Exact travelling wave solutions of three-species competition--diffusion systems

Chen, Chiun-Chuan; Hung, Li-Chang; Mimura, Mamoru; Ueyama, Daishin

doi:10.3934/dcdsb.2012.17.2653

Cited by 41 publications

(69 citation statements)

References 5 publications

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“…We now address the naive question of whether non-trivial traveling-wave solutions of (19) and (20) exist. This has not yet been answered, except for certain exact non-trivial travelingwave solutions found by Chen et al (2012). With the help of the AUTO numerical method (Doedel et al, 2010), they obtained the global solution structure of traveling wave solutions of (19) and (20) when b 23 varies globally, as shown in Fig.…”

Section: Three-species Traveling-wave Solutionsmentioning

confidence: 97%

Dynamic coexistence in a three-species competition–diffusion system

Mimura

Tohma

2015

Ecological Complexity

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Section: Three-species Traveling-wave Solutionsmentioning

confidence: 97%

Dynamic coexistence in a three-species competition–diffusion system

Mimura

Tohma

2015

Ecological Complexity

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“…Let (u(y, t), v(y, t)) be the solution of (1.3) with the entire space R replaced by a bounded domain in R under the zero Neumann boundary conditions. Then for initial conditions u(x, 0),v(x, 0) > 0, we have (i) a 1 < 1 < a 2 ⇒ lim t→∞ (u(y, t), v(y, t)) = (1, 0);…”

Section: Introductionmentioning

confidence: 99%

A maximum principle for diffusive Lotka–Volterra systems of two competing species

Chen

Hung

2016

Journal of Differential Equations

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Using an elementary approach, we establish a new maximum principle for the diffusive Lotka-Volterra system of two competing species, which involves pointwise estimate of an elliptic equation consisting of the second derivative of one function, the first derivative of another function, and a quadratic nonlinearity. This maximum principle gives a priori estimates for the total mass of the two species. Moreover, applying it to the system of three competing species leads to a nonexistence theorem of traveling wave solutions.

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“…It is worth noting that the nonlinear system (1) was studied under the restriction e 1 = 0, otherwise the system reduces to the three-component diffusive Lotka-Volterra system (DLVS). Lie symmetries of the three-component DLVS are completely described in [9], while its exact solutions were constructed in [9,8,20].…”

Section: Discussionmentioning

confidence: 99%

A hunter-gatherer–farmer population model: Lie symmetries, exact solutions and their interpretation

Cherniha¹,

Davydovych²

2018

Eur. J. Appl. Math

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The Lie symmetry classification of the known three-component reaction-diffusion system modelling the spread of an initially localized population of farmers into a region occupied by hunter-gatherers is derived. The Lie symmetries obtained for reducing the system in question to systems of ODEs and constructing exact solutions are applied. Several exact solutions of traveling front type are found, their properties are identified and biological interpretation is discussed.

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Exact travelling wave solutions of three-species competition--diffusion systems

Cited by 41 publications

References 5 publications

Dynamic coexistence in a three-species competition–diffusion system

Dynamic coexistence in a three-species competition–diffusion system

A maximum principle for diffusive Lotka–Volterra systems of two competing species

A hunter-gatherer–farmer population model: Lie symmetries, exact solutions and their interpretation

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