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A rational dynamical-systems approach to plankton population modelling

Edwards, Andrew Michael (1997) A rational dynamical-systems approach to plankton population modelling. PhD thesis, University of Leeds.

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Abstract

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.

Item Type: Thesis (PhD)
Keywords: Zooplankton
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.247891
Depositing User: Ethos Import
Date Deposited: 02 Aug 2018 14:44
Last Modified: 02 Aug 2018 14:44
URI: http://etheses.whiterose.ac.uk/id/eprint/21072

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