| Synopsis: This page presents a general overview of the use of models in evolutionary biology. The differences between adaptational and neutral models are briefly discussed. The bulk of the material deals with an overview of two types of adaptationalist models -- optimality and game theory-- and ends with a comparison between them. Later pages will present a more detailed explanation of game theory. |
Contents
Assumptions -- Adaptation vs. Neutral Models of Evolution
Adaptational Models: Following the success of Darwin and Wallace's theory of natural selection, most modern biologists believe that most of the aspects of the morphology, behavior and physiology of an organism represent adaptations. That is, these aspects of the phenotype exist in a population because in the recent past they allowed their possessors to reproduce more successfully than individuals with alternative traits.
Neutralist Models: However, one must keep in mind that adaptation is often only an assumption. In the 1930s Sewell Wright (see reference) developed the main theoretical underpinnings of an alternative means of accounting for evolution based on genetic drift. This process works well in small populations especially when competing traits confer little relative survival advantage over each other (these traits are said to be adaptively neutral).
Extending Wright's work, others have shown that the particular traits found in a population can be the result of historical accidents. For instance, Ernst Mayr (certainly an adaptationalist) pointed out the importance of the genetic makeup of a small number of progenitors or founders (reference) of a population. More recently, Stephen Jay Gould has written extensively about the role of history and accident ("contingency") in determining the present phenotypes of members of a population and the actual range of organisms that exist at a given time (see reference).
It is fair to say that much of the value of the work of Gould and others has been to force biologists to acknowledge that all aspects of the phenotype need not represent specific adaptations. The phenotype is in part an accident and not an ideal design. In many cases a number of competing versions of a phenotype might all do equally well, especially given the nature of environmental change. Thus, it is necessary to provide evidence that a particular phenotypic feature represents an adaptation, rather than provide circular "just-so" stories that purport to show adaptation by assuming it and then spinning out an explanation based on the (untested) assumption of adaptation.
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Mathematical Models and Rigorous Science
Mathematical models are abstractions that an investigator hopes will have, to varying degrees of precision, predictive power. Models represent the scientist's best informed guess as to:
Mathematical models are useful because they take a scientist's ideas and produce a more complex abstraction (because a model consists of parts and their interaction). Complex, multi-element models often yield new insights and novel predictions. Mathematical models have the advantage that they yield quantitative predictions. Quantitative predictions are often less ambiguous than other types of predictions. Since tests of hypotheses are attempts to show the predictions are incorrect (tests attempt to falsify the model), quantitative predictions are usually easier to test. Did the model behave exactly as predicted or not? If not, how different was it from prediction? How could the model be modified to make it more consistent with the results and then re-tested?
Inability to falsify the model does not validate it. Inability to falsify means nothing more than that. Not showing that a model is wrong means only tentative acceptance, not proof of its truth. A model that has not been falsified is nothing more than a useful working hypothesis. For example, a telling observation was made by Dr. David Norman about restoration mounts of dinosaurs (these mounts are, of course, nothing more than hypotheses) when he wryly observed "we've got it right -- for now". Much of the formalism of testing and describing the scientific process can be traced to the work of the English philosopher Sir Karl Popper (see reference) . (For a reference to Ernst Mayr's extended and fascinating treatment of biological methodology press here).
Most commonly, models are modified as the result of (i) experimental falsification of one or more of their components or (ii) independent refinements in our understanding of the variables and interactions that make up a model. You have probably noticed that this same process is normally followed throughout the scientific process; a main difference is that hypotheses in the form of mathematical models are often more concrete and quantitatively predictive than are other types of hypotheses. However, keep in mind that working with experimentally supported quantitative models is fraught with the same dangers as with less quantitative models -- like any hypothesis models should always be viewed with skepticism and only trusted to the extent that they have been strongly tested.
One other note about mathematical models in evolutionary biology: they can spring from either an adaptationalist (review material above) or a neutralist (review material above) view point. Two important types of mathematical adaptation-based models are optimality and game theory models. The next two sections compare these two approaches. Since both are adaptation models, both will look for behavioral characteristics that maximize an individual's reproductive success or some related variable.
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Often a behaviorist is interested in predicting the best way (in terms of its fitness consequences) for a particular animal to behave irrespective of what other individuals are doing. To illustrate, suppose we are trying to understand how loudly an animal should make an advertisement call (one designed to attract a mate). Thus, we are looking at a general behavior (producing an advertisement call) and we are trying to understand the selective forces that determine the best way to perform a particular part of the behavior -- in this case its loudness. In this particular example (and in all optimality models) we start from the assumption that the loudness of other callers has nothing to do with predicting the loudness of a given individual. So we might imagine a situation where an animal calls without others nearby (as would be the case in many species of crickets).
