On nonstandard chemotactic dynamics with logistic growth induced by a modified complex Ginzburg–Landau equation

Abstract

In this paper, we derive a variant of the classical Keller-Segel model of chemotaxis incorporating a growth term of logistic type for the cell population n(t,x), say nu n(1-n) with nu>0, and a nonstandard chemical production-degradation mechanism involving first- and second-order derivatives of the logarithm of the cell density, say f lambda ab(n,nx,nxx)=lambda n+anxxn+bnx2n2 with lambda,a,b is an element of R, via the (n,S)-hydrodynamical system associated with a modified Ginzburg-Landau equation governing the evolution of the complex wavefunction psi=neiS. In a chemotactic context, S(t,x) will play the role of the concentration of chemical substance. Then, after carrying out a detailed analysis of the modulational stability of uniform-in-space plane waves, dark soliton-shaped traveling wave densities of the former system are constructed from solitary wave solutions of the latter.