The non-linear structure of deep, stochastic, gyrotactic bioconvection is explored. A linear analysis is reviewed and a weakly non-linear analysis justifies its application by revealing the supercritical nature of the bifurcation. An asymptotic expansion is used to derive systems of partial differential equations for long plume structures which vary slowly with depth. Steady state and travelling wave solutions are found for the first order system of partial differential equations and the second order system is manipulated to calculate the speed of vertically travelling pulses. Implications of the results and possibilities of experimental validation are discussed.