Optimality theory is adaptational - thus it springs from the theory of
natural selection which predicts that an animal
should behave so as to maximize its fitness. All behaviors can be viewed as having both
fitness benefits and fitness costs. Since optimality models
are quantitative, a first step will be to establish the relationships between
the variable of interest (the decision variable) and the associated
costs and benefits. Benefits (B) and Costs (C) are kept strictly separate
(just as is practiced in business bookkeeping) and thus two separate relationships
(benefit vs. the decision variable and cost vs. the decision variable) are
implicitly part of the process of setting up an optimality model. Notice
that since we assume that the behavior (decision variable) has consequences
on fitness, the behavior is the independent variable for each of these relationships
with benefit or cost being the dependent variables.
Now,
since the fitness consequence of behavior is:
Net Change in Fitness = Benefit - Cost
the solution to an optimality model is to find the point where B- C is maximized.
In principle this formulation is easy to understand. However in practice, it can be more complex. Let's return to our example of call loudness to illustrate the process of constructing and solving a simple optimality model:
In acoustic communication, producing a loud call is energetically expensive (see reference). However, louder calls tend to attract more mates, (for example, see reference dealing with mole crickets). We want to construct an optimality model in order to try to predict how loudly to call with the "goal" of maximizing lifetime fitness.
Let's start with the benefits of calling more loudly for a certain period of time. In theory there are a number of possible relationships between loudness and benefit. One would be linear -- get louder and proportionately more mates will come (graph I, below). A little reflection would suggest that this cannot go on forever -- at some point increasing loudness brings in so many mates that the focal animal can't handle all of them and so there is no further increase in fitness with loudness (graph II, below). Alternately, one might assume that the rate at which matings increase drop off before finally reaching a maximum (graph III, below) -- i.e.,because of other things the caller has to attend to, as the number of mating opportunities increase, the percentage that are actually consummated becomes less:
There are several things to note about these plots:
Costs Curves: It is logical to assume that loud calling will have consequences on fitness since it costs considerable amounts of energy and therefore could weaken the caller. This leads directly to expressing cost as energy loss (energy is something which I, as a behavioral physiologist, feel very comfortable with -- it can be measured directly and relatively easily). But absolute energy terms such as calories or joules may not be the most relevant way to measure the energy portion of the cost of calling. For instance, if the cost of getting louder is viewed relative to energy stores, low cost calls might have little impact on reserves and therefore,up to a point, increasing loudness might have little accompanying increase in cost. However, when the energy demands of making louder calls increases beyond a certain point, they may significantly affect an animal's energy reserves and therefore force it to either call for a shorter time period (perhaps thereby lowering its fitness) or eat less (also lowering its fitness). Notice that once again, the exact position and shape of the curve would depend on many factors -- for instance size of food reserves and the ease with which they are replaced relative to the incremental costs of louder calls.
Moreover, another important cost of calling is increased chance of falling victim to predation (e.g., see Mike Ryan's work on bats preying on calling frogs). What units do you use to measure this cost? The cost could be expressed as the chance of being killed or injured, or better yet the number of future matings or future offspring lost as a result of injury or death associated with a certain loudness of call.
Let's assume that we believe there are just two important costs to calling -- energy and risk. We need to combine these into one cost relationship. Both are already related to the same decision variable (loudness) but these two types of costs are usually expressed in units that are different from each other (for example joules for energy and probability of death for predation risk).
If we want to combine all costs into one curve, we need to put them into a common unit of measurement (a common currency). This currency must also be the same one used in the benefits function. This can be very difficult, but lets say that we have found a way to express both energy and predation risk as lost future matings. And let's say that our best understanding is that weak calls do not attract predators at all and confer no additional feeding demands. Nevertheless, eventually a point is reached where predation starts to increase and eventually significant increases in feeding must occur. Here might be a graphical example of our costs hypothesis:
Now let's complete and solve our model. Let's say that we decide that benefits model III (review graph of benefits) is the best one to use. If we now express benefits and costs of call loudness in a currency of number of successful matings, then we can superimpose the benefit and costs plot on the same axis and then solve for the loudness that gives the greatest increase in fitness as measured by the greatest number of matings:
Notice that the model predicts that the best loudness to call is not the loudest (which is what you would predict if only benefits had been considered). Notice also that in this case, the model does not keep costs to an absolute minimum.
We can use the B and C curves (above) to make another graph, this time of B- C vs. loudness, to illustrate the optimum another way:
This type of depiction clearly shows that the greatest lifetime fitness in our hypothetical situation is achieved by calling with some type of intermediate loudness.
Finally, realize that the model is nothing more than an integrated set of hypotheses that together attempt to predict the best loudness to call, irrespective of what others are doing. As we saw above (press to review general info. on mathematical models), this prediction will need to be tested (for instance by attempting to determine the lifetime mating curve for animals of different loudness). In practice, this essential step is often very difficult (think how hard it would be to determine the curve given above for some type of animal) and much of the art and hard work of behavioral science lies in these tests of the models.
