Abstracts for published papers only and not fully up-to-date - see here for details of more recent papers, fisheries stock assessments, technical reports and other articles.
1. Mathematical modelling is playing an increasing role in studies of ecological systems. This requires the communication of the details of a mathematical model, including the use of mathematical notation to represent ecological variables, parameters and processes. 2. In our experience, the clarity of mathematical notation varies between papers and can often be inconsistent with general conventions. Poor notation can impede communication and understanding of ideas, and make models appear more complicated than necessary.
3. Here, we present some guidelines, including: (a) define every term in an equation, (b) use fonts appropriately (italicise mathematical symbols, use bold lower case for vectors and bold upper case for matrices), (c) use subscripts appropriately (to index quantities, for example, by year), (d) use superscripts appropriately (to indicate a power, the transpose of a matrix or the steady-state value of a quantity), (e) avoid multiletter variable names, and (f) revisit notation early on in a project to see if it should be refined.
4. Although we focus mainly on ecology, our guidelines should be of interest to researchers applying models in evolutionary biology and broader areas of biology.
1. The size spectrum of an ecological community characterizes how a property, such as abundance or biomass, varies with body size. Size spectra are often used as ecosystem indicators of marine systems. They have been fitted to data from various sources, including groundfish trawl surveys, visual surveys of fish in kelp forests and coral reefs, sediment samples of benthic invertebrates and satellite remote sensing of chlorophyll. 2. Over the past decades, several methods have been used to fit size spectra to data. We document eight such methods, demonstrating their commonalities and differences. Seven methods use linear regression (of which six require binning of data), while the eighth uses maximum likelihood estimation. We test the accuracy of the methods on simulated data.
3. We demonstrate that estimated size-spectrum slopes are not always comparable between the seven regression-based methods because such methods are not estimating the same parameter. We find that four of the eight tested methods can sometimes give reasonably accurate estimates of the exponent of the individual size distribution (which is related to the slope of the size spectrum). However, sensitivity analyses find that maximum likelihood estimation is the only method that is consistently accurate, and the only one that yields reliable confidence intervals for the exponent.
4. We therefore recommend the use of maximum likelihood estimation when fitting size spectra. To facilitate this, we provide documented R code for fitting and plotting results. This should provide consistency in future studies and improve the quality of any resulting advice to ecosystem managers. In particular, the calculation of reliable confidence intervals will allow proper consideration of uncertainty when making management decisions.
Fishing pressure on coral reef ecosystems has been frequently linked to reductions of large fishes and reef fish biomass. Associated impacts on overall community structure are, however, less clear. In size-structured aquatic ecosystems, fishing impacts are commonly quantified using size spectra, which describe the distribution of individual body sizes within a community. We examined the size spectra and biomass of coral reef fish communities at 38 US-affiliated Pacific islands that ranged in human presence from near pristine to human population centers. Size spectra 'steepened' steadily with increasing human population and proximity to market due to a reduction in the relative biomass of large fishes and an increase in the dominance of small fishes. Reef fish biomass was substantially lower on inhabited islands than uninhabited ones, even at inhabited islands with the lowest levels of human presence. We found that on populated islands size spectra exponents decreased (analogous to size spectra steepening) linearly with declining biomass, whereas on uninhabited islands there was no relationship. Size spectra were steeper in regions of low sea surface temperature but were insensitive to variation in other environmental and geomorphic covariates. In contrast, reef fish biomass was highly sensitive to oceanographic conditions, being influenced by both oceanic productivity and sea surface temperature. Our results suggest that community size structure may be a more robust indicator than fish biomass to increasing human presence and that size spectra are reliable indicators of exploitation impacts across regions of different fish community compositions, environmental drivers, and fisheries types. Size-based approaches that link directly to functional properties of fish communities, and are relatively insensitive to abiotic variation across biogeographic regions, offer great potential for developing our understanding of fishing impacts in coral reef ecosystems.
