Confronting the Complexity of HAB Dynamics

[This is a paper prepared for the LIFEHAB Workshop in Mallorca and was last edited on 25 October 2001. Not to be quoted or cited without permission of the author.]

Traditionally the modelling of Harmful Algal Blooms (HABs) has been viewed as a branch of physical oceanography, and the resultant numerical simulation models have been based on the concept that plankton are passive elements in a complex fluid environment. Adding information on the dynamics of algal life stages makes these models much more complex, but rarely adds to the performance and reliability of the models. We should consider alternate approaches that are more appropriate for complex systems and address their complexity directly, rather than hoping that by putting more and more details into the models we will automatically get better models. In particular, given the scarce and sometimes unreliable data contained in short time series for HABs, rule-based models may provide a good starting point for understanding and modelling these systems.

I am not an expert on modelling HAB dynamics, and successful models tend to be very site specific. This talk addresses some general questions of how to approach modelling, pitfalls to avoid, and possible new approaches that have not been tried before.

There isn’t much that modellers agree on, but one point about HAB dynamics that almost everyone must support is that they are very complex. This is not the usually motherhood statement about the complexity of natural ecosystems that one often hears – in fact, most ecosystems can be modelled quite simply (Silvert 1996) – but rather a reflection of the way that physical, chemical and biological factors interact in ways that are often erratic and unpredictable. Cysts are transported between the sediments and the euphotic zone, some organisms have alternate reproductive strategies and dozens of different life stages, and undetectable traces of rare elements may play a critical role in triggering a bloom. In some cases it may even happen that the associated bacteria are more important than the algae themselves! The frequently poor reproducibility of HABs, reflected in the fact that they often fail to appear under what seem to be the same circumstances in which they appeared previously, makes it hard to identify important causal factors and to build robust predictive models.

A good example of the convoluted nature of the processes which control HABs is the recent proposal by Walsh and Steidinger (in press) that Saharan dust blown across the Atlantic Ocean promotes production on the west Florida shelf by cyanobacteria (Trichodesmium erythraeum), and that the nitrogen these extract from the atmosphere fuels blooms of Gymnodinium breve. This hypothesis, which the editor of Harmful Algae News called an "inspired but quite unbelievable conjecture" (Wyatt 2000), is now working its way into wide acceptance. This example shows that a modelling strategy based on the idea that we know all the relevant processes and need only build suitable models of them simply cannot be comprehensive enough to address the subtleties of complex systems.

In fact our understanding of primary production in general is poor, and HABs introduce the additional complication that sometimes a bloom of a particular alga is harmful (e.g., toxic) and sometimes it is not. Often blooms are harmful only during a particular stage of their development, and the intensity and duration of this stage is difficult to model and predict.

This is not to say that all HAB models confront nearly impossible challenges – some HABs are routine events, like the return of the swallows to Capistrano, but this is not common. One may have apparently identical conditions season after season, in which the only notable difference is that sometimes there is a major HAB, sometimes a minor HAB, and sometimes nothing at all happens (this is usually the case when you have the funding and other resources to carry out an extensive monitoring program, with the usual result that there is nothing to monitor!).

There are several different ways to approach the modelling of complex systems (Silvert 1981a, 2001b). The most common and widely accepted method is to:

This approach is highly reductionistic and is based on the idea that a model is the sum of submodels that represent all the various processes going on in the system. To build such a model you simply model everything that you can think of, put all the parts together, and thereby construct a massive simulation model.

Such models are popular for various reasons:

1. They are straightforward to construct. The challenging part of modelling is knowing what to include and what to leave out, and if you include everything, no judgement is required. Building such a model is like making a pile of bricks – it is hard work, but so simple that anyone can do it.
2. They are relatively immune to criticism. Because everything that the modeller knows about is included, the modeller cannot be accused of leaving anything out. Because they are so massive, sometimes with thousands of equations, no one can understand them, so it is hard to find fault.
3. They attract funding. Large models that require vast amounts of computing power are a safe bet for funding agencies, and they provide good public relations for supercomputer centres that spend most of their time developing missile shields and nuclear reactors.
4. They are easy to publish. Referees are reluctant to admit that they do not understand a paper, so unless they can find a concrete flaw (usually something from their own speciality that got left out), they just shrug and approve.

There is in fact only one major flaw with these massive reductionistic models:

1. They usually do not work.

It is an unfortunate fact that most really big models do not produce realistic results (Lee 1973). Of course the usual explanation is, "We must have left something out", but numerous studies have indicated that as models of this sort get bigger and bigger with more submodels and other details, they do not generally converge on an accurate solution.

