Can chemotaxis speed up or slow down the spatial spreading in parabolic–elliptic Keller–Segel systems with logistic source?

Salako, Rachidi B.; Shen, Wenxian; Xue, Shuwen

doi:10.1007/s00285-019-01400-0

Cited by 33 publications

(31 citation statements)

References 43 publications

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“…Vanishing-spreading result [2, Theorem 1.2] indicate there is a separating value l * , which is independent of the chemotactic sensitivity coefficients χ 1 , χ 2 , such that in the vanishing scenario the limiting moving boundary h ∞ < l * and when the initial habitat h 0 > l * the spreading guaranteed. The dependence of the dynamics of the system on the chemotactic sensitivity coefficients is another important and interesting questions [30], [41]. We also have the following question in this direction.…”

Section: Numerical Logistic Type Chemotaxis Systems With a Free Boundary 1087mentioning

confidence: 98%

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

Yang¹,

Bao²

2021

Discrete &Amp; Continuous Dynamical Systems - B

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The current paper is to investigate the numerical approximation of logistic type chemotaxis models in one space dimension with a free boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front (see Bao and Shen [1], [2]). The main challenges in the numerical studies lie in tracking the moving free boundary and the nonlinear terms from the chemical. To overcome them, a front-fixing framework coupled with the finite difference method is introduced. The accuracy of the proposed method, the positivity of the solution, and the stability of the scheme are discussed. The numerical simulations agree well with theoretical results such as the vanishing spreading dichotomy, local persistence, and stability. These simulations also validate some conjectures in our future theoretical studies such as the dependence of the vanishing-spreading dichotomy on the initial value u 0 , initial habitat h 0 , the moving speed ν and the chemotactic sensitivity coefficients χ 1 , χ 2 .1. Introduction. The current paper is to study, in particular, numerically, the spreading and vanishing dynamics of the following attraction-repulsion chemotaxis

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Section: Numerical Logistic Type Chemotaxis Systems With a Free Boundary 1087mentioning

confidence: 98%

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

Yang¹,

Bao²

2021

Discrete &Amp; Continuous Dynamical Systems - B

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“…Very recently, the authors of the paper [39] improved the results in [37]. It is proved that in the case of constant growth rate r > 0, if b > χµ and 1…”

Section: Introductionmentioning

confidence: 94%

Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments

Shen

Xue

2020

Journal of Differential Equations

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The current paper is concerned with the forced waves of Keller-Segel chemoattraction systems in shifting environments of the form, u t = u xx − χ(uv x) x + u(r(x − ct) − bu), x ∈ R 0 = v xx − νv + µu, x ∈ R, (0.1) where χ, b, ν, and µ are positive constants, c ∈ R, the resource function r(x) is globally Hölder continuous, bounded, r * = sup x∈R r(x) > 0, r(±∞) := lim x→±∞ r(x) exist, and either r(−∞) < 0 < r(∞), or r(±∞) < 0. Assume that b > 2χµ. In the case that r(−∞) < 0 < r(∞), it is shown that (0.1) has a forced wave solution connecting (r * b , µ ν r * b) and (0, 0) with speed c provided that c > χµr * 2 √ ν(b−χµ) − 2 r * (b−2χµ) b−χµ. In the case that r(±∞) < 0, it is shown that (0.1) has a forced wave solution connecting (0, 0) and (0, 0) with speed c provided that χ is sufficiently small and λ ∞ > 0, where λ ∞ is the generalized principal eigenvalue of the operator u(•) → u xx (•) + cu x (•) + r(•)u(•) on R in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.

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“…travelling wave solutions are investigated in a series of works [30][31][32] for both cases τ = 0 and τ = 1, where χ > 0 denotes the chemotactic coefficient. The existence of traveling wave solutions with minimal wave speed depending on a and χ was obtained, the asymptotic wave speed as χ → 0 as well as the spreading speed were examined in details in [30][31][32][33] where the major tool used therein to prove the existence of traveling wave solutions is the parabolic comparison principle. Except traveling wave solutions, the chemotaxis-growth system (1.4) can also drive other complex patterning dynamics (cf.…”

Section: Introductionmentioning

confidence: 99%

Traveling wave solutions for a bacteria system with density-suppressed motility

Lui

Ninomiya²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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To understand the "self-trapping" mechanism inducing spatio-temporal pattern formations observed in the experiment of [19] for bacterial motion, the following density-suppressed motility modelwas proposed in [6,19], where u(x, t) and v(x, t) represent the densities of bacteria and the chemical emitted by the bacteria, respectively; γ(v) is called the motility function satisfying γ ′ (v) < 0 and a, b > 0 are positive constants accounting for the growth and death rates of bacterial cells. The analysis of the above system is highly non-trivial due to the cross-diffusion and possible degeneracy resulting from the nonlinear motility function γ(v) and mathematical progresses on the global well-posedness and asymptotics of solutions were just made recently. Among other things, the purpose of this paper is to consider a specialized motility function γ(v) = MSC2020: 35B51, 35C07, 35K57, 35K65, 35Q92, 92C17.

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Can chemotaxis speed up or slow down the spatial spreading in parabolic–elliptic Keller–Segel systems with logistic source?

Abstract: The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system,where χ, a, b, λ, and µ are positive constants. Assume b > χµ. Then if in addition, 1 + *

Cited by 33 publications

References 43 publications

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

Persistence and spreading speeds of parabolic-elliptic Keller-Segel models in shifting environments

Traveling wave solutions for a bacteria system with density-suppressed motility

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