Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system

Guo, Jong‐Shenq; Nakamura, Kenichi; Ogiwara, Toshiko; Wu, Chin-Chin

doi:10.1016/j.jmaa.2018.12.007

Cited by 10 publications

(11 citation statements)

References 22 publications

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“…The analytical results for the discrete type of reaction diffusion systems have been reported in Bates et al (2001), Chow (2003). The results related to the traveling wave solutions in the system of the discrete models are reported in Guo et al (2019), Hupkes and Sandstede (2010), Straatman and Hupkes (2019).…”

Section: Introductionmentioning

confidence: 99%

A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

Ishii

Sato

et al. 2020

J. Math. Biol.

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In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta–Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.

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Section: Introductionmentioning

confidence: 99%

A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

Ishii

Sato

et al. 2020

J. Math. Biol.

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“…Finally, since cooperative systems enjoy the (strong) comparison principle, the conclusion follows by the same proof as that of [14,Theorem 4.2]. We safely omit the details.…”

mentioning

confidence: 58%

“…It is one of the questions to be addressed in this paper. For system (6)- (8), the existence of traveling front connecting (1, 0, 1) and (0, 1, 0) is derived in [13], while the stability and uniqueness of traveling fronts were addressed in [14]. Since our main concern here is the sign of wave speed for traveling fronts of bistable type, we shall not address the existence of traveling fronts to (3)-(5) connecting (1, 0, 1) and (0, 1, 0) with (9) here.…”

mentioning

confidence: 99%

“…Proof. Indeed, as in [14], it is more convenient to transform system (21)-(23) to the following cooperative system…”

mentioning

confidence: 99%

“…whereÛ := 1 − U ,V := V andŴ := 1 − W . Then, using (25) and (26), the same super-sub-solutions as in [14] with j + ct replaced by x − st based on a given traveling wave solution (s, U, V, W ) of (3)-(5) can be constructed (see (3.10)-(3.12) in [14]), where the operator D 2 is replaced by the second derivative and the monotone functions…”

mentioning

confidence: 99%

See 2 more Smart Citations

The sign of traveling wave speed in bistable dynamics

Guo

Nakamura

Ogiwara

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we first review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two different circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.

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Propagation Direction of the Traveling Wave for the Lotka–Volterra Competitive Lattice System

Wang

2020

J Dyn Diff Equat

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Stability and uniqueness of traveling waves for a discrete bistable 3-species competition system

Cited by 10 publications

References 22 publications

A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices

The sign of traveling wave speed in bistable dynamics

Propagation Direction of the Traveling Wave for the Lotka–Volterra Competitive Lattice System

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