This failure of many large models is somewhat surprising, since in other areas, notably engineering, they work quite well. The modelling literature contains many discussions of the model size issue, but one factor stands out – in engineering and other fields where large models are used successfully, the underlying scientific principles are usually well known and can be specified very accurately. The classic example of this is in calculation of the trajectories of spacecraft, a tremendously complicated problem for which NASA and other space agencies use massive banks of supercomputers, but one which is based on Newton’s laws, simply F=ma. When addressing problems in which the exact relationships are not well known, the large number of interconnections between the parts of a massive model simply multiplies the individual uncertainties and leads to results that are often meaningless.

Furthermore, even in situations where the laws governing the system are well-known, reductionistic models do not always provide correct results, especially when the system exhibits emergent behaviour. The classic example, as pointed out by Loschmidt (1876) and many others, is the irreversible behaviour of the classical gas, which is inexplicable since Newton’s laws of motion are perfectly reversible in time and therefore any reductionistic calculation could not produce an irreversible result (the phrase "Arrow of Time" is often used in discussing this paradox, which has preoccupied physicists for more than a century).

Another problem with large models is that all modellers start from a personal background with a point of view determined by their educational perception of the system being modelled. If they don’t understand what they are modelling, they may make a real mess of it, or, at best, leave out something that is important.

There have been numerous cases, in ecology and other fields, where a mathematically gifted modeller made a complete mess of a model because he misunderstood how the system worked (c.f. Cuff 1983). Modelling is a bit like language translation, in that it involves translating scientific knowledge into mathematics, and it only works if the modeller understands the science, or the scientists understand the mathematics – imagine what would happen if a speech in Spanish by someone who didn’t know German were translated into German by someone who didn’t know Spanish – the result might be perfectly grammatical German, but it would have very little relationship to what was originally said in Spanish.

The same problem often arises in ecological modelling, and in particular in the modelling of HABs. Most modellers in this field are physical oceanographers, and while the physical aspects of their models are generally good, the chemical and biological components tend to be oversimplified. For example, while nutrient budgets for water bodies are often correct, the internal chemical processes, such as remineralisation of nitrogen compounds, are often neglected. Biological considerations are often – I might say "usually" – poorly modelled. One of the most common aspects of such models is that they treat plankton as passive tracers that are transported by water masses like dye patches, when in fact there are several processes that can cause plankton to move relative to the water. Perhaps the most important of these is buoyancy, and the way that phytoplankton buoyancy changes with condition (some models include a constant sinking rate, but I do not know of any in which the sinking rate varies realistically). Also of course, planktonic organisms – including some phytoplankters as well as zooplankters – can swim. While the speed of their motion, whether driven by buoyancy or by flagella, is slow, the vertical motion of plankton in stratified flow fields can have a significant impact on their transport, and this may explain why some physical models of primary production produce very unrealistic results.

This is not to say that physical models are not useful – so long as they deal with biological considerations adequately they play an important role in our understanding of HABs. Even models with no biology can be valuable if, for example, they identify gyres and other regions where plankton are likely to be retained, or if they tell us about stratification and vertical transport.

Nor would it be fair to single out physical oceanographers as the only culprits committing crimes of omission in this matter of including all relevant factors. Biologists also tend to ignore important factors – for example, phytoplanktologists often seem oblivious to the effects of grazing and nutrient recycling by zooplankton, and the possibility that at different times and different conditions one might have both top-down (i.e., grazing) and bottom-up (i.e., nutrients) control of primary production is rarely countenanced by biological specialists. The unfortunate fact is that we all have our blind spots, and we must take this human failing into account when we design our models.

An alternative to the ground-up development of models based on what we know (or assume) about the dynamical structure of complex systems is to start from the top down and build models that behave in the way that is actually observed, rather than first building the model and then hoping that it generates reasonable behaviour. I originally referred to these models as "top-down", but because of the confusion this caused with models of top-down control of ecosystems, I now prefer the term "data-driven" (similarly one should not confuse ground-up models with modèles farcis or gehackte Modellen).

