Growth-factor depletion travelling wave ($b_t=b_{xx}+gb,g_t=-gb$)

The dimensionless bacterial growth and substrate depletion system $b_t=b_{xx}+gb$, $g_t=-gb$ has travelling-wave equations $b\prime \prime +cb\prime +gb=0$, $cg\prime =gb$ in coordinate $y=x-ct$. For a front with $(b,g)\to (b_0,0)$ behind and $(b,g)\to (0,1)$ ahead, its first integral is $b\prime +c(b+g)=c$, forcing $b_0=1$. Positivity of the leading edge requires $c\ge 2$.

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.