1. Week 6 Applications of ODEs to the evolution game theory
1. A two-strategy game: the hawks and the doves
2. The dynamics of games
3. Stability of equilibrium populations

2. 1. A two-strategy game: the hawks and the doves
۞ A population consists of individuals.
۞ Individuals compete for resources needed for their reproduction through elementary pair-wise contests.
۞ The outcome of a contest is measured by the payoffs to the contestants:
win (W) = 6
loss (L) = 0
injury (I) = –10
waste of time (WoT) = –1

3. ۞ Individuals follow one of the two strategies: hawks or doves.
Strategy1
Strategy2
Payoff1
Payoff2
Comment H D 6 H –2
= ½ (W + I) D 2
= ½ W + WoT
Remark:
A “strategy” implies either a species or a certain type of behaviour within a species.

4. ۞ The payoff matrix P is
Remark:
We shall denote matrices either in boldface (P) or with subscripts (Pij, where i and j take values of H or D), e.g.

5. Q: Which strategy is better?
A: It depends on the population.
Example 1:
In a populations of 99% doves, a hawk does better than a dove (the mean payoffs ≈ 6 and 2, respectively).
In a populations of 99% hawks, a hawk does worse than a dove (the mean payoffs ≈ –2 and 0, respectively).

6. ۞ Let’s characterise the composition of a population by the vector
where
Clearly,

7. For a given population x, the expected payoffs to the species are
(1b)
۞ FH and FD are called the fitnesses of hawks and doves. We also introduce the vector fitness,

8. In vector/matrix form,
i.e.
After carrying out the matrix multiplication, this formula coincides with (1).
۞ The fitness of the whole population is defined by

9. An important assumption:
Let’s assume that the payoffs Pij have been defined in such a way that the fitnesses represent the reproductive abilities of the corresponding species (basically, their birth rates).
Then, a population can only be in equilibrium (i.e. maintain the same composition in time) if and only if the fitnesses of all the species are equal.
۞ The equilibrium population (EP) is such that the fitnesses of the hawks and doves are equal, i.e.
where the asterisks indicate that the corresponding variables apply to the equilibrium population.

10. FH* FD* According to formulae (1), the equality FH*=FD* implies
By definition xH + xD = 1 (for any x, not necessarily x*) – hence,
Now, solve for xD*, then use the equality xH* = 1 – xD*:

11. ۞ EPs (and, generally, equilibria in physical, chemical, biological and other systems) can be stable, neutrally stable or unstable.
If a stable EP is perturbed, the system returns to its original state.
If a neutrally stable EP is perturbed, the system oscillates about the original state.
If an unstable EP is perturbed, the system drifts away, no matter how small the initial perturbation was.

12. Example 2:
A billiard ball on a ‘hilltop’ is unstable.
A billiard ball in a ‘dip’ is stable (provided friction between the ball and the supporting surface is taken into account).
If the friction is neglected, a billiard ball in a ‘dip’ can be regarded neutrally stable.

13. Q: Is the EP of the hawk–dove game stable?
A: Yes, because
which can be verified by a calculation or (better still) a diagram.

14. Things to pay attention to when drawing the diagram:
The horizontal axis represents both xH and xD (because these are related by the condition xH + xD = 1).
The vertical axis represents the fitnesses FH and FD, but these are not related (unlike xH and xD) – hence, there will be two separate graphs.
To draw the graphs of FH and FD, use formulae (1a,b).
Since formulae (1) are linear, the graphs of FH and FD are straight lines – hence, they are fully determined by just two points (e.g. the endpoints).
The intersection of the graphs of FH and FD represents the EP.

16. ۞ Stable EPs will be referred to as evolutionary stable populations (ESPs).
Our model is based on the following assumptions:
there are only two strategies,
a hawk’s always a hawk, a dove’s always a dove (never change strategies),
asexual reproduction,
a hawk always gives birth to a hawk, a dove always gives birth to dove,
resources grow proportionally with the population,
pair-wise contests,
the population is large.