Traveling waves for a diffusive Lotka-Volterra  competition model I: singular perturbations

Hosono, Yuzo

doi:10.3934/dcdsb.2003.3.79

Cited by 32 publications

(24 citation statements)

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“…Conley and Gardner [5] and Gardner [8] showed that (1.5) has travelling front solutions connecting the equilibria E 1 and E 2 , where Conley index and degree theory methods have been developed. Other related results can be found in Gourley and Ruan [9], Hosono [11,12], Kan-on [17], etc. We shall establish the existence of travelling waves in system (1.4), thus in systems (1.2) and (1.3), that connect the trivial equilibrium E 0 and the positive equilibrium E * .…”

Section: Introductionmentioning

confidence: 95%

Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems

2006

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This paper is concerned with the existence of travelling wave solutions in a class of delayed reaction-diffusion systems without monotonicity, which concludes two-species diffusion-competition models with delays. Previous methods do not apply in solving these problems because the reaction terms do not satisfy either the so-called quasimonotonicity condition or non-quasimonotonicity condition. By using Schauder's fixed point theorem, a new cross-iteration scheme is given to establish the existence of travelling wave solutions. More precisely, by using such a new cross-iteration, we reduce the existence of travelling wave solutions to the existence of an admissible pair of upper and lower solutions which are easy to construct in practice. To illustrate our main results, we study the existence of travelling wave solutions in two delayed two-species diffusion-competition systems.Mathematics Subject Classification: 35K57, 35R20, 92D25

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Section: Introductionmentioning

confidence: 95%

Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems

2006

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“…Systems responding in this way to perturbations are said to have a threshold. Systems with excitable behavior appear in extended chemical reactions [1], in the propagation of action potentials along nerve axons [2,3], and in population dynamics models [4,5].…”

Section: Introductionmentioning

confidence: 99%

Excitability in a model with a saddle-node homoclinic bifurcation

Dilão¹,

Volford²

2004

Discrete &Amp; Continuous Dynamical Systems - B

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In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.

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“…If the growth rates of both populations are decreased, then the interaction is called a competition. If each population's growth rates are enhanced, then the interaction is called mutualism or symbiosis [VP09,Gop82,Hos03,Wan78].…”

Section: Reaction-diffusion Equation For Generalized Fisher Equationmentioning

confidence: 99%

“…The existence of at least one travelling wave at minimal wave speed is proven in [FC03]. Perturbation methods are used in [Hos03] to study the travelling wave solutions for this model asymptotically, when there is only one stable equilibrium solution and the diffusion coefficient is small. In our present work we will study a reaction-diffusion system for competing and cooperating species.…”

Section: Travelling Wave Solutions Of Diffusion Lotka-volterra Systemmentioning

confidence: 99%

A Reaction Diffusion Model for Inter-Species Competition and Intra-Species Cooperation

Rasheed

Billingham

2013

Math. Model. Nat. Phenom.

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This thesis deals with a two component reaction-diffusion system (RDS) for competing and cooperating species. We have analyse in detail the stability and bifurcation structure of equilibrium solutions of this system, a natural extension of the Lotka-Volterra system. We find seven topologically different regions separated by bifurcation boundaries depending on the number and stability of equilibrium solutions, with four regions in which the solutions are similar to those in the Lotka-Volterra system. We study RDS in the small parameter of the range 0 < λ 1 (fast diffusion and slow reaction), and in a few cases we assume λ = O(1). We consider three types of initial conditions, and we find three types of travelling wave solutions using numerical and asymptotic methods. However, neither numerical nor asymptotic methods were able to find a particular travelling wave solution which connects a coexistence state say, (u 0 , w 0 ) to an extinction state (0, 0) when 0 < λ 1. This type can be found when the reaction-diffusion system satisfy the symmetry property and λ = 1.From asymptotic methods, we find that RDS is a singular perturbation problem (one inner and two outer regions) when one of the equilibrium solutions associated with the travelling wave has u = 0, whereas, when u = 0 we get a regular perturbation problem.We find the numerical and asymptotic solutions for RDS. We have published the aboveWe study the stability of travelling wave solutions for RDS in two dimensions using numerical and asymptotic methods. We also used the Evans function as a tool for computing the stability of the three types of wave. We find that all the types of travelling wave solutions are stable for all values of the parameters.ii

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Traveling waves for a diffusive Lotka-Volterra competition model I: singular perturbations

Cited by 32 publications

References 0 publications

Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems

Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems

Excitability in a model with a saddle-node homoclinic bifurcation

A Reaction Diffusion Model for Inter-Species Competition and Intra-Species Cooperation

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