The Extended Phenotype: The Long Reach of the Gene (Popular Science) (26 page)

Although there is no polymorphism of diggers and enterers here, something mathematically equivalent to frequency-dependent selection can go on. Here is how it would work. As before, there is a critical population frequency of digging,
p*
, at which entering would yield exactly the same ‘pay-off’ as digging.
p
* is, then, the evolutionarily stable digging probability.
If the stable probability is 0.7, programs instructing wasps to follow a different rule, say ‘Dig with probability 0.75’, or ‘Dig with probability 0.65’, would do less well. There is a whole family of ‘mixed strategies’ of the form ‘Dig with probability
p
, enter with probability 1 –
p
’, and only one of these is the ESS.

I said that the two extremes were joined by a continuum. I meant that the stable population frequency of digging,
p
* (70 per cent or whatever it is), could be achieved by any of a large number of combinations of pure and mixed individual strategies. There might be a wide distribution of
p
values in individual nervous systems in the population, including some pure diggers and pure enterers. But, provided the total frequency of digging in the population is equal to the critical value
p*
, it would still be true that digging and entering were equally successful, and natural selection would not act to change the relative frequency of the two subroutines in the next generation. The population would be in an evolutionarily stable state. The analogy with Fisher’s (1930a) theory of sex ratio equilibrium will be clear.

Proceeding from the conceivable to the actual, Brockmann’s data show conclusively that these wasps are not in any simple sense polymorphic. Individuals sometimes dug and sometimes entered. We could not even detect any statistical tendency for individuals to specialize in digging or entering. Evidently, if the wasp population is in a mixed evolutionarily stable state, it lies away from the polymorphism end of the continuum. Whether it is at the other extreme—all individuals running the same stochastic program—or whether there is some more complex mixture of pure and mixed individual programs, we do not know. It is one of the central messages of this chapter that, for our research purpose, we did not
need
to know. Because we refrained from talking about individual success, but thought instead about subroutine success averaged over all individuals, we were able to develop and test a successful model of the mixed ESS which left open the question of where along the continuum our wasps lay. I shall return to this point after giving some pertinent facts and after outlining the model itself.

When a wasp digs a burrow, she may either stay with it and provision it, or she may abandon it. Reasons for abandoning nests were not always obvious, but they included invasion by ants and other undesirable aliens. A wasp who moves into a burrow that another has dug may find that the original owner is still in residence. In this case she is said to have
joined
the previous owner, and the two wasps usually work on the same nest for a while, both independently bringing katydids to it. Alternatively, the entering wasp may be fortunate enough to hit upon a nest that has been abandoned by its original owner, in which case she has it to herself. The evidence indicates that entering wasps cannot distinguish a nest that has been abandoned from one that is still occupied by its previous owner. This fact is not so surprising as it may seem, since both wasps spend most of their time out hunting, so two
wasps ‘sharing’ the same nest seldom meet. When they do meet they fight, and in any case only one of them succeeds in laying an egg in the nest under dispute.

Whatever had precipitated the abandoning of a nest by the original owner seemed usually to be a temporary inconvenience, and an abandoned nest constituted a desirable resource which was soon used by another wasp. A wasp who enters an abandoned burrow saves herself the costs associated with digging one. On the other hand, she runs the risk that the burrow she enters has not been abandoned. It may still contain the original owner, or it may contain another entering wasp who got there first. In either of these cases the entering wasp lets herself in for a high risk of a costly fight, and a high risk that she will not be the one to lay the egg at the end of a costly period of provisioning the nest.

We developed and tested a mathematical model (Brockmann, Grafen & Dawkins 1979) which distinguished four different ‘outcomes’ or fates that could befall a wasp in any particular nesting episode.