1. Ontogenetic niche shifts are widely prevalent in nature and are important in shaping the structure and dynamics of ecosystems. Stable isotope analysis is a powerful tool to assess these shifts, with δ15N providing a measure of trophic level and δ13C a measure of energy source. 2. Previous applications of stable isotopes to study ontogenetic niche shifts have not considered the appreciable time lag between diet and consumer tissue associated with isotopic turnover. These time lags introduce significant complexity into field studies of ontogenetic niche shifts.
3. Juvenile Chinook salmon (Oncorhynchus tshawytscha) migrate from freshwater to marine ecosystems and shift their diet from feeding primarily on invertebrates to feeding primarily on fish. This dual ontogenetic habitat and diet shift, in addition to the long time lag associated with isotopic turnover, suggests that there is potential for a disconnect between the prey sources that juvenile salmon are consuming, and the inferred prey sources from stable isotopes.
4. We developed a model that considered ontogenetic niche shifts and time lags associated with isotopic turnover, and compared this `ontogeny' model to one that considered only isotopic turnover. We used a Bayesian framework to explicitly account for parameter uncertainty.
5. Data showed overwhelming support for the ontogeny model relative to the isotopic turnover model. Estimated variables from best model fits indicate that the ontogeny model predicts a much greater reliance on fish prey than does the stomach content data. Overall, we found that this method of quantifying ontogenetic niche shifts effectively accounted for both isotopic turnover and ontogenetic diet shifts; a finding that could be widely applicable to a variety of systems.
As the oceans absorb anthropogenic CO2 they become more acidic, a problem termed ocean acidification (OA). Since this increase in CO2 is occurring rapidly, OA may have profound implications for marine ecosystems. In the temperate northeast Pacific, fisheries play key economic and cultural roles and provide significant employment, especially in rural areas. In British Columbia (BC), sport (recreational) fishing generates more income than commercial fishing (including the expanding aquaculture industry). Salmon (fished recreationally and farmed) and Pacific Halibut are responsible for the majority of fishery-related income. This region naturally has relatively acidic (low pH) waters due to ocean circulation, and so may be particularly vulnerable to OA. We have analyzed available data to provide a current description of the marine ecosystem, focusing on vertical distributions of commercially harvested groups in BC in the context of local carbon and pH conditions. We then evaluated the potential impact of OA on this temperate marine system using currently available studies. Our results highlight significant knowledge gaps. Above trophic levels 2-3 (where most local fishery-income is generated), little is known about the direct impact of OA, and more importantly about the combined impact of multi-stressors, like temperature, that are also changing as our climate changes. There is evidence that OA may have indirect negative impacts on finfish through changes at lower trophic levels and in habitats. In particular, OA may lead to increased fish-killing algal blooms that can affect the lucrative salmon aquaculture industry. On the other hand, some species of locally farmed shellfish have been well-studied and exhibit significant negative direct impacts associated with OA, especially at the larval stage. We summarize the direct and indirect impacts of OA on all groups of marine organisms in this region and provide conclusions, ordered by immediacy and certainty.
Lévy flights have gained prominence for analysis of animal movement. In a Lévy flight, step-lengths are drawn from a heavy-tailed distribution such as a power law (PL), and a large number of empirical demonstrations have been published. Others, however, have suggested that animal movement is ill fit by PL distributions or contend a state-switching process better explains apparent Lévy flight movement patterns. We used a mix of direct behavioural observations and GPS tracking to understand step-length patterns in females of two related butterflies. We initially found movement in one species (Euphydryas editha taylori) was best fit by a bounded PL, evidence of a Lévy flight, while the other (Euphydryas phaeton) was best fit by an exponential distribution. Subsequent analyses introduced additional candidate models and used behavioural observations to sort steps based on intraspecific interactions (interactions were rare in E. phaeton but common in E. e. taylori). These analyses showed a mixed-exponential is favoured over the bounded PL for E. e. taylori and that when step-lengths were sorted into states based on the influence of harassing conspecific males, both states were best fit by simple exponential distributions. The direct behavioural observations allowed us to infer the underlying behavioural mechanism is a state-switching process driven by intraspecific interactions rather than a Lévy flight.
