Traveling wave front in diffusive and competitive Lotka–Volterra system with delays

Lv, Guangying; Wang, Mingxin

doi:10.1016/j.nonrwa.2009.02.020

Cited by 53 publications

(49 citation statements)

References 17 publications

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“…(1.2) and obtained the existence of traveling waves connecting the trivial equilibrium to the coexistence equilibrium by using Schauder's fixed point theorem and the cross iteration scheme. Since then, lots of authors investigated the various properties on kinds of traveling waves for 2-species competitive Lotka-Volterra systems with discrete delays or spatio-temporal delays, see [18,22,34,36] . Moreover, Yu and Yuan [35] investigated the existence of traveling waves connecting the trivial equilibrium to the coexistence equilibrium for 3-species competitive Lotka-Volterra systems with delays.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves for n-species competitive system with nonlocal dispersals and delays

Xia¹,

Dong³

et al. 2016

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 99%

Traveling waves for n-species competitive system with nonlocal dispersals and delays

Xia¹,

Dong³

et al. 2016

Applied Mathematics and Computation

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“…Some related problems on traveling wave solutions for the Lotka-Volterra competitive system with spatial diffusion 6) were investigated in [8-10, 14-16, 31, 33] and the references therein, where a i , b i , r i , D i (i = 1, 2) are positive constants. For the Lotka-Volterra competitive system with delays and spatial diffusion, the existence of traveling wave solutions was also admitted in [4,21,22,25,40]. Huang and Zou [12] naturally considered the system with two mutualistic species…”

Section: Du(t) Dt = U(t) B -Au(t) -Du(t -1) (13)mentioning

confidence: 99%

Existence and asymptotic of traveling wave fronts for the delayed Volterra-type cooperative system with spatial diffusion

Meng

Zhang

2018

Adv Differ Equ

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In the paper, we are concerned with the existence and the exponential asymptotic behavior of traveling waves for the delayed Volterra-type cooperative system with nonquasimonotone conditionmodeling the variation of the populations. Here, the major contribution to population model is the introduction of population self-regulation depending on not only the populations at time t, but also on the earlier population time t -τ i (i = 1, 3). By constructing a pair of suitable upper and lower solutions we obtain the existence of traveling wave fronts connecting the trivial equilibrium and the positive equilibrium, which indicates that there is a transition zone moving the steady state with no species to the steady state with the coexistence of two species. Furthermore, with the help of Ikehara's theorem, the exponential asymptotic behavior of traveling wave front is exactly derived for this system without quasi-monotone conditions. The results are not only an extension of existing results for the known logistic or cooperative system, but also can extend another type of delayed logistic equation with spatial diffusion ∂u (x, t) ∂t x, t) ∂x 2 + ru(x, t) 1 -au(x, t) -bu(x, t -τ ) . MSC: 35C07; 92D25; 35B35

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“…In ecology, the existence of travelling wave solutions connecting two equilibria means that the unstable equilibrium is taken over by the stable one in space as time increases. Studies on existence of traveling waves in such system have received considerable attention, and many noteworthy findings have come out of this field, e.g., [2,3,5,6,7,8,9,10,17,18,19,20,21,22].…”

Section: Parametermentioning

confidence: 99%

Traveling Wave Solutions in a Stage-Structured Delayed Reaction-Diffusion Model With Advection

Zhang

Zhao

2015

Mathematical Modelling and Analysis

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We investigate a stage-structured delayed reaction-diffusion model with advection that describes competition between two mature species in water flow. Time delays are incorporated to measure the time lengths from birth to maturity of the populations. We show there exists a finite positive number c * that can be characterized as the slowest spreading speed of traveling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium. The model and mathematical result in [J.F.M. Al-Omari, S.A. Gourley, Stability and travelling fronts in Lotka-Volterra competition models with stage structure, SIAM J. Appl. Math. 63 (2003) 2063-2086] are generalized.

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Traveling wave front in diffusive and competitive Lotka–Volterra system with delays

Cited by 53 publications

References 17 publications

Traveling waves for n-species competitive system with nonlocal dispersals and delays

Traveling waves for n-species competitive system with nonlocal dispersals and delays

Existence and asymptotic of traveling wave fronts for the delayed Volterra-type cooperative system with spatial diffusion

Traveling Wave Solutions in a Stage-Structured Delayed Reaction-Diffusion Model With Advection

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