An important distinction should be drawn between data-driven modelling and statistical curve-fitting, although the two sound very similar. The goal of the statistician is just to fit the data, without concern for the structure of the model – that is why polynomial models are often used, even though nothing in nature generates polynomials. Data-driven modelling seeks to uncover the structure of the system by tracing its behaviour back to the processes that generate that behaviour. This distinction is particularly important in the study of HAB dynamics, since statistical models are generally only useful for interpolation – they can describe events that have happened before, or that resemble ones that have occurred in the past, but once you go beyond the historical record you have no way of knowing what will happen. Many HAB occurrences are extreme events, going well beyond what has been recorded in the past, and we need to understand the underlying dynamics in order to understand these processes. Furthermore, there are seldom sufficient data to fit a statistical model, given the multivariate complexity of algal blooms and the relatively short and incomplete time series that are usually available.

I do not mean to downplay the importance of statistical analysis, although in data-driven modelling it serves more as an exploratory tool than as a way to build working models. Some extensions of statistical modelling, such as path analysis (Li 1975), can also provide powerful tools for recognising patterns in complex data sets. An interesting example of how statistics can contribute to data-driven modelling comes from a study of mosquito population dynamics with a simulation model (Miller 1974). There was much more variability in the actual populations than could be accounted for by the model. On examining the data the author observed that the length of the larval stage was very sensitive to rainfall conditions at the time, and while he was not able to identify a mechanism that could be incorporated into the simulations, it was clear from his top-down analysis that this factor was more important than the processes that he had originally built into the model. It would not be surprising if something like this happened with HABs.

It is worth pointing out that even with systems that we understand well, careful examination of the data may show us that something is happening that we didn’t believe was possible (Silvert 1981b), and may even demonstrate that the behaviour of the system is governed by processes quite different form what was expected (Vandermeulen et al. 1983). The more complex the system, the greater the likelihood of this happening, and yet we tend to apply reductionistic reasoning to just these kinds of systems.

There are many ways of building data-driven models, but many of these suffer from the same excessive need for data that statistical models do. "Inverse Modelling" (Silvert and Cembella 1999) refers to the process of building models of known structure, but fitting the parameters of the model to the data rather than estimating the parameters and looking at the output (fitting a statistical regression is actually a form of inverse modelling, although the expression is more commonly used for systems of differential equations). "System Identification" is another approach for developing time-series models from empirical time series. I am unaware of any HAB models that were developed using these techniques, and I have not seen many data sets that are large enough to support such approaches.

Rule-based models are quite different, and although they lead to very different types of models, they have proved very effective in developing understanding of complex systems. They have a charming simplicity that belies their power, and because they avoid many of the hidden assumptions of statistical and other more explicitly quantitative techniques, they actually have far greater flexibility.

Consider for example a common problem in applied biology, determining the yield of a fish farm as a function of water temperature. If we want to apply statistical techniques we might begin with a linear regression of fish yield on mean water temperature over the growing season. A more sophisticated approach might be to construct a moving average model to calculate growth throughout the year. But all of these modelling approaches are trumped by the basic rule that if the water temperature falls below –2° for more than a couple of minutes, the fish all die (the precise critical temperature depends on the species of fish). Statistical models can therefore give disastrously wrong results if not constructed with good insight, while very simple rule-based models let us focus on the most important aspects of the system, and makes up for their lack of quantitative precision by drawing attention to the most critical qualitative features.

The relevance of rule-based modelling to HAB dynamics can best be understood by skipping the formal papers at scientific conferences and listening to the conversations during the coffee breaks. Often you will hear a scientist who has made no claim to understand the dynamics of blooms say that this should be a big year for a certain type of harmful alga because of this and that and something else that he observed. The problem in relating this to modelling is that to many scientists it just doesn’t look like a model – it may have predictive value, but without a veneer of mathematical objectivity it doesn’t count.

The major problem with rule-based systems, and the probable reason why they have made minimal inroads in ecological modelling, is that it is difficult to state the rules in the precise way that scientists expect. The earliest successful applications of rule-based models were to discrete systems, where a crisply defined rule led to clearly defined outcomes. One of the earliest uses was in the design of computer systems, where the configuration choices could be determined by nice clear rules. In modern terms, we might use rules like this to configure a personal computer:

IF the client wants to be able to play music on his computer

THEN include a CD drive.

Unfortunately, this kind of rule does not help much in understanding how ecosystems function.

A better example of how a rule-based model can be developed, and one with clear application to modelling HAB dynamics, is in the area of medical diagnostics. Physicians also deal with complex and poorly understood systems, they almost never have adequate data with long time series on the objects they study (imagine if your physician knew your blood pressure and heart rate for the past five years!), and yet they have to interpret a small set of observations to make life and death decisions. Consider the kinds of models that a physician uses, and compare them with those that might be used in studying HABs, as illustrated by Table 1:

Table 1. Rule-Based Prediction.

