Edwards A.M. (2006)
Simple models of the plankton ecosystem have been usefully analysed and understood using dynamical-systems techniques. These techniques have addressed important ecological questions and have provided insight into how models should be constructed. Edwards and Brindley (1996, Dynamics and Stability of Systems, 11:347-370) used such methods to investigate the dynamics of a model that represented the concentrations of nutrients, phytoplankton and zooplankton. Halanay (2003, Dynamical Systems, 18:227-229) asserted that Edwards and Brindley incorrectly determined the stability of one of the model's steady states. Halanay's assertion requires the unrealistic consideration of negative zooplankton concentrations, and so, although mathematically correct, it is not relevant to the biological meaning of the model.
Note that the journal's original two attempts at .pdf files contained errors (and did not match the published version), but that this latest version seems okay, especially if you do not select `fit to page' when printing.
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 homoclinic 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.
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The presence of phytoplankton in a body of water affects the penetration of irradiance through the water column. This influences the temperature and hence the density distribution of the water. If the phytoplankton concentration varies horizontally, then the consequent density distribution will result in a horizontal pressure gradient. Here we consider a long band (or strip) of high phytoplankton biomass, flanked on either side by clearer water containing little biomass. By means of a simple model we present calculations of the velocities induced by the pressure gradients, to show under what conditions the differential heating effects may become significant. The model's momentum equations assume a steady state, and include effects of Coriolis and vertical eddy viscosity. An analytical solution is obtained, and the induced velocities are shown graphically. Further calculations investigate the potential for the biologically-induced vertical velocities to transport nutrients into the surface waters and subsequently influence new primary production. This work demonstrates the capacity for feedbacks from the biological component of the ecosystem to the physical component (and back again).
In high-nutrient, low-chlorophyll (HNLC) regions of the ocean, phytoplankton biomass remains low despite an abundance of major nutrients. Platt et al. [2003, Proc. Roy. Soc. A, 459:1063-1073] constructed a simple two-component (chlorophyll and nitrate) model of HNLC regions and used it to determine analytically the upper bound on chlorophyll and the lower bound on nitrate in terms of the bio-optical and physical properties of the system. Subsequently, Platt et al. [2003, Mar. Ecol. Prog. Ser., 254:3-9] showed that the response of HNLC regions to iron addition could be captured through the effect of iron on the parameters of the growth term in the model. Here, we extend this approach to derive a procedure for less-conservative bounds on chlorophyll and nitrate. We also examine the consequences of replacing the linear loss term in the original model with a quadratic loss term. The application of the model is illustrated using parameters typical of the eastern Equatorial Pacific in its unperturbed state and also after enrichment with iron. The results are consistent with observations made during an experimental manipulation of the region by addition of iron (IronEx I). Our work emphasises the value of simple mathematical models as tools to address complex issues in biological oceanography. The method has generality as well as simplicity: it could be applied by non-modellers to investigate other problems in other regions, and to facilitate this we make our computer programs freely available.
We define a new dimensionless number S to be the ratio of nitrogen supply to nitrogen demand of new primary production in the pelagic ecosystem. When S>1, we expect high-nutrient, low-chlorophyll (HNLC) conditions. Using the results of a new model of nitrogen input and consumption for the mixed layer of the ocean, we calculate S for selected oceanic regimes. Those generally accepted to be HNLC are characterised by S>1. The bio-optical terms in this model (specific absorption of pigments, parameters of the light-saturation curve) are known to respond to addition of iron. Using these known responses, we recalculate the expected value of S under hypothetical enrichments of the selected regimes with iron. In each case, the magnitude of S is reduced, but not always below unity. The maximum value of chlorophyll biomass that can be sustained in a given mixed layer may be calculated from consideration of either the bio-optics or the nitrogen supply. The maximum realised biomass will be the smaller of these two estimates.
We develop and analyse a simple, two-compartment (chlorophyll and nitrate) model of the surface mixed layer of the ocean. The mixed-layer depth is modulated intermittently to simulate the effects of storms. The optical properties of the water column are linked to changes in the chlorophyll biomass. The model can be treated analytically. Mathematical bounds are found for the autotrophic biomass and residual nitrate in terms of the intensity and frequency of storms and the bio-optical properties of the phytoplankton. The results are discussed in the context of the high-nutrient, low-chlorophyll regimes where unconsumed nitrate is a persistent occurrence.
