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.
"Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system." Hiroshima Math. J. 49 (2) 251 - 271, July 2019. https://doi.org/10.32917/hmj/1564106547