(Note how cleverly I avoided a charge of practising either medicine or planktology without a license!)

There is an evident parallel here, but there is also an evident problem – what do the words "high" and "low" mean? The vagueness of these terms has somehow been accepted by the medical profession, but it has been a stumbling block to the adoption of rule-based systems in other fields of science (and of course there are those who argue that medical diagnostics is an art, not a science!). It is only in the past few decades that we have developed the mathematical tools to make this very common-sense approach have the appearance of rigour that scientists demand.

These tools arise from fuzzy set theory, a well-known but much deprecated branch of mathematics that has been around for about 35 years (Silvert 1997, 2000, 2001a). The strength of fuzzy set theory is that it brings mathematical reasoning into correspondence with common sense, while its weakness is that it sounds too much like common sense and not enough like mathematics. It is in fact very simple, so simple that we tend to say "it is too simple to be right" rather than "oh, the right answer is simpler than I thought". When we need to define high and low body temperatures, the classic method is to draw a line, and say that temperatures over, say, 40° are high, and temperatures below 40° are not high. In mathematical terms, this means that the set of high temperatures includes all temperatures over 40°. This is rigorous but unsatisfactory, since it implies no difference between 37° and 39.9°, or between 40.1° and 43°, but a big difference between 39.9° and 40.1°. In fuzzy set theory we define the set of high temperatures to be a fuzzy set with the "membership" like that shown in Figure 1, which can be zero for temperatures below 37°, one for temperatures above 43°, and with linear interpolation between these two points. This can be interpreted as meaning that temperatures over 43° are definitely high, those under 37° are definitely not high, and values in between are represented by a fractional value, or percentage – for example, 40° is just 50% high, and temperatures close to 40°, such as 39.9° and 40.1°, are close to 50% instead of being drastically different in classification (they would have values 47% and 53% high respectively).

Figure 1. Fuzzy Membership for High Body Temperature

While this may seem like a pointless attempt to add mathematical complications to matters of common sense, the use of fuzzy set theory to classify qualitative observations can facilitate the understanding and modelling of complex phenomena. One of the primary purposes of trying to model HAB dynamics is to be able to predict HABs, either by predicting the probability of a HAB, or, with greater sophistication, to be able to predict the probability distribution for HABs of different levels of harmfulness (e.g., cell concentrations and levels of toxicity). Consider a fuzzy extension of the medical diagnostic procedure shown in Table 1 of the form shown in Table 2, which deals with marginal cases. This obviously can be extended to the problem of HAB prediction, where we also usually lack clear-cut guidelines to tell us whether there will be a bloom.

Table 2. Fuzzy Rules for Medical Diagnosis.

While it may seem that this is not an efficient way to deal with the dynamics of complex systems, we should not overlook the important, although paradoxical, principle that simplicity is often the best way to deal with complexity. The diagnostic rule expressed in Table 2 actually has more power than a more rigorously quantitative rule, since it leaves the physician with enough flexibility to incorporate factors that are likely to be overlooked in a more elaborate model. For example, age is an important consideration in medical diagnostics, and since children’s temperatures tend to fluctuate more than that of adults, a temperature of 40° could have a different significance for a 5-year-old than for a 50-year-old patient (that is, the membership of 40° in the set of high temperatures could depend on age). The simpler the formulation of the model, the easier it is to incorporate such factors.

Consider how a system of fuzzy rules might be developed for HAB prediction. To make things very simple, consider just the single rule, "A warm spring is usually followed by a HAB." (This rule was actually proposed to me by a well-known HAB specialist who was shocked that I then referred to this as a model!) While most of us feel comfortable making statements like "This spring was warm" or "Last fall was cold", translating these statements into the objective language of science is difficult. The problem is not only one of defining "warm", which for a single temperature measurement can be addressed by defining a membership function analogous to the one shown in Figure 1, but also involves deciding what aspect of the spring time series is relevant – is it the mean temperature, the peak temperature, or what? And what is spring? If we define spring to be the three-month period between the vernal equinox and the summer solstice, do we ignore a spell of warm temperatures in February and early March? Not only is "warm" a fuzzy concept, but so is "spring", and unless we are careful to use a mathematical language that reflects our best scientific insights into the way the system works, we can easily get caught up in misleadingly formal and rigorous approaches that have little correspondence to reality.

