 Growth-factor depletion travelling wave

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 $c\ge 2$.