Understanding the dynamics of plankton populations is of major importance since
plankton form the basis of marine food webs throughout the world's oceans and play
a significant role in the global carbon cycle. In this thesis we examine the dynamical
behaviour of plankton models, exploring sensitivities to the number of variables
explicitly modelled, to the functional forms used to describe interactions, and to the
parameter values chosen. The practical difficulties involved in data collection lead to
uncertainties in each of these aspects of model formulation.
The first model we investigate consists of three coupled ordinary differential equations,
which measure changes in the concentrations of nutrient, phytoplankton and
zooplankton. Nutrient fuels the growth of the phytoplankton, which are in turn grazed
by the zooplankton. The recycling of excretion adds feedback loops to the system. In
contrast to a previous hypothesis, the three variables can undergo oscillations when
a quadratic function for zooplankton mortality is used. The oscillations arise from
Hopf bifurcations, which we track numerically as parameters are varied. The resulting
bifurcation diagrams show that the oscillations persist over a wide region of parameter
space, and illustrate to which parameters such behaviour is most sensitive. The
oscillations have a period of about one month, in agreement with some observational
data and with output of larger seven-component models. The model also exhibits fold
bifurcations, three-way transcritical bifurcations and Bogdanov-Takens bifurcations,
resulting in homo clinic connections and hysteresis. Under different ecological assumptions, zooplankton mortality is expressed by a
linear function, rather than the quadratic one. Using the linear function does not
greatly affect the nature of the Hopf bifurcations and oscillations, although it does
eliminate the homoclinicity and hysteresis. We re-examine the influential paper by
Steele and Henderson (1992), in which they considered the linear and quadratic mortality
functions. We correct an anomalous normalisation, and then use our bifurcation
diagrams to interpret their findings.
A fourth variable, explicitly modelling detritus (non-living organic matter), is then
added to our original system, giving four coupled ordinary differential equations. The
dynamics of the new model are remarkably similar to those of the original model, as
demonstrated by the persistence of the oscillations and the similarity of the bifurcation
diagrams. A second four-component model is constructed, for which zooplankton can
graze on detritus in addition to phytoplankton. The oscillatory behaviour is retained,
but with a longer period. Finally, seasonal forcing is introduced to all of the models, demonstrating how our dynamical systems approach aids understanding of model
behaviour and can assist with model formulation.