This is not to say that a more typical type of mathematical model might not be equally good in making predictions in some cases. It may well be that a detailed simulation model could give the same results, or even better results, and could accurately predict HABs based on hourly time series of air temperature and other factors. This would obviate the need for fuzzy rule-based models, just as microbiological analysis of sputum samples can be more conclusive than rule-based medical diagnostics, but reliable models are not always available.

In the long run, our objective should be to use the results of rule-based models to develop the best possible models, and they can help by identifying patterns which should appear in the output of these models. Our understanding of how HABs develop should not stop with identification of simple patterns, such as a correlation with warm spring temperatures – we need to ask why these patterns arise. Is it stratification? Is it the chemical change caused by a non-harmful spring bloom? Does it have anything to do with grazing? The important aspect of any type of modelling is that we use all the information that we have, and use it effectively, which means that we should be prepared to apply any analytical tools that might lead to better understanding.

Developing a model based on rules, especially fuzzy rules, is very different from taking a mountain of data and dumping it on the desk of a mathematician who will convert it into differential equations to be solved on a supercomputer. The essence of this type of modelling is that it is based not only on the data that scientists collect, but also on their understanding, interpretation, and expertise. It is based on the philosophy that the skill of good scientists consists not only in their ability to make good measurements, but also on their ability to understand what they are measuring and to interpret their data. This is actually the core of what modelling is, and the translation of this understanding into systems of equations is only a small part of the modelling process.

The most obvious consequence of this point of view is that modelling cannot be separated from measurement, and if the modeller is not the same person as the one who makes the measurements, at least the two individuals must interact closely and continuously. The "modeller" – and by this I mean the person who codifies the information and writes it down in mathematical form – must understand the work of the scientist who actually makes the measurements and collects the data, and whose understanding of the system is really the model that we want to capture. This means that the modeller must be involved from the beginning of the research program and should be part of the team, rather than a separate specialist working independently. While a modeller who works independently may not always produce bad results, the risk of doing so is substantial (Silvert 2001b).

Fortunately one of the great advantages of building rule-based models is that the development process is incremental, and permits ongoing interaction as the parallel processes of empirical observation and model-building proceed. Rules are first introduced as hypotheses, to be tested against both existing and new data, and exceptions to the rules can be discussed and analysed in different ways – sometimes exceptions invalidate a rule, sometimes they suggest additional rules that are needed as qualifiers. This is similar to what sometimes happens with more typical mathematical models, but can be more productive. For example, we may have a model – either a rule or a simulation – that predicts a bloom, but one year the predicted bloom does not appear. In the subsequent discussion someone usually comes up with an explanation, such as an unusual storm event or some rare biological phenomenon. For a simulation model it may be difficult and impractical to include such a rare event, but it is easy to add a rule to cover this and any other type of special case.

It is worth keeping the parallels to medical diagnostics in mind, both because of the extensive research that has gone into building rule-based medical expert systems and because of the familiarity that we all have with the diagnostic process. The "modellers" who develop these models work closely with medical teams, and they are well aware that none of their models (i.e. rules) work perfectly – there are always exceptions. This makes validation in the traditional statistical sense difficult, or perhaps even impossible, and yet we increasingly trust our lives to medical judgement that is based on fuzzy rules.

Conclusion

Although it seems intuitively clear that complex systems require complex models, in practice this is not always the case. In fact, the more details a model contains, the more likely it seems to be that the results of the model will turn out to be affected by details that were omitted, and the performance record of complex multi-disciplinary models is not good.

The field of modelling of complex systems has drifted away from the classic engineering approach where a computer programmer simply pumps vast quantities of differential equations into a powerful computer. There is increasing awareness of the importance of simply sitting down and looking at the system, looking for patterns, and trying to understand what is happening in qualitative terms. As J. Maynard Smith (1974) puts it, "Ecology is still a branch of science in which it is usually better to rely on the judgement of an experienced practitioner than on the predictions of a theorist." I think that this is an approach that we should apply to modelling HABs.

I have described just a few of the tools for doing this, rule-based and fuzzy-rule-based models, but there are many approaches to the problem of organising subjective and even intuitive insights into a useful understanding of how complex systems work.. In fact a whole field of "knowledge engineering" has grown up around this type of analysis, using tools such as Delphi analysis to "extract" information from the minds of experts who might not be prepared to put their ideas into equation form, and to translate this information into models known as "expert systems". Clearly one of the advantages of this type of approach is that it bridges the gap between people with understanding and people with mathematical expertise, a gap which is often manifest at meetings devoted to HABs and other complex phenomena.

