= Solution
For the modified strain the <travelling wave> equations are
$$
b''+cb'+gb(1-\eta b)=0,\qquad cg'=gb.
$$
Integrating the second across the entire <travelling wave> gives $\int_{\mathbb R}gb\,dy=c$, since the <biological growth factor> profile still goes from $0$ to $1$. Integrating the first, using the constant asymptotic states and vanishing <derivative> at infinity, gives
$$
cb_1=\int_{\mathbb R}gb(1-\eta b)\,dy
=c-\eta\int_{\mathbb R}gb^2\,dy.
$$
Consequently
$$
\boxed{b_1=1-\frac\eta c\int_{\mathbb R}g(y)b(y)^2\,dy<1=b_0.}
$$
The subtracted <integral> is strictly positive because the profiles are positive and $\eta>0$. It is finite: $b$ is bounded by continuity and its finite endpoint limits, and $\int gb=c$. The additional density-dependent inhibition therefore reduces the <bacterial density> left behind; positivity of $b_1$ is part of the assumed positive-wave solution.
Back to article page
