= Growth-factor depletion travelling wave
{title2=$b_t=b_{xx}+gb,\qquad g_t=-gb$}
The <nondimensionalization>[dimensionless] <bacteria>[bacterial] growth and substrate depletion system $b_t=b_{xx}+gb$, $g_t=-gb$ has <travelling wave>[travelling-wave equations] $b\prime\prime+cb\prime+gb=0$, $cg\prime=gb$ in coordinate $y=x-ct$. For a <travelling wave>[front] with $(b,g)\to(b_0,0)$ behind and $(b,g)\to(0,1)$ ahead, its <first integral of an ordinary differential equation>[first integral] is $b\prime+c(b+g)=c$, forcing $b_0=1$. Positivity of the leading edge requires $c\ge2$.
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