 Solution

For a spatially uniform steady state with nonzero bacteria abundance, $b(1-b)=0$ forces $b_{\ast }=1$. The nutrient balance then givesTo test spatially uniform linear stability, writeAt the positive equilibrium, the Jacobian matrix isIts eigenvalues are $\boxed{-1,-\mu }$, both negative, so the positive equilibrium is asymptotically stable against spatially uniform perturbations. This remains true at $\mu =1$, where a repeated eigenvalue may produce a decaying $t{e}^{-t}$ contribution. The printed stability claim must be understood in this spatially uniform sense: sufficiently strong chemotaxis can destabilize nonuniform perturbations, as the next answer shows.