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Product Details
| ISBN-13: | 9780521831321 |
|---|---|
| Publisher: | Cambridge University Press |
| Publication date: | 06/28/2007 |
| Pages: | 242 |
| Product dimensions: | 6.30(w) x 9.25(h) x 0.79(d) |
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9780521831321 - MODELLING FOR FIELD BIOLOGISTS AND OTHER INTERESTING PEOPLE - by Hanna Kokko
Excerpt
1
Modelling philosophy
where we get momentarily lost in a forest, but emerge intact
Figure 1.1 shows three different kinds of model. A supermodel like Naomi Campbell presents, to some of us at least, an ‘idealized’ concept of a human being (Fig. 1.1a). The miniature model shown in Fig. 1.1b was built by my uncle to show what his home town Kuopio looked like in the 1930s. Finally, Fig. 1.1c is a mathematical description of the dynamics of a two-species system of a predator and a prey species.
The models all look very different. They also differ a lot in how scary they look to the average behavioural or evolutionary ecologist – most will have to resist the temptation to close their eyes when encountering Fig. 1.1c, together with its equations, much more than when looking at Fig. 1.1a. My aim in this chapter is to show that there is indeed a reason why the word ‘model’ is used to describe all these figures, to rectify some common misconceptions about models (especially the mathematical ones), and to make life a little less scary for those who know they should be more familiar with modelling in behavioural or evolutionary ecology than they currently are.
Ecology is defined as a science that investigates theabundance and distribution of organisms. This may first sound a little boring, but it gets more interesting once one notices that interactions between organisms play a crucial role here. This means that ecology must study the causalities that underlie the changes in individual numbers, rather than merely providing us with simplistic methodologies of bookkeeping. When trying to understanding those causalities, evolutionary aspects must be taken into account, because evolution underlies everything that organisms do. Very often, the interactions manifest themselves in the behaviour of the
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Fig. 1.1 Models. (a) Naomi Campbell; (b) a street corner from a miniature model of Kuopio, a town in Eastern Finland; (c) hare–lynx cycle described by Lotka–Volterra equations dV/dt = rV − aVP and dP/dt = –qP + baVP. Here V is hare density; P is lynx density; t marks time such that dV/dt gives the rate of change in the hare density and dP/dt similarly for lynx; r, a, b and q are parameters that determine how quickly hares reproduce (r), how many hares are eaten (a) for a given density of lynx and hare, how efficiently eaten hares are converted into lynx offspring (b) and how quickly lynx die when food is in short supply (q). The limit cycle is drawn using values r = 0.1, a = 0.2, b = 0.1, and q = 0.1. The predator–prey cycle shows qualitatively nice cycling, but the model is meant to be conceptual only: parameters are hardly realistic when at the peak of lynx density there are only twice as many hares as there are lynx in the forest. The Lotka–Volterra cycle can be found in almost all textbooks on ecology. I have used the notation of Odenbaugh (2005), who uses the predator–prey cycle to illustrate several important philosophical issues on the need to simplify when modelling ecological phenomena.
organism in question. Behavioural and evolutionary ecology are the sciences that study the evolutionary and ecological causalities that cause individuals to behave the way they do – and while saying this, one should not forget the physiological and physical mechanisms that shape and constrain the ways in which individuals can behave.
This definition places behavioural and evolutionary ecology in a scarily complex web of interactions. It appears that everything interacts with everything else: a migrating bird has to combat weather and winds, manage its energy reserves, find the flight speed that is appropriate for the wing shape the bird has, find its way using perhaps several different orientation mechanisms, avoid predators on the way, find good stopover sites, arrive in a sensible time of the year and compete with conspecifics for breeding localities and possibly mates too. And, all of this is governed by genes that influence the bird’s behaviour in a multitude of ways. Why should we ever be interested in modelling such a system – or even if we were, what hope do we have of ever capturing the complexity of the situation the bird finds itself in?