Background. Ecologists are collecting extensive data concerning movements of animals in marine ecosystems. Such data need to be analysed with valid statistical methods to yield meaningful conclusions.Principal Findings. We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (Nature 451:1098; Science 332:1551). The first study analysed vertical movement data to conclude that diverse marine predators (Atlantic cod, basking sharks, bigeye tuna, leatherback turtles and Magellanic penguins) exhibited “Lévy-walk-like behaviour”, close to a hypothesised optimal foraging strategy. By reproducing the original results for the bigeye tuna data, we show that the likelihood of tested models was calculated from residuals of regression fits (an incorrect method), rather than from the likelihood equations of the actual probability distributions being tested. This resulted in erroneous Akaike Information Criteria, and the testing of models that do not correspond to valid probability distributions. We demonstrate how this led to overwhelming support for a model that has no biological justification and that is statistically spurious because its probability density function goes negative. Re-analysis of the bigeye tuna data, using standard likelihood methods, overturns the original result and conclusion for that data set. The second study observed Lévy walk movement patterns by mussels. We demonstrate several issues concerning the likelihood calculations (including the aforementioned residuals issue). Re-analysis of the data rejects the original Lévy walk conclusion.
Conclusions. We consequently question the claimed existence of scaling laws of the search behaviour of marine predators and mussels, since such conclusions were reached using incorrect methods. We discourage the suggested potential use of “Lévy-like walks” when modelling consequences of fishing and climate change, and caution that any resulting advice to managers of marine ecosystems would be problematic. For reproducibility and future work we provide R source code for all calculations.
There is an urgent need for indicators that anticipate changes in populations of exploited marine species. We modelled the complex patterns of variability found in fisheries time series that are not detected using classical models. We applied fractal analyses to detect time-invariant scaling symmetries in daily catch time series from the smooth pink shrimp (Pandalus jordani) fishery from the west coast of Vancouver Island, British Columbia, Canada. A universal multifractal model, which accounts for intermittent fluctuations and extreme values in a time series, provided a better fit to daily catches than a monofractal model. Multifractality is an indication of multiple scaling patterns and suggests that more than one process is affecting the variability of catches. Fractal dynamics in catch time series were found for a range of scales between 16 and 120 fishing days. To our knowledge, this is the first time that multifractality has been demonstrated for an invertebrate fishery. The multifractal model has the potential to provide an in-season estimate (up to 120 fishing days) of the variability of shrimp catches based on the variability at time scales less than 1 month. Changes in these fractal patterns may provide an early warning that conditions underlying this fishery are changing.
The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent µ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.
Here we therefore address the open question of whether the previous data still support the Lévy flight hypothesis, and thus determine whether Lévy flights really are so ubiquitous in ecology. We present a comprehensive re-analysis of 17 data sets from seven previous studies for which Lévy flight behaviour had been concluded, covering marine, terrestrial and experimental systems from four continents. We use the modern likelihood and Akaike weights approach to test whether simple alternative models are more supported by the data than Lévy flights.
The previously estimated values of the power-law exponent µ do not match those calculated here using the accurate likelihood approach, and almost all of them lie outside of the likelihood-based 95% confidence intervals. Furthermore, the original power-law Lévy flight model is overwhelmingly rejected for 16 out of the 17 data sets when tested against three other simple models. For one data set, the data are consistent with coming from a bounded power-law distribution (a truncated Lévy flight). For three other data sets, an exponential distribution corresponding to a simple Poisson process is suitable. Thus, Lévy flight movement patterns are not the common phenomena that was once thought, and are not suitable for use as ecosystem indicators for fisheries management, as has been proposed.
2. The exponent that parameterizes the power-law tail, denoted µ, has repeatedly been found to be within the Lévy range of 1 < µ ≤ 3. Here, we use Monte Carlo simulations to show that the methods used to infer the value of µ are inaccurate.
