Dynamical systems analysis of a two-level trophic food web in the southern oceans

A theoretical model developed by Stone describing a two level trophic system in the Ocean is analysed, for the case in which there is unlimited supply of nutrients. It is shown that spontaneous oscillations in population numbers are possible, but they do not arise from a Hopf bifurcation. Seasonal forcing of the model is also investigated, and it is shown that resonances can occur, in addition to highly nonlinear behaviour including high period oscillations, quasi-periodicity and chaos. The model is then extended to include the case in which nutrient concentrations are allowed to vary. In this model seasonal forcing is not considered. Nevertheless a Hopf bifurcation is found for a critical value of the bifurcation parameter which is chosen as the non-dimensional reproductive rate of bacteria. The Hopf bifurcation gives rise to oscillatory solutions appearing as limit cycles. The stability of the limit cycles found is determined using Floquet theory, where it is observed that the periodic solutions arise from the Hopf bifurcation as stable orbits. As the bifurcation parameter is varied the branch of oscillatory solutions loses stability. This is due to a fold bifurcation, giving rise to regions in the parameter space where two different oscillatory solutions are possible for the same parameter values.