Biology Reference
In-Depth Information
Hawks in the population. Therefore, the proportion of Doves must be (1
h ). The
average pay-off for a Hawk is the pay-off for each type of contest multiplied by the
probability of meeting each type of contestant. Therefore:
H average
= −
25 h
+
50 (1
h )
Similarly, for Dove the average pay-off will be:
D average
=
0 h
+
25 (1
h )
At the stable equilibrium, H average must equal D average . Solving the equation above
by setting H average
=
D average gives h
=
½; therefore, the proportion of Doves (1
h )
must also be ½. In general, if V
<
C, the stable proportion of Hawks in this game is given
by V/C.
The ESS in Table 5.1 could come about in two ways.
A mixture of
Hawk and Dove is
evolutionarily
stable
(1) There is an evolutionarily stable polymorphic state, with individuals all playing
pure strategies, half of them Hawk and half of them Dove.
(2) Individuals all adopt a mixed strategy, playing Hawk randomly with probability ½
and Dove with probability ½.
It is instructive to note that at the ESS, the average pay-off per contest is
+
12.5. If only
everyone had agreed to be Doves, the pay-off would be
25! As with our human crowd
example, the optimal strategy to maximize everyone's fitness is often higher than the
pay-offs at the ESS. Nevertheless, we expect evolution to lead to stable strategies because,
in the words of Richard Dawkins, 'they are immune to treachery from within'. The
Hawk ~ Dove game makes another general point. At the stable equilibrium there is often
variation in the population; either between or within individuals. Variation is, therefore,
not always noise about a population norm. Rather, it is often the expected stable outcome
when individuals compete.
The assumption that V
+
C will apply to many contests in nature. For example, it will
rarely be worth the risks of serious injury from a fight simply to win an item of food or
access to a shelter. However, if V
<
The ESS solution
does not
maximize every
individual's
fitness
C then Hawk is an ESS. Intuitively, it is easy to see why
this is so. We need to consider the consequences of the current contest for lifetime
reproductive success. This will involve a balance between the value of the contested
resource and the expected value from future contests. When the value of the resource is
similar to, or greater than the value of the future, we would expect individuals to risk more
in contests, even at the cost of serious injury or death. Indeed, if the value of the future is
close to zero then contestants should, in theory, never give up after starting a fight, so the
contest should be fatal for at least one of the opponents (Enquist & Leimar, 1990).
As predicted, fatal fighting has been reported in cases where individuals have a
short  lifespan and few reproductive opportunities. For example, male bowl and
doily  spiders, Frontinella pyramitela , (Austad, 1983) and male fig wasps (Hamilton,
1979; Murray, 1987) often fight to the death for their once-in-a-lifetime chance to win
a receptive female.
The Hawk-Dove game is clearly too simple to apply to any real cases in nature; there
are likely to be more than just two strategies, strategies will vary with individual
strength, encounters will not be random and so on. In Chapter 14, we will show how
>
If V > C, Hawk is
an ESS
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