The quick answer is that there is no hope. Our model simply will never be able to deal with such complexity. The perhaps more surprising elaboration of this statement is that if a model did capture all of this – perhaps computers in the future could stomach it all? – the outcome would not be desirable at all. Why? The famed author J. A. Suarez Miranda said it all in his book Travels of Praiseworthy Men, already in 1658:
… In that Empire, the craft of Cartography attained such Perfection that the Map of a Single province covered the space of an entire City, and the Map of the Empire itself an entire Province. In the course of Time, these Extensive maps were found somehow wanting, and so the College of Cartographers evolved a Map of the Empire that was of the same Scale as the Empire and that coincided with it point for point. Less attentive to the Study of Cartography, succeeding Generations came to judge a map of such Magnitude cumbersome, and, not without Irreverence, they abandoned it to the Rigours of sun and Rain. In the western Deserts, tattered Fragments of the Map are still to be found, Sheltering an occasional Beast or beggar; in the whole Nation, no other relic is left of the Discipline of Geography.
If this passage makes you want to find the rest of Suarez Miranda’s book in a library (instead of learning more about modelling), prepare to be disappointed. The book or its author never existed except in the above quote written by Jorge Luis Borges and Adolfo Bioy Casares (see Borges
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Fig. 1.2 A map is a model too. How would you plan your route in this landscape?
1975). Nevertheless, it marks a perfect beginning for a tour of the virtual world of modelling.
Why do we make maps of landscapes? These exist at a range of different scales, and the detail included may also vary irrespective of scale: a mining company probably requires different information about an area than a passing tourist or a hiker. Using the map in Fig. 1.2, a hiker is probably quite keen to have a look at the contour lines: they tell her the easiest route through a mountain pass and predict the direction of the flow of the river she will encounter. This can be important for a variety of reasons, including orientation purposes. Now, to predict the flow of the river, it is not necessary to mark every tuft of grass that borders the river, or every tree in the forest our hiker is passing through, on the map. These would only make it more difficult to grasp the overall shape of the landscape. Taken to the extreme, if the ultimate goal was to include all the detail of the forest in the map, our poor hiker would have to carry the whole landscape with her. Even if this was physically possible, the gain from doing so would be zero: if lost in a forest, she would not become any wiser by looking at a too vastly detailed map than by staring at the original forest. Such maps indeed deserve to be left rotting in the rain.
This, of course, is exactly the reason why models are ‘not real’. They are idealized versions of the real world. My uncle is now building a street with shops in his miniature model of Kuopio. He knows that to create the visual impression one would get by looking down a particular street in Kuopio, it is important to shape the fonts in shop windows to match the original ones, and he has made countless visits to the local library’s archives to verify all this detail. However, it is far less important to use exactly the same chemical composition of the paint. This is the art of modelling: to know which aspects of reality one can sacrifice, and which ones are crucial to retain. Any medical simulation of drug concentrations in various tissues of the human body is likely to include the fact that the blood vessels from the gut lead to the liver, since that organ rids us of strange compounds in the bloodstream. The whole machinery involved is never included. For example, the developmental fact that the DNA in of our cells contain instructions on how to build the blood vessels is simply not relevant – until, perhaps, we find patients who have a genetic disorder that disrupts this process and makes them react to medicines in surprising ways. But even then, we should not go to the other extreme and start including every possible gene action in the model, from eye colour determination to how fast one’s toenails grow.
George Box, a well-known statistician, once quipped ‘All models are false, but some models are useful’. Have a look again at Fig. 1.1c. Here, V might denote the numbers of hares in an area, and P are lynx. The equations may look scary, but all they do is to describe some numerical rules of species interaction: dV/dt is the change of hare numbers over time, and this is larger if there are lots of hares (large V ) – since many hares can produce more leverets (baby hares) than few hares can. Likewise, lots of lynx can produce lots of new lynx. How lynx and hare populations respond to each other, however, differs between the species. Lynx populations tend to increase when hares are abundant, but hare populations respond negatively to the abundance of lynx, for rather obvious reasons. This is reflected in a negative sign in front of P in the hare equation, and in a positive sign in front of V in the lynx equation. The squiggles in Fig. 1.1c are called Lotka–Volterra equations, and they predict – when solved – that lynx and hare numbers will cycle up and down, hares always declining before their predators do.
