Falcons and pigeons
Imagine a population of falcons and pigeons . When they meet an opponent, the falcons fight until they either win or are seriously hurt. The pigeons only threaten their opponents but withdraw if the other wants to fight. Let us suppose also that the long-term survival of the two types of bird depends on the results obtained in these fights and that the winnings of each possible fight are as follows: 50 points for winning, -100 points for being seriously hurt and -10 for wasting time making threats. In other words, (1) if a falcon meets a pigeon, the pigeon withdraws and obtains a zero result and the falcon gets 50 points; (2) if a falcon meets another falcon, they fight to the bitter end when one wins 50 points and the other loses -100; and (3) if a pigeon meets another pigeon, both make threats for a long time until one of them gives in with a probability of 0.50, so that both win on average 15 points (= [50 - 10] 0.50 + [-10] 0.50).
a) Let us suppose, first, that there is no way of knowing in advance if an individual is a falcon or a pigeon until the fight begins. Start with a population of pigeons. What result is obtained? Imagine that a mutant falcon appears. What would the result be? Suppose that, with their better results, the mutant falcons proliferate and the population is now made up of falcons, what result would they obtain and what result would a mutant pigeon obtain? What would the composition of an equilibrium population be? Why would it be an equilibrium population? Show graphically the results of the falcons and pigeons in terms of the proportion of pigeons.
b) Let us now suppose that it is possible to distinguish whether an individual is a falcon or pigeon before the fight starts, that both types of animals can choose before fighting whether to undertake or not the detection, and that the costs of detection amount to 5 units.
Based on Dawkins (1986, pp. 104-109) following a model by Maynard Smith
a) Starting with a population of pigeons, these obtain 15 and the mutant falcon 50. If the population then comprises falcons, their average result would be -25 (= 50 0.50-100 0.50), which is much lower than the zero result of the mutant pigeon. The equilibrium proportion is 7/12 of falcons and 5/12 of pigeons (or, which would amount to the same thing, animals behaving 7/12 times as falcons and 5/12 times as pigeons), and in this situation both types obtain 6.25 on average and any deviation would lead to changes in the opposite direction which would lead the population back to equilibrium. The functions are: Rp = 15 rp and Rh = 75 rp - 25 for falcons and pigeons respectively.
b) In this case, a pigeon will never decide to incur the detection cost of 5 units. If she decides not to detect, she can obtain 15 (in case of facing a pigeon) or 0 (in case of facing a falcon). If she detects, she can obtain 15-5=10 (in case of facing a pigeon) or -5 (in case of facing a falcon). Therefore, for a pigeon, Rp=15 rp (the same as in (a)).
A falcon will decide to detect or not depending on which is the proportion of pigeons. If he does not detect, he can obtain 50 or -25 (that is, he would get the same function as in (a)) If he decides to detect, he can obtain 45 (in case of facing a pigeon or -5 (in case of facing a falcon, he will decide not to attack). Therefore, Rf = 50rp-5. The falcon’s decision is to detect if and only if: 50rp-5 > 75rp-25; that is, he will detect if and only if rp < 0.8.
Drawing Rp = 15rp and Rf = 50rp-5 when rp < 0.8, and Rf = 75rp-25 when rp > 0.8, the new equilibrium is in rp = 1/7 (where 15 rp=50rp-5). Since in the new equilibrium falcons detect, it is logic that the proportion of pigeons is lower.
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