Hawk-Dove Invasion Stability
Compare a rare invader with a resident population and discover when aggression stabilizes at a mixture rather than a pure strategy.
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Hawk-Dove Invasion Stability
The Hawk-Dove game is a foundational model in evolutionary game theory used to study the evolution of aggressive behavior. It asks whether a rare mutant or invader can successfully integrate into a resident population, and determines what proportion of strategies is stable against all such invasions.
Worked Example
Let's consider a resident population containing 25% Hawks and 75% Doves (x=0.25). We set the resource value to V=1 and the injury cost to C=2. What happens if a rare mutant that strictly plays Hawk (q=1) attempts to invade?
- Calculate the average payoffs of being a Hawk (f_H) and being a Dove (f_D) in the resident population:
- f_H = V(1-x) + x(V-C)/2 = 1(0.75) + 0.25(1-2)/2 = 0.75 - 0.125 = 0.625
- f_D = V(1-x)/2 = 1(0.75)/2 = 0.375
- Determine the resident's mean fitness:
E(x,x) = x f_H + (1-x) f_D = 0.25(0.625) + 0.75(0.375) = 0.15625 + 0.28125 = 0.4375 - Determine the fitness of the rare invader (pure Hawk, q=1):
E(q,x) = q f_H + (1-q) f_D = 1(0.625) + 0(0.375) = 0.625 - The rare invader's advantage over the resident is:
0.625 - 0.4375 = 0.1875. Since this is positive, the pure Hawk can successfully invade a population composed mostly of Doves. The resident is therefore not an Evolutionarily Stable Strategy (ESS).
Experiment
Use the interactive Hawk-Dove Invasion Stability model below to compare fitness and advantages for different combinations of Resource value (V) and Injury cost (C).
- Start by setting Resource value to 1 and Injury cost to 2.
- Set the Resident hawk fraction to 0.5 and the Invader hawk fraction to 1.
- Observe that the Rare invader advantage becomes exactly 0. At this point, the first-order condition is zero. We must inspect the second ESS condition via Second ESS comparison to see if the mutant can drift to take over.
- Adjust the Resident hawk fraction back and forth to see how the rare invader advantage changes on the plotted curve. When V < C, the system stabilizes at a mixture of V/C.
Limitations
This basic Hawk-Dove replicator model relies on several key simplifications. It assumes infinite, well-mixed populations where individuals encounter one another perfectly randomly without spatial structure. In finite biological populations, stochastic drift or kin selection effects can allow mutants to either die out purely by chance or succeed despite a small initial disadvantage.
References
The payoff structures and the distinction between neutral first-order invasion and the second ESS condition are based on foundational lectures on game theory mechanics.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Hawk-Dove Invasion Stability module.
Try Hawk-Dove Invasion Stability on Riano →