Hiroshima Mathematical Journal

Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system

Kanako Noda and Koichi Osaki

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We study the global-in-time existence and the asymptotic behavior of solutions to a one-dimensional chemotaxis system presented by Deneubourg (Insectes Sociaux 24 (1977)). The system models the self-organized nest construction process of social insects. In the limit as a time-scale coefficient tends to 0, the Deneubourg model reduces to a parabolic-parabolic Keller-Segel system with linear degradation. We first show the global-in-time existence of solutions. We next define the dynamical system of solutions and construct the global attractor. In addition, under the assumption of a large resting rate of worker insects, we construct a Lyapunov functional for the unique homogeneous equilibrium, which indicates that the global attractor consists only of the equilibrium.

Article information

Hiroshima Math. J., Volume 49, Number 2 (2019), 251-271.

Received: 6 February 2018
Revised: 8 February 2019
First available in Project Euclid: 26 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B25: Lyapunov functions and stability; attractors, repellers 35K57: Reaction-diffusion equations
Secondary: 35A01: Existence problems: global existence, local existence, non-existence 35B41: Attractors 35B45: A priori estimates

chemotaxis partly dissipative system global existence attractors Lyapunov function


Noda, Kanako; Osaki, Koichi. Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system. Hiroshima Math. J. 49 (2019), no. 2, 251--271. doi:10.32917/hmj/1564106547. https://projecteuclid.org/euclid.hmj/1564106547

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