Traveling Wavefronts in a Delayed Food‐limited Population Model

Ou, Chunhua; Wu, Jianhong

doi:10.1137/050638011

Cited by 36 publications

(29 citation statements)

References 25 publications

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“…(H 1 ) The equilibria E 1 and E 2 are hyperbolic in the sense that Λ 0 0 (iv) = 0, Λ K 0 (iv) = 0 for any real number v. instead of (2.2), see [21,42] and [45]. In this case, our main results below remain valid as long as…”

Section: Persistence Of Traveling Wavefronts: Monostable Casementioning

confidence: 86%

Persistence of wavefronts in delayed nonlocal reaction–diffusion equations

2007

Journal of Differential Equations

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116

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We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors.

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Section: Persistence Of Traveling Wavefronts: Monostable Casementioning

confidence: 86%

Persistence of wavefronts in delayed nonlocal reaction–diffusion equations

2007

Journal of Differential Equations

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116

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“…This leads to the development of partial differential equation (PDE) models in the investigation of population dynamics in food chain; see Refs. [5,[7][8][9]20,32,34,35,38]. PDE models for chemostat food chain with diffusions were also considered in Refs.…”

Section: Introductionmentioning

confidence: 99%

How many consumer levels can survive in a chemotactic food chain?

Liu

2009

Front. Math. China

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We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels.

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“…From now onwards, we will use the same argument as that in Ou & Wu (2007) to proceed with our proof. It contains five steps as follows.…”

Section: ( (mentioning

confidence: 99%

“…By adopting the asymptotic method developed by Faria et al (2006) and Ou & Wu (2007), which is a combination of perturbation methods, the Fredholm theory and the Banach fixed point theorem, we have theoretically justified the existence of a travelling wave solution with large wave speed in system (1.1). In order to illustrate the validity of the theoretical result obtained in §2, we perform numerical calculations using the software MATLAB.…”

Section: Simulationmentioning

confidence: 99%

“…The purpose of this paper is to revisit the competition equation with a new motility term modelling contact inhibition between cell populations, theoretically justifying the existence of a travelling wave solution with large wave speed in system (1.1) by adopting the asymptotic method developed by Faria et al (2006) and Ou & Wu (2007). Although here we use the same idea as that in Faria et al (2006) and Ou & Wu (2007), there are crucial differences in that the singularly perturbed terms are nonlinear and thus result in particular difficulties in transforming the differential system into an integral system so that the fixed point theorem can be applicable.…”

Section: Introductionmentioning

confidence: 99%

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Justification for wavefront propagation in a tumour growth model with contact inhibition

Zhu

Wei

2008

Proc. R. Soc. A.

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Cancer is a disease characterized by abnormal division of cells combined with the malignant behaviour of these cells. Malignant cancer cells tend to spread, either by direct growth into adjacent tissue through invasion and dispersal or by implantation into distant sites by metastasis. In this paper, we will concentrate on the first type of tumour growth and consider a reaction-diffusion model for competition cells with nonlinear diffusion term modelling contact inhibition between normal and tumour cell populations for which wave propagation is usually observed in clinical data. Mathematically, based on a combination of perturbation methods, the Fredholm theory and the Banach fixed point theorem, we theoretically justify the existence of the travelling wave solution. Numerical simulations are finally illustrated to confirm our rigorous results.

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Traveling Wavefronts in a Delayed Food‐limited Population Model

Cited by 36 publications

References 25 publications

Persistence of wavefronts in delayed nonlocal reaction–diffusion equations

Persistence of wavefronts in delayed nonlocal reaction–diffusion equations

How many consumer levels can survive in a chemotactic food chain?

Justification for wavefront propagation in a tumour growth model with contact inhibition

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