Can you use realistic numbers to produce a pure Hawk ESS? Pure Dove? Or are totally unbelievable numbers required?
One situation to look at is where the payoff to winning the resource is less than zero. But you need to ask yourself the question "would any animal work for a such a payoff since it lowers its fitness from what it would get without exhibiting the behavior?"
Another situation would be to set the injury or display costs to a low (near zero) value.
Think about what these sorts of payoffs mean -- could you translate any of them into situations that deal with the behavior of real animals?
When there is a pure ESS, where do the fitness lines for H and D converge relative to the frequency of Hawk?
At values greater 1.0 or less than 0.0 -- in other words, at impossible frequencies!
Why is it that the relative fitness of H does not change (up or down) in simulations when H increases in frequency? Recall that we learned earlier that the absolute fitness of both H and D decrease as the frequency of H increases.What does this tell you about measures of relative fitness?
It tells you that measures of relative fitness are just that -- relative. Evolution is a game of relative advantage (assuming that there is at least enough reproduction so that there is another generation!). In this two strategy game, whenever Hawk is more fit than Dove, it has a relative fitness of 1.0 (and the same is true of Dove).
Also, either strategy can maintain a relative fitness of 1.0 even though its absolute fitness decreases with a change in its frequency (for example, the absolute fitness of Hawk decreases with increasing numbers of hawks). All that has to be true is that it is more fit than the alternative. Thus, using the default payoffs, H will increase at low frequencies even though whenever it increases it actually results in a lowering of the average reproduction of the next generation of hawks.
Why should there be cases where the fitness of a strategy increases as its frequency decreases?
If the strategy is at a greater frequency than its equilibrium, its fitness is lowered as a result of a relatively large number of unfavorable interactions -- for instance, using default payoffs, H vs. H interactions are highly unfavorable to H. Thus, as freq of H decreases and fewer of these contests occur, the fitness of H increases (proportionately it has more favorable interactions with Doves). The value where relative fitness stops increasing is the mixed ESS point -- in the case of Hawk, the problems with running into other hawks are exactly balanced by the benefits of intimidating doves!
Imagine a situation where losing a fight causes severe injury but that fighting is the only way to procure fitness. (An example would be that winning a fight would be the only way to mate).
In this situation, what is the fitness of an individual playing a strategy that does not fight and lives a long time?
This individual will have a fitness that is a very, very negative number. It doesn't matter how long an individual lives, if it doesn't reproduce (or somehow gain indirect fitness which is a separate issue), its fitness is zero. Notice that this is a Dove-like strategy that cannot work! It also is not a very realistic one since there are usually plenty of ways for an animal to gain a critical resource short of fighting, even when fighting is common.
Notice also that we are in a bit of a mathematical quandary here. By adopting the convention that benefits are positive numbers and costs negative, we cannot easily assign this strategy the fitness it deserves -- zero. Recall that in the system we are using, a payoff of zero simply means no effect on fitness, not zero fitness. Zero fitnesses are payoffs that are infinitely negative. This problem is obviated somewhat by using some positive number as a baseline fitness, assigning benefits in addition to that baseline, and assigning costs as values below it with zero as the zero fitness point. But, if you think about it a moment, you'll see that now the problem exists in assigning costs which are now constrained between 0 and the baseline positive value for fitness.
Compared to the individual just discussed,what would the average relative payoff be to an individual playing a strategy that fights others like itself for the resource. Assume that most individuals of this strategy die early without procuring the resource. Nevertheless, some of them are successful and leave offspring.
This is a Hawk-like strategy. There are some problems with the math in this case (see next question) but let's use it to make an important point. Death does not matter to the allele responsible for the strategy so long as someone carrying the allele succeeds in reproducing and does so at as good of a rate as the alternatives. So, as long as some individuals do win and reproduce, even though the costs are high, this payoff will be higher than the payoffs to those who never fight and never reproduce.
If death occurs as a result of a contest, can sequential contest games like Hawk and Dove really be used?
Generally, no. The mathematics of the game assumes that individuals fight a number of sequential contests. If they die in the contests, then their strategy's frequency decreases over time. So, in our Hawk and Dove game, if Hawks kill each other, the frequency of Dove (and therefore contests involving Dove) will increase over time during the game. Notice that when we determined the fitness of a strategy, we multiplied the payoff of each type of encounter by the frequency of the encounter, which we took to be constant. Thus, the game assumes that no one dies. Injury just lowers your success at reproduction.
Now, there is one way that death can be allowed in a game (short of recalculating strategy frequencies after every contest). If everyone simply engages in one contest with an opponent picked at random, and if no further contests occur after (so that strategy fitnesses are the aggregates of these one-time encounters), then individuals can die and the game will still work. In this case, death simply equates to a fitness of zero (or since not everyone dies, a very, very high injury cost). (Alternatively, the number of living individuals could be used to recalculate the strategy frequencies after each contest, but that is not how we set up the mathematics of the game.)
There are many cases where there are highly escalated contests involving serious fights that can cause death of one of the contestants. For example, male elephant seals engage in such contests over sections of a beach and "breeding rights" with the females in this area. Does that mean that males that lose such fights or that do not engage in fighting have no fitness? What does this tell you about simple games like Hawk and Dove?
In a simple world where fighting was the only way to gain access to a mate, then males that lose or do not engage in fights would have no fitness. If we defined two strategies, "fight" (Hawk-like) and display or "don't fight" (Dove-like) the payoffs for the "don't fight" strategy would truly be zero or, using the numeration scheme we have selected, the benefit would be zero and the costs (exclusion from mating) infinitely large and negative. Most of the fighters would not succeed and we could define the costs of losing as large. But the benefits to winning would be immense and "fight" would be a pure ESS since E(fighter vs. fighter) would be greater than E(don't fight vs. fight).
However, it should be easy to envision a strategy that could invade a population of fighters. They could also eschew all fighting, but try to sneak matings. As long as they were successful sometimes, even though the benefits they received would be different than those of the fighter, they would not suffer any injuries and they might be able to invade.
In northern elephant seals things like this happen. Large males do defend sections of the beach using displays and fights that can escalate to death. Nonetheless, other males will try to sneak matings, at least during certain parts of their lives. Whether or not this sneaking strategy is an ESS or simply a matter of trying to do the best you can will depend on whether or not the fitnesses are equal. Compared to our simple Hawk and Dove game this situation is much more complex; and if it were an ESS (to my knowledge no one has looked at it), it might well involve mixes of strategies over a lifetime that were conditional on an animal's size, age and general physical condition. So simple games like Hawk and Dove may help us to learn about some of the factors involved with animal contests, but they need lots of refinement before we can really understand the often highly sophisticated behavior of animals.
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Copyright © 1998 by Kenneth N. Prestwich
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