Of course, the model is incomplete. It is wrong. It does not include density-dependent responses in the absence of predation: even if hares bred like, well, rabbits, neither species can in reality fill every square metre of the world with their offspring – food would run out long before that. Also, the model does not include the fact that both hares and lynx need some time before they mature. Or that both species reproduce sexually. Or that hares could possess antipredator responses. Or that lynx might use different hunting techniques in the summer and the winter. And so on ad infinitum. All these points are true, but the important point is: would one have predicted that the simple statements “hares beget hares; lynx beget lynx; and lynx eat hares” can, by themselves, lead to predator–prey cycles? Perhaps some people’s intuition tells them immediately that this is the case. But perhaps someone else disagrees and argues that some other process, perhaps some cyclic weather pattern, must interfere before cycles can occur. How could we ever solve the case, without writing the matter down in equations? By carefully looking at patterns in nature? Not easy: we simply cannot find idealized hare and lynx populations that would not have interfering factors of all sorts. By conducting clever experiments? Yes, this definitely plays a large part: experiments very typically try to strip down the messy nature of reality to get at the pure effect of a particular biological factor. Nevertheless, there is always the chance that some other factor that we did not think about at all was influencing our results. In the end, the only way to find out if an argument ‘from X follows Y’ is a valid statement to offer, one has to put it in a mathematical form.
The analogy with an experimenter’s work is very important here. Experiments could also be (and often indeed are) criticized for being unrealistic, as they do not fully reflect the natural setting. For example, a subordinate male fish might prefer large females over small ones in a simultaneous choice test, yet in nature this could be irrelevant if large fecund females tend to be paired to dominant males who defend their territory (and mate) vigorously. The defence of the experimenter to this challenge is similar to that of the modeller, too: the whole point of the experiment is to tell us that, all other things being equal, the large female is preferred over the small one. Likewise, the hare–lynx dynamics assumes that, all other things being equal, more lynx are bad news to hare populations, and this can run cyclic dynamics.
But there is an important distinction to be made between experimental approaches and modelling studies. An experimentalist is trying to find out if something (say, a male preference for large females) really does occur in nature. A modeller will never achieve this, which is good news to field ecologists were they scared of modellers taking over the scientific world. Models do not investigate nature. Instead, they investigate the validity of our own thinking, i.e. whether the logic behind an argument is correct. Are you interested in finding out if the simple fact that predators eat prey can in principle lead to population cycles? If so, go and model it. But whether this fact actually does cause cycles in real populations in Canada or Scandinavia or the vicinity of Kuopio must be investigated out there in nature – perhaps by removing predators from an area, or seeing if cycles are more often seen where the assumptions of this model appear to be better met, or by some other clever methods that the author of this book is not an expert on.
Once one begins to think of models as ‘thinking aids’ rather than investigations of natural phenomena, one could even go as far as to say that we only need models because our brains suffer from too many limitations and are not able to consider all sides of a complicated argument in a balanced way. Take, for example, R. A. Fisher’s (1930) idea that sexually selected traits such as extravagantly long tails in birds exist because the following has happened. Initially a longer than average tail could have conferred a viability benefit to the male, perhaps because it improves flight performance. This means that females who prefer long-tailed males as mates will have offspring who have inherited a nice long tail. The genes for female preferences have now become statistically linked with longer tails in males, and long-tailed male offspring now have the additional advantage of being preferred by females. So, even if too long tails no longer give the viability benefit but instead diminish male offspring survival, the system may end up in a ‘runaway’ where both preferences and tail lengths evolve to ever more extreme values … Is it all now crystal clear in your head?
Don’t worry, the confused feeling is shared by countless others who have read Fisher’s account of the process. If only we all had brains capable of mathematically accurate split-second imagination, keeping precise track of all the relevant pros and cons of these genes – then Fisher would just have had to state his idea, and every person in the world would instantly have seen his point. Perhaps Fisher was like that, but most of us are certainly not. We rely on intuition and common sense, both of which can sometimes perform dismally badly.
