Analytical solutions for distributions of chemotactic bacteria (Q1057209)

The present paper reports general and specialized results on the analytical solutions of the equations \[ \partial b/\partial t=\nabla \cdot (\mu (s)\nabla b-\chi (s)b\nabla s)+Yk(s)b,\quad \partial s/\partial t=D\nabla^ 2s-k(s)b \] where \(b=b(x,t)\) denotes the number of bacteria cells per unit volume, \(s=s(x,t)\) denotes the concentration of the critical substrate chemotactic agent, \(\mu\) (s) is the motility function, \(\chi\) (s) is the chemotactic flux coefficient, k(s) denotes the consumption rate function, Y is the yield coefficient for growth of the bacterial population and D is the substrate diffusivity. First the latter nonlinear coupled equations are considered to hold through an arbitrary spatial region R with a boundary \(\partial R\) that is impermeable to bacteria cell transport, i.e.: \[ n(x)(\mu (s)\nabla b- \chi (s)b\nabla s)=0\quad for\quad x\in \partial R, \] where n(x) is the unit outward normal vector at the boundary point x. The steady-state distributions of chemotactic bacteria with negligible population growth, the distributions of weakly chemotactic bacteria with population growth governed by substrate diffusion and steadily propagating plane - wave solutions are studied.