Consider a frontal region that has high phytoplankton biomass on one side, and low biomass on the other. Irradiance penetrates deeply through the water column on the low-biomass side, but is attenuated nearer the surface on the biomass-rich side due to absorption by phytoplankton. Thus the near-surface water is heated more on the biomass-rich side than on the clearer side, resulting in lower-density surface water on the biomass-rich side. At greater depths, the situation is reversed, with lower-density water occurring on the biomass-poor side. We model this situation, and examine the resulting perturbations to the frontal circulation. Our aim is to provide an order-of-magnitude estimate of the feedbacks from the biological component of the ecosystem to the current field. The model consists of the steady-state momentum equations, including Coriolis, pressure-gradient and viscous effects. We compute induced vertical velocities of up to 0.2 mm/s, commensurate with field measurements and previous modelling estimates of vertical velocities at fronts. The horizontal along-frontal velocities are of order 2 cm/s or less, and so will not represent a major contribution to the overall flow field; however, such values are certainly not insignificant.
The dynamics of two plankton population models are investigated to examine sensitivities to model complexity and to parameter values. The models simulate concentrations of nutrients, phytoplankton, zooplankton and detritus in the oceanic mixed layer. In Model 1, zooplankton can graze only upon phytoplankton, whereas in Model 2 zooplankton can graze upon phytoplankton and detritus. Both feeding strategies are employed by zooplankton in the ocean, and both are features of models in the literature. Each model here consists of four coupled ordinary differential equations, and can exhibit unforced oscillations (limit cycles) of the four concentrations. By constructing diagrams that show how steady states and oscillations persist as each parameter is varied, a general picture of the dynamics of each model is built up. The addition of the detritus pool to an earlier nutrient-phytoplankton-zooplankton model appears to have little influence on the dynamics when the zooplankton cannot graze upon the detritus (Model 1), but if the zooplankton can graze upon the detritus (Model 2) then the dynamics are affected in a significant way. These results, obtained using the theory of dynamical systems, enhance our knowledge of the factors governing the dynamics of plankton population models.
Low-dimensional plankton models are used to help understand measurements of plankton in the world's oceans. The full dynamics of these models and the effects of varying the functional forms are not completely understood. Moreover, the effects of small-scale physical influences are only recently becoming apparent. In particular, turbulence may play a pivotal role in the strategies adopted by predators of zooplankton, and thus may alter the so-called closure term, which models predation on zooplankton when the predators themselves are not being explicitly simulated. We investigate the use of a closure term with a non-integer exponent, allowing determination of the dynamics as the closure term varies continuously between the commonly-used linear and quadratic forms. We determine which characteristics of the dynamics are generic, in that they occur for any exponent of closure, and which are purely a consequence of the usual integer exponents. A three-way transcritical bifurcation of three steady states is the generic situation, occurring for all except the purely linear closure term. Hopf bifurcations, consequent limit cycles, and chaotic attractors appear to be generic across all exponents of closure. Oscillations, and hence chaos, had been hypothesised to be eliminated with the use of quadratic closure.
(2000)
Zooplankton mortality in plankton population models is often represented by the so-called closure term. Recently, much attention has been paid to the choice of functional form used for the closure term, primarily due to the influential paper by Steele and Henderson (1992; J. Plankton Res., 14, 157-172). Here we reveal an inconsistency in the normalisation of Steele and Henderson's models, and show that unforced short-term oscillations (limit cycles) can occur when a quadratic closure term is used. Furthermore, we contradict the hypothesis regarding the relationship between nutrient steady-state values and the choice of closure term: using the seven-component plankton model of Fasham (1993; pp. 457-504 of The Global Carbon Cycle, ed. M. Heimann) with four alternative closure terms, we find the nutrient value to depend more upon the choice of parameter values than on the choice of closure term. However, our results agree with and strengthen the general conclusion of Steele and Henderson's work - that the choice of closure term can strongly influence the dynamics of models.
We investigate the dynamical behaviour of a simple plankton population model, which explicitly simulates the concentrations of nutrient, phytoplankton and zooplankton in the oceanic mixed layer. The model consists of three coupled ordinary differential equations. We use analytical and numerical techniques, focusing on the existence and nature of steady states and unforced oscillations (limit cycles) of the system. The oscillations arise from Hopf bifurcations, which are traced as each parameter in the model is varied across a realistic range. The resulting bifurcation diagrams are compared with those from our previous work, where zooplankton mortality was simulated by a quadratic function - here we use a linear function, to represent alternative ecological assumptions. Oscillations occur across broader ranges of parameters for the linear mortality function than for the quadratic one, although the two sets of bifurcation diagrams show similar qualitative characteristics. The choice of zooplankton mortality function, or closure term, is an area of current interest in the modelling community, and we relate our results to simulations of other models.
We examine the qualitative behaviour of an NPZ (nutrient - phytoplankton - zooplankton) model for parameter ranges consistent with values used in the literature. The wide range of values partly reflects variations of conditions in different environments for the plankton, but in many cases is a measure of the difficulties in making observations and consequent uncertainties. We pay particular attention to the bifurcational behaviour of the system, and to the regions of parameter space for which oscillatory behaviour is possible; in some regions of parameter space we find that multiple attractors occur. Finally we examine in more detail the behaviour for a range of values of nutrient input.
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