On the stability of homogeneous steady states of a chemotaxis system with logistic growth term

Mark Andrew Joseph Chaplain; J. I.  Tello

doi:10.1016/j.aml.2015.12.001

On the stability of homogeneous steady states of a chemotaxis system with logistic growth term

Mark Andrew Joseph Chaplain, J. I. Tello

Applied Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider a nonlinear PDEs system of Parabolic-Elliptic type with chemotactic terms. The system models the movement of a population “n” towards a higher concentration of a chemical “c” in a bounded domain Ω. We consider constant chemotactic sensitivity χ and an elliptic equation to describe the distribution of the chemicaln_t − d_nΔn = −χdiv(n∇c) + μn(1−n), −d_cΔc + c = h(n) for a monotone increasing and lipschitz function h. We study the asymptotic behavior of solutions under the assumption of 2χ∣h′∣ < μ. As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.

Original language	English
Pages (from-to)	1-6
Journal	Applied Mathematics Letters
Volume	57
Early online date	11 Jan 2016
DOIs	https://doi.org/10.1016/j.aml.2015.12.001
Publication status	Published - Jul 2016

Keywords

Chemotaxis
Stability
Steady state
Lower and upper solutions

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Chaplain_Homogeneous_APL_AM
© 2016, Publisher / the Author(s). This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://dx.doi.org/10.1016/j.aml.2015.12.001
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title = "On the stability of homogeneous steady states of a chemotaxis system with logistic growth term",

abstract = "We consider a nonlinear PDEs system of Parabolic-Elliptic type with chemotactic terms. The system models the movement of a population “n” towards a higher concentration of a chemical “c” in a bounded domain Ω. We consider constant chemotactic sensitivity χ and an elliptic equation to describe the distribution of the chemicalnt − dnΔn = −χdiv(n∇c) + μn(1−n), −dcΔc + c = h(n) for a monotone increasing and lipschitz function h. We study the asymptotic behavior of solutions under the assumption of 2χ∣h′∣ < μ. As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.",

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AU - Tello, J. I.

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N2 - We consider a nonlinear PDEs system of Parabolic-Elliptic type with chemotactic terms. The system models the movement of a population “n” towards a higher concentration of a chemical “c” in a bounded domain Ω. We consider constant chemotactic sensitivity χ and an elliptic equation to describe the distribution of the chemicalnt − dnΔn = −χdiv(n∇c) + μn(1−n), −dcΔc + c = h(n) for a monotone increasing and lipschitz function h. We study the asymptotic behavior of solutions under the assumption of 2χ∣h′∣ < μ. As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.

AB - We consider a nonlinear PDEs system of Parabolic-Elliptic type with chemotactic terms. The system models the movement of a population “n” towards a higher concentration of a chemical “c” in a bounded domain Ω. We consider constant chemotactic sensitivity χ and an elliptic equation to describe the distribution of the chemicalnt − dnΔn = −χdiv(n∇c) + μn(1−n), −dcΔc + c = h(n) for a monotone increasing and lipschitz function h. We study the asymptotic behavior of solutions under the assumption of 2χ∣h′∣ < μ. As a result of the asymptotic stability we obtain the uniqueness of the strictly positive steady states.

KW - Chemotaxis

KW - Stability

KW - Steady state

KW - Lower and upper solutions

U2 - 10.1016/j.aml.2015.12.001

DO - 10.1016/j.aml.2015.12.001

M3 - Article

SN - 0893-9659

VL - 57

SP - 1

EP - 6

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

ER -

On the stability of homogeneous steady states of a chemotaxis system with logistic growth term

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Keywords

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