Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 13B / iv / Solution

At the threshold, the determinant quadratic has a double zero. Its vertex lies at $s=(\chi \alpha n_0-\beta D_n-\gamma n_0{D}_c)/(2D_n{D}_c)$. Substituting the equality case of part (iii) givesThe mode is neutral exactly at onset and grows when the chemotactic coefficient increases beyond that threshold.