1 She could be forced to abandon the nest, say by an ant raid.

2 She could end up alone, in sole charge of the nest.

3 She could be joined by a second wasp.

4 She could join an already incumbent wasp.

Outcomes 1 to 3 could result from an initial decision to dig a burrow. Outcomes 2 to 4 could result from an initial decision to enter. Brockmann’s data enabled us to measure, in terms of probability of laying an egg per unit time, the relative ‘payoffs’ associated with each of these four outcomes. For instance, in one study population in Exeter, New Hampshire, Outcome 4, ‘joining’, had a payoff score of 0.35 eggs per 100 hours. This score was obtained by averaging over all occasions when wasps ended up in that outcome. To calculate it we simply added up the total number of eggs laid by wasps who, on the occasion concerned, had joined an already incumbent wasp, and divided by the total time spent by wasps on nests that they had joined. The corresponding score for wasps who began alone but were subsequently joined was 1.06 eggs per 100 hours, and that for wasps who remained alone was 1.93 eggs per 100 hours.

If a wasp could control which of the four outcomes she ended up in, she should ‘prefer’ to end up alone since this outcome carried the highest payoff rate, but how might she achieve this? It was a key assumption of our model that the four outcomes did not correspond to decisions that were available to a wasp. A wasp can ‘decide’ to dig or to enter. She cannot decide to be joined or to be alone any more than a man can decide not to get cancer. These are outcomes that depend on circumstances beyond the individual’s control. In this case they depend on what the other wasps in the population
do. But just as a man may statistically reduce his chances of getting cancer, by taking the decision to stop smoking, so a wasp’s ‘task’ is to make the only decision open to her—dig or enter—in such a way as to maximize her chances of ending up in a desirable outcome. More strictly, we seek the stable value of
p, p*
, such that when
p
* of the decisions in the population are digging decisions, no mutant gene leading to the adoption of some other value of
p
will be favoured by natural selection.

The probability that an entering decision will lead to some particular outcome, such as the desirable ‘alone’ outcome, depends on the overall frequency of entering decisions in the population. If a large amount of entering is going on in the population, the number of available abandoned burrows goes down, and the chances go up that a wasp that decides to enter will find herself in the undesirable position of joining an incumbent. Our model enables us to take any given value of
p
, the overall frequency of digging in the population, and predict the probability that an individual that decides to dig, or an individual that decides to enter, will end that episode in each of the four outcomes. The average payoff to a wasp that decides to dig, therefore, can be predicted for any named frequency of digging versus entering in the population as a whole. It is simply the sum, over the four outcomes, of the expected payoff yielded by each outcome, multiplied by the probability that a wasp that digs will end up in that outcome. The equivalent sum can be worked out for a wasp that decides to enter, again for any named frequency of digging versus entering in the population. Finally, making certain plausible additional assumptions which are listed in the original paper, we solve an equation to find the population digging frequency at which the average expected benefit to a wasp that digs is exactly equal to the average expected benefit to a wasp that enters. That is our predicted equilibrium frequency which we can compare with the observed frequency in the wild population. Our expectation is that the real population should either be sitting at the equilibrium frequency or else in the process of evolving towards the equilibrium frequency. The model also predicts the proportion of wasps ending up in each of the four outcomes at equilibrium, and these figures too can be tested against the observed data. The model’s equilibrium is theoretically stable in that it predicts that deviations from equilibrium will be corrected by natural selection.

Brockmann studied two populations of wasps, one in Michigan and one in New Hampshire. The results were different in the two populations. In Michigan the model failed to predict the observed results, and we concluded that it was quite inapplicable to the Michigan population, for unknown reasons as discussed in the original paper (the fact that the Michigan population has now gone extinct is probably fortuitous!). The New Hampshire population, on the other hand, gave a convincing fit to the predictions of the model. The predicted equilibrium frequency of entering
was 0.44, and the observed frequency was 0.41. The model also successfully predicted the frequency of each of the four ‘outcomes’ in the New Hampshire population. Perhaps most importantly, the average payoff resulting from digging decisions did not differ significantly from the average payoff resulting from entering decisions.