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A Brief Introduction to Game Theory
OK, we just went through a long introduction to an important topic in behavior, physiology, and ecology: optimality theory. But you thought this website was about game theory. It is. The stuff on optimality (above) was important because you need to know something about optimality theory to understand game theory and especially to understand which techniques to use when trying to understand animal behavior.
So, what is game theory? In optimality theory it is assumed that we can predict the best behavior for a particular (focal) animal irrespective of what others are doing. However, the frequency with which others are performing a particular behavior is often highly relevant to the fitness consequences of acting a certain way. Thus, the crucial aspect of game theory is that selection among alternative behaviors depends to a large degree on what others are doing. A behavior that works very well when rare in a population (for instance, some type of deception) may not be nearly as advantageous to its actors when it becomes common. Thus explorations of game theory involve studies of a form of frequency-dependent selection.
Here's a brief example. Let's return to calling behavior. Let's assume that females are attracted to calling males and travel to them to mate. But let's also assume that arriving females are not infallible in spotting the male that actually was making the call to which they were attracted. Or, females may be intercepted by males as they approach and sometimes the intercepting males were not the ones that attracted the female in the first place.
What's the best thing for a male to do? If no one is calling, it is probably best to call and become conspicuous to females interested in mating. However, if there are many callers, it might be best to keep quiet (avoiding energy and predation risk costs -- review previous material) and try to intercept females as they approach a calling male. Such a male is termed a satellite.
As a strategy satellite could well be as or more fit than calling, depending on its frequency relative to callers. Thus, even though a satellite might have fewer opportunities per day to mate (lower benefit), the fact that his costs are less means that he might have as many if not more lifetime opportunities for mating. However, as should be obvious, his relative success does not simply reduce to costs and benefits as in optimality theory -- instead, his fitness depends very much on the frequency of callers (vs. satellites) in the population. If few others call, satellite is probably not a good strategy and relatively speaking, calling is an excellent strategy (or taking it to an extreme, if no one is calling, the silent satellite strategy makes no sense since females are attracted to calling males). On the other hand, as more and more callers are present, satellite works better and at some frequencies might over a lifetime payoff better than calling. Thus, frequency dependence distinguishes this example from simple optimality.
| ! Note: don't get the idea that I am arguing here that satellite is generally a more or less fit strategy than the alternative, call. Depending on conditions, at any moment in time all of these are possible. Later we will consider the most interesting outcome of evolutionary games, an evolutionary stable strategy (ESS). We will see that one type of ESS (mixed ESS) predicts that both strategies would coexist at constant relative proportions to each other and at these frequencies the fitness of individuals of each strategy are equal. |
Summary : Notice the similarities and differences between optimality and game theory. Both deal with the benefits and costs of a behavior. In situations where optimality theory is applicable, that is all that is needed. However, as in the example above, there are many cases where the fitness associated with a particular behavior depends on what others are doing -- that is, fitness is frequency-dependent. Thus, what is best will depend on what everyone else in the population is doing (and therefore on the likelihood of certain types of interactions with certain types of fitness consequences).
You now have a basic idea about the uses of mathematical modeling in studies of behavioral evolution. You should also be familiar with the basics of game and optimality models. You may now continue on to a more detailed introduction to game theory especially in regards to an important concept, the evolutionary stable strategy, or you may continue reading this page and learn a bit about the history of game theory.
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A Brief Sketch of the History of Game Theory
Game theory is equally useful in studies of learned and innate behavior. In fact, when originally developed by von Neumann and Morgenstern (reference) during the mid 20th century, it's primary purpose was to understand the most rational way for humans to make decisions between alternative courses of action, in particular as they applied to economics.
However, any technique that will allow us to study the payoffs of a learned behavior can be used equally well to study innate behavioral strategies. John Maynard Smith (1982) points out that the idea of rational interest in economics is simply replaced by the concept of fitness. He traces the use of game theory in biology first to the work of Lewontin (see reference) (1961) and later Slobodkin and Rapoport (see reference). These workers applied game theory to situations of "species or group survival" (Did you ever play the board game "Extinction"?). The present mainstream of game theory began with Hamilton (1967) (see reference) and then a few years later with Maynard Smith and Price (1972) (see reference) and then Maynard Smith (1974) (see reference). In the intervening years, game theory has been adapted by a number of biologists to examine evolutionary problems-- hardly an issue of Animal Behaviour passes when one does not see an article using game theory.
Perhaps the most thorough introduction to the use of game theory in biology is by one of the pioneers in applying it to biology -- John Maynard Smith (1982) (see reference). The interested and mathematically inclined student is urged to read his classic introduction to the field. The material at this site is based largely on the the first few chapters of his book. However, most students will find that Riechert and Hammerstein's 1983 article in the Annual Review of Ecology and Systematics is more readable and is highly informative.
Continue on to a Detailed Introduction to Game Theory
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Copyright © 1999 by Kenneth N. Prestwich
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