Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Liu, Qian; Liu, Shuang; Lam, King‐Yeung

doi:10.3934/dcds.2020050

Cited by 20 publications

(9 citation statements)

References 65 publications

(153 reference statements)

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“…For the Cauchy problem, Lin and Li [31] considered system (1.1) with compactly supported initial functions and obtained the spreading speed of the faster species and some estimates on the speed of the slower species. More recently, Liu, Liu and Lam [32,33] obtained rather complete results.…”

Section: Resultsmentioning

confidence: 98%

Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: the strong-weak type

Wu¹,

Xiao²,

Zhou³

2022

Preprint

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We consider the classical two-species Lotka-Volterra competition-diffusion system in the strong-weak competition case. When the corresponding minimal speed of the traveling waves is not linear determined, we establish the precise asymptotic behavior of the solution of the Cauchy problem in two different situations: (i) one species is an invasive one and the other is a native species; (ii) both two species are invasive species.

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Section: Resultsmentioning

confidence: 98%

Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: the strong-weak type

Wu¹,

Xiao²,

Zhou³

2022

Preprint

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“…Standard linearization near the equilibrium point (0, 1) gives that the minimal speed c min of travelling wave front satisfies c min c 0 := 2 √ 1 − k 1 , where c 0 is always called the critical speed. And Lam et al in [8,12] also pointed out c min c 0 . As introduced in [2,9], if c min = c 0 , then we say that the minimal wave speed is linearly selected, otherwise, if c min > c 0 , we say that the minimal wave speed is nonlinearly selected.…”

Section: Y Wang H LI and X Limentioning

confidence: 93%

Travelling wave fronts of Lotka-Volterra reaction-diffusion system in the weak competition case

Wang¹,

Li²,

Li³

2021

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed $c$ . Specially, when the competition rate exerted on one species converges to zero, then for any $c>c_0$ , where $c_0$ is the critical speed, the travelling wave front with the speed $c$ is globally exponentially stable.

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“…Concerning the large time behavior of the Cauchy problem, some first estimates were obtained by Lin and Li [17]. More recently, Liu, Liu and Lam [18,19] obtained some rather complete results.…”

Section: Introductionmentioning

confidence: 99%

Lotka-Volterra competition-diffusion system: the critical competition case

Alfaro

Xiao²

2021

Preprint

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We consider the reaction-diffusion competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the non-existence of ultimately monotone traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the "faster" species excludes the "slower" one (with a known spreading speed), but also provide a sharp description of the profile of the solution, thus shedding light on a new bump phenomenon.

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Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Cited by 20 publications

References 65 publications

Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: the strong-weak type

Sharp estimates for the spreading speeds of the Lotka-Volterra competition-diffusion system: the strong-weak type

Travelling wave fronts of Lotka-Volterra reaction-diffusion system in the weak competition case

Lotka-Volterra competition-diffusion system: the critical competition case

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