Biomass yield identity for a depletion front ($b_1=1-\frac{\eta }{c}{\int }_{\mathbb{R}}g{b}^{2}dy<1$)

With inhibited bacterial growth $b_t=b_{xx}+gb(1-\eta b)$ and unchanged depletion $g_t=-gb$, a positive travelling wave leaving density $b_1$ obeys $c{b}_1=c-\eta {\int }_{\mathbb{R}}g{b}^{2}dy$. This follows by integrating both travelling-wave equations between their constant endpoint states. Thus $b_1<1$ for $\eta >0$: the yield behind the front is smaller than for uninhibited growth.