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

At the leading edge $b\to 0$ and $g\to 1$. The linearized bacterial equation iswith characteristic exponentsFor $0<c<2$ they form a nonreal complex conjugate pair. A nonzero profile approaching the travelling front then has an exponentially decaying oscillatory leading term, and repeatedly changes sign. This is incompatible with the assumed positive bacterial density. The leading edge must therefore have real characteristic exponents, requiringAt $c=2$ the repeated characteristic exponent is $-1$ and polynomial factors in the asymptotic profile may occur. The argument supplies a necessary lower bound; it is not an independent existence proof for every speed satisfying it.