Past exam of the topics in applied mathematics course of the University of Oxford / 2008 / A / 3 / e / iii / Solution

For the modified strain the travelling wave equations areIntegrating the second across the entire travelling wave gives ${\int }_{\mathbb{R}}gbdy=c$, 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, givesConsequentlyThe 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.