So I haven’t presented my own HAB models here, and I haven’t offered a critical review of anyone else’s models. What I have tried to do is present an alternative way of attacking the problem, by developing rule-based models to enhance our understanding of these systems, in order to attack the problem in a different way.

It is not my intention to suggest that only rule-based models can be applied to HAB phenomena, and certainly there is no question that detailed simulations based on reductionistic fine-scale models of physical phenomena, going back at least as far as the early work of Gordon Riley (1946), have told us a great deal about primary production and the development of HABs in particular. The key issue is not whether any particular modelling approach has merit, but whether there is an approach so powerful and general that it is the only one we need – and the answer I suggest is clearly "NO".

References

Most of the ideas expressed in this paper are my own personal views, and I have not sought to relate them to references in the standard literature. The following references are provided more for background than for anything else, as they include some elaboration of the ideas on which this paper is based. Annotations are included where appropriate.

Cuff, W. R. 1983. "An evaluation of the Port Hacking Project from the Viewpoint of Applied Science Synthesis and modelling of Intermittent Estuaries. " In: Synthesis and Modelling of Intermittent Estuaries, edited by W. R. Cuff and M. Tomczak Jr., Springer, Berlin, pp. 273-292. [A real horror story about what happened when an ecosystem was modelled by someone with no ecological background but good credentials as a pharmacological modeller]

Lee, D. B. Jr. 1973. "Requiem for large-scale models," Journal of the American Institute of Planners, 39: 163-178. [Best known for the insightful observation that "bigger computers simply permit bigger mistakes"]

Li, C. C. 1975. Path Analysis – a primer. Boxwood Press, California.

Loschmidt, J. 1876. Uber den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft, 1. Teil, Sitzungsber. Kais. Akad. Wiss. Wien Math. Naturwiss. Classe 73: 128­142. [An early criticism of Boltzmann’s theory of irreversible thermodynamics]

Maynard Smith, J. 1974. Models in Ecology. Cambridge University Press. p. xi.

Miller, D. R. 1974. Sensitivity analysis and validation of simulation models. J. Theor. Biol. 48: 345-360.

Riley, G.A. 1946. Factors controlling phytoplankton populations on Georges Bank. J. Mar. Res. 6: 54-73. [A pioneering paper in the use of physical theory to study primary production]

Silvert, William. 1981a. Principles of ecosystem modelling. In: Analysis of Marine Ecosystems, edited by A. R. Longhurst, Academic Press, London, pp. 651-676.

Silvert, William. 1981b. Top-down modelling in marine ecology. In: Progress in Ecological Engineering and Management by Mathematical Modelling, edited by D. M. Dubois, Editions Cebedoc, Liege, pp. 259-270. [Includes an example of how an important factor in an experiment, bacterial contamination, was inferred from data-driven modelling long before it was confirmed experimentally (and reluctantly)]

Silvert, William. 1996. Complexity. J. Biol. Syst. 4: 585-591. [How complex do models of complex systems have to be?]

Silvert, William. 1997. Ecological impact classification with fuzzy sets. Ecological Modelling 96: 1-10.

Silvert, W. L., and A. D. Cembella. 1999. Inverse modelling of toxic algal bloom dynamics and cell toxicity by back-calculation from shellfish toxicity. In: Proc. 6th Can. Workshop on Harmful Marine Algae, St. Andrew's, N. B., May 1998, edited by J. L. Martin and K. Haya. Can. Tech. Rep. Fish. Aquat. Sci. 2261: 44-51.

Silvert, William. 2000. Fuzzy indices of environmental conditions. Ecological Modelling 130: 111-119.

Silvert, William. 2001a. Fuzzy aspects of system science. In: Integrative Systems Approaches to Natural and Social Dynamics : Systems Science 2000, edited by M. Matthies, H. Malchow and J. Kriz, Springer-Verlag, Berlin, p. 73-81.

Silvert, William. 2001b. Modelling as a discipline. Int. J. General Systems 30: 261-282.

Vandermeulen, J. H., W. Silvert, and A. Foda. 1983. Sublethal hydrocarbon toxicity in the marine unicellular alga Pavlova lutheri Droop. Aquatic Toxicology 4: 31-49. [In this paper a model based on standard toxicological principles simply could not fit the data, but a very simple model with surprising implications could]

Walsh, J. J. , and K. A. Steidinger. Saharan dust and Florida red tides: the cyanophyte connection. J. Geophys. Res. (in press). Presented at Ninth International Conference on Harmful Algal Blooms, Tasmania, Australia, Book of conference abstracts and participants, p.66.

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