3. The widely used method of simply logarithmically transforming a standard histogram of movement lengths has been shown elsewhere to be problematic. Here, we further demonstrate how poor it is, and show that it actually biases estimates of µ towards the Lévy range of 1 < µ ≤ 3, and can bias estimates towards the value of µ = 2. Thus, previous reports of animals undergoing Lévy flights, or of µ being close to the reported optimal value of µ = 2, may simply be a consequence of the bias generated by this method.
4. A technique that has been recently recommended is to logarithmically bin the data and then normalize the resulting histogram. We show that this technique also produces biased results, and suffers from similar problems as those just outlined, although to a lesser extent.
5. The proposed solution is to use likelihood. We find that calculating the maximum likelihood estimate of µ gives the most accurate results (having also tested the rank/frequency method). Likelihood has the further advantages of being the easiest method to implement, and of yielding accurate confidence intervals. Results are applicable to power-law distributions in general, and so are not restricted to inference of Lévy flights.
6. We also re-analyse a data set of grey seal movements that was originally reported to demonstrate Lévy flight behaviour. Using Akaike weights, we test four models, and find no evidence for Lévy flights. Overall, our results suggest that Lévy flights might not be as common as previously thought.
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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A surprisingly diverse variety of foragers have previously been concluded to exhibit movement patterns known as Lévy flights, which are a special type of random walk. These foragers range in size from microzooplankton in experiments, to fishermen in the Pacific Ocean and the North Sea. The Lévy flight conclusion implies that all the foragers have similar scale-free movement patterns that can be described by a single dimensionless parameter, the exponent µ of a power-law (Pareto) distribution. However, the previous conclusions have been made using methods that have since been shown to be problematic - inaccurate techniques were used to estimate µ, and the power-law distribution was usually assumed to hold without testing any alternative hypotheses.
Weakly density-dependent effects, characterized by fractional scaling exponents close to one, are rarely studied in the ecological literature. Here, we consider the effect of an additional weakly density-dependent term on a simple competition model. Our investigation reveals that weak density-dependence opens up an "invisible niche". This niche does not constitute a new mechanism for coexistence, but is a previously unexplored consequence of known mechanisms. In the invisible niche a weaker competitor can survive at very low density. Coexistence thus requires large habitat size. Such niches, if found in nature, would have a direct impact on species-area laws and species-abundance curves and should therefore receive more attention.
1. Ecologists are obtaining ever-increasing amounts of data concerning animal movement. A movement strategy that has been concluded for a broad variety of animals is that of Lévy flights, which are random walks whose step lengths come from probability distributions with heavy power-law tails.
The study of animal foraging behaviour is of practical ecological
importance, and exemplifies the wider scientific problem of
optimizing search strategies. Lévy flights are random walks, the
step lengths of which come from probability distributions with
heavy power-law tails, such that clusters of short steps are connected
by rare long steps. Lévy flights display fractal properties,
have no typical scale, and occur in physical and chemical
systems. An attempt to demonstrate their existence in a natural
biological system presented evidence that wandering albatrosses
perform Lévy flights when searching for prey on the ocean
surface. This well known finding was followed by similar
inferences about the search strategies of deer and bumblebees.
These pioneering studies have triggered much theoretical work in
physics, as well as empirical ecological
analyses regarding reindeer, microzooplankton, grey seals,
spider monkeys and fishing boats. Here we analyse a new,
high-resolution data set of wandering albatross flights, and find
no evidence for Lévy flight behaviour. Instead we find that flight
times are gamma distributed, with an exponential decay for the
longest flights. We re-analyse the original albatross data using
additional information, and conclude that the extremely long
flights, essential for demonstrating Lévy flight behaviour, were
spurious. Furthermore, we propose a widely applicable method
to test for power-law distributions using likelihood and Akaike
weights. We apply this to the four original deer and bumblebee
data sets, finding that none exhibits evidence of Lévy flights, and
that the original graphical approach is insufficient. Such a graphical
approach has been adopted to conclude Lévy flight movement
for other organisms, and to propose Lévy flight analysis
as a potential real-time ecosystem monitoring tool. Our results
question the strength of the empirical evidence for biological Lévy
flights.
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.
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.
Andrew.Edwards
dfo-mpo.gc.ca
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