Volterra Type

Strugarek¹,

Vauchelet²

2016

SIAM J. Appl. Math.

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We consider general models of coupled reaction-diffusion systems for interacting variants of the same species. When the total population becomes large with intensive competition, we prove that the frequencies (i.e. proportions) of the variants can be approached by the solution of a simpler reaction-diffusion system, through a singular limit method and a relative compactness argument. As an example of application, we retrieve the classical bistable equation for Wolbachia's spread into an arthropod population from a system modeling interaction between infected and uninfected individuals.

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Section: Lemmamentioning

confidence: 97%

“…We recall that the initial data are assumed to satisfy (11). Then the sequence (p init, ) is bounded uniformly in in L 2 (R d ) and (n init, − h(0)) is bounded uniformly in in L 2 ∩ L ∞ (R d ).…”

Section: Proof Of Convergencementioning

confidence: 99%

Reduction to a Single Closed Equation for 2-by-2 Reaction-Diffusion Systems of Lotka--Volterra Type

Strugarek¹,

Vauchelet²

2016

SIAM J. Appl. Math.

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“…We exploit now the localization of the infinitely fast reaction on the free boundary (similar calculations have been done e.g. in [11,17]). …”

Section: Proofmentioning

confidence: 99%

“…More recently, several papers, cf. for instance [10,5,16,11], and [18], have considered problems of this nature under various structural assumptions on the system. We collect the assumptions behind our results and give the concept of the weak formulation in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of diffusion-limited fast reactions

Seidman

Muntean

2017

Quart. Appl. Math.

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Abstract. We are concerned with the fast-reaction asymptotics λ → ∞ for a semi-linear coupled diffusion-limited reaction system in contact with infinite reservoirs of reactants. We derive the system of limit equations and prove the uniqueness of its solutions for equal diffusion coefficients. Additionally, we emphasize the structure of the limit free boundary problem. The key tools of our analysis include (uniform with respect to λ) L 1 -estimates for both fluxes and products of reaction and a balanced formulation, where combinations of the original components which balance the fast reaction are used.

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“…In the last decade, both the asymptotics and the qualitative properties of the limit segregated profiles have been the object of an intensive study, mostly in the symmetric case, by different teams [7,8,9,1,13,24,22]. Similarly, the dynamics of strongly competing species has been addressed as a singularly perturbed parabolic reaction-diffusion system in connection with spatially segregated limit profiles in [12,14,16,17]. In the quoted papers, a special attention was paid to the structure of the common zero set of these limit profiles.…”

Section: Introductionmentioning

confidence: 99%

Spiraling Asymptotic Profiles of Competition‐Diffusion Systems

Terracini

Verzini

Zilio

2019

Comm Pure Appl Math

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This paper describes the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction‐diffusion systems with strongly competitive interactions of Lotka‐Volterra type, when the matrix of the interspecific competition coefficients is asymmetric and the competition parameter tends to infinity. Unlike the symmetric case, when it is known that the nodal set consists of a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, the asymmetric case shows the emergence of spiraling nodal curves, still meeting at locally isolated points with finite vanishing order. © 2019 Wiley Periodicals, Inc.

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Singular limit of a competition–diffusion system with large interspecific interaction

Cited by 12 publications

References 32 publications

Reduction to a Single Closed Equation for 2-by-2 Reaction-Diffusion Systems of Lotka--Volterra Type

Reduction to a Single Closed Equation for 2-by-2 Reaction-Diffusion Systems of Lotka--Volterra Type

Asymptotics of diffusion-limited fast reactions

Spiraling Asymptotic Profiles of Competition‐Diffusion Systems

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