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Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system. (English) Zbl 1423.37068
Summary: 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.

MSC:
37N25 Dynamical systems in biology
35K57 Reaction-diffusion equations
35B41 Attractors
35B45 A priori estimates in context of PDEs
92C17 Cell movement (chemotaxis, etc.)
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