So, models only exist because we need them to help us: none of us are born with such supercomputer brains that we could evaluate arbitrarily complex arguments immediately and without external help. What is the optimal complexity of a model, then? Once again, it depends on the question. Reflect for a moment that there are maps with different scales. In the context of scientific models, it is useful to be reminded of the ultimate reason we do science: it is the joy of understanding something. If we could visualize and memorize much more detailed maps than we currently do, useful maps would include more detail than they currently do. Exactly analogously, if we could grasp much more complex processes without getting headaches than we currently do, models would look different too. Given the way our brains are built, a good guideline is that a model should include all the relevant details for the particular question at hand, but it should be kept so simple that it can be understood (if with joy, then still better). In other words, a model is not particularly helpful if it predicts that under conditions A the animal should do X, while under conditions B it should do X 30% of the time and Y in the rest of instances, and then there are 17 other parameters that interact with each other in producing a diversity of outcomes – but when asked why the model produces these effects of A and B, we still have no answer that can be expressed in a language that anyone’s intuition can understand. Removing some additional detail from the model can then be surprisingly helpful: the effects of A and B could still be the same, but with far simpler equations.
For example, we might have spent a lot of time modelling the distribution of body condition in a population of migratory birds, ending up with very cumbersome equations, when a far simpler way to grasp the conceptual issue is to divide up the population into two classes of individuals, ‘hungry’ and ‘satisfied’. To show a conceptual point, this might be sufficient. Results could be far simpler to derive this way than with a more complete model, and if our understanding of the biology advances faster this way, the simplification is justified. But how to know, then, that the division has not caused some artifacts? Perhaps an exact shape of the body condition distribution would have produced a totally different answer? The answer is … we don’t really know, unless we build the more complicated model too. (Which means that modellers rarely run out of models to study.
In the above – and indeed in the rest of this book – I am mainly dealing with conceptual models. They are typically models that aim to answer questions, ‘Does from A follow B?’ Or in a little more complicated way: ‘Under which biologically relevant conditions can we claim that from A follows B?’ Such models usually aim at relatively broad taxonomic applicability, which also means that details of the behaviour of a specific species, no matter how exciting and important they appear, should usually be considered irrelevant. A modeller should not necessarily be judged as arrogant or ignorant if she brushes over such detail. When we want to know under what conditions, in general, one expects female preferences to evolve based on indirect genetic benefits, it is not very wise to consider details of the energetics of black grouse leks. Predictions of conceptual models, therefore, tend to be qualitative rather than quantitative. For example, we may predict that sexually selected male traits can be very costly to their bearer, whereas large costs of female preferences are not expected (because if we assume such costs, preferences evolve to zero, see Kokko et al. (2006a) for a review). The model is not intended to predict exactly how costly the trait will be in the case of a male black grouse.
Instead, a good model should in the end say: these are the conditions that have to be met before we believe we have a logically consistent argument that explains male trait evolution; now go and find out if it really is the case. The need to test assumptions and not just predictions of a model can hardly be overemphasized. But as we have just learnt, assumptions never reflect the system completely faithfully, and that is perhaps the most important reason why an empiricist should have some grasp of mathematical modelling, even if not interested in pursuing it as a career. A model that is based on the assumption that the moon is a flat Roquefort cheese is obviously so out of this world that any conclusions drawn from it, no matter how mathematically solid, will have no relevance whatsoever. But a good and useful model will still appear ‘false’ in the sense that many aspects of reality are necessarily ignored, and here one must be able to judge if the assumptions nevertheless capture the essence of the biological argument.
Not all models refrain from making numerically explicit predictions, however. Some are much more applied and number oriented; perhaps we are not interested in the causalities at all, but we would really like to predict the number of rats left in an area after an eradication programme, extrapolating from past knowledge of rat behaviour. In such a case, it is probably not particularly important to know exactly why rats retain memories of bad food experiences for as long a time as they do. We can simply assume this happens, take an exact numerical value from experiments and build this into our model. Such models are predictive and may be precise (if we are lucky), but they do not try to aid our conceptual understanding of evolutionary processes. Of course, if we are really lucky, we might get at that too; some famous success stories where evolutionary insight combines with numerically accurate predictions come from studies that link sex ratios to local mate competition. There are some others too.
There are also models that fail to be simple enough for us to understand fully their inner workings when they churn out results, yet they are not precise either in the sense of predicting the numbers of rats on an island. Often such models are complex computer simulations, tracking the state and behaviour of a large number of virtual individuals, and examining the emergent properties of the system. Such results can be fascinating to watch – who wouldn’t like to play god, or have their own study species repeating their fascinating behaviour on a computer screen – but I would like to warn against the overuse of such methods. It is quite
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