The point of my telling this story in the present book has now finally arrived. I want to claim that we would have found it difficult to do this research if we had thought in terms of
individual
success, rather than in terms of strategy (program) success averaged over all individuals. If the mixed ESS had happened to lie at the balanced polymorphism end of the continuum, it would, indeed, have made sense to ask something like the following. Is the success of wasps that dig equal to the success of wasps that enter? We would have classified wasps as diggers or enterers, and compared the total lifetime’s egg-laying success of individuals of the two types, predicting that the two success scores should be equal. But as we have seen these wasps are not polymorphic. Each individual sometimes digs and sometimes enters.

It might be thought that it would have been easy to do something like the following. Classify all individuals into those that entered with a probability less than 0.1, those that entered with a probability between 0.1 and 0.2, those with a probability between 0.2 and 0.3, between 0.3 and 0.4, 0.4 and 0.5, etc. Then compare the lifetime reproductive successes of wasps in the different classes. But supposing we did this, exactly what would the ESS theory predict? A hasty first thought is that those wasps with a
p
value close to the equilibrium
p
* should enjoy a higher success score than wasps with some other value of
p:
the graph of success against
p
should peak at an ‘optimum’ at
p
*. But
p
* is not really an optimum value, it is an evolutionarily stable value. The theory expects that, when
p
* is achieved in the population as a whole, digging and entering should be equally successful. At equilibrium, therefore, we expect no correlation between a wasp’s digging probability and her success. If the population deviates from equilibrium in the direction of too much entering, the ‘optimum’ choice rule becomes ‘always dig’ (not ‘dig with probability
p
*’). If the population deviates from equilibrium in the other direction, the ‘optimum’ policy is ‘always enter’. If the population fluctuates at random around the equilibrium value, analogy with sex ratio theory suggests that in the long run genetic tendencies to adopt exactly the equilibrium value,
p*
, will be favoured over tendencies to adopt any other consistent value of
p
(Williams 1979). But in any one year this advantage is not particularly likely to be evident. The sensible expectation of the theory is that there should be no significant difference in success rates among the classes of wasps.

In any case this method of dividing wasps up into classes presupposes that there is some consistent variation among wasps in digging tendency. The
theory gives us no particular reason to expect that there should be any such variation. Indeed, the analogy with sex ratio theory just mentioned gives positive grounds for expecting that wasps should not vary in digging probability. In accordance with this, a statistical test on the actual data revealed no evidence of inter-individual variation in digging tendency. Even if there were some individual variation, the method of comparing the success of individuals with different
p
values would have been a crude and insensitive one for comparing the success rates of digging and entering. This can be seen by an analogy.

An agriculturalist wishes to compare the efficacy of two fertilizers, A and B. He takes ten fields and divides each of them into a large number of small plots. Each plot is treated, at random, with either A or B, and wheat is sown in all the plots of all the fields. Now, how should he compare the two fertilizers? The sensitive way is to take the yields of all the plots treated with A and compare them with the yields of all the plots treated with B, across all ten fields. But there is another, much cruder way. It happens that in the random allocation of fertilizers to plots, some of the ten fields received a relatively large amount of fertilizer A, while others happened to be given a relatively large amount of B. The agriculturalist could, then, plot the overall yield of each of the ten fields against the proportion of the field that was treated with fertilizer A rather than B. If there is a very pronounced difference in quality between the two fertilizers, this method might just show it up, but far more probably the difference would be masked. The method of comparing the yields of the ten fields would be efficient only if there is very high between-field variance, and there is no particular reason to expect this.

In the analogy, the two fertilizers stand for digging and entering. The fields are the wasps. The plots are the episodes of time that individual wasps devote either to digging or to entering. The crude method of comparing digging and entering is to plot the lifetime success of individual wasps against their proportionate digging tendency. The sensitive way is the one we actually employed.