A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation

Abstract

The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square n(t,x) and the argument S(t,x) of a wavefunction ψ = √n exp(iS) that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. This connection is then exploited to construct a family of quasi-stationary solutions to the Keller-Segel system under the influence of no-flux and anti-Fick laws