Evoluvon de cooperacion!

Question 3: Evolution of co-operation
The model for the evolution of co-operation discussed in the lectures can be
rewritten in terms of continuous behavioural strategies. Let 0 _ r _ 1 be thedegree of investment in co-operation by the resident and 0 _ m _ 1 be the
investment by the mutant. Assuming these costs and bene_ts of co-operation
increase linearly with investment, then the invasion_tness of a small number
of mutants can be written as
S(r;m) = b(r + m)
where b and c are constants.
1. Assuming a mutant invades if S(r;m) > 0 draw pairwise invasibility
plots for the casewhere b > c and the case where b < c.
2. Derive a continuous time population dynamics model (similar to the
replicator equation in the lecture notes) for a population of residents
and mutants withinvasion _tness S(r;m). To do this, assume that x
and xm are proportions of residents and mutants respectively and write
down equations for the rate of growth of each type. List the biologicalassumptions which your model makes.
3. Generalise the model so that instead of constants b and c, costs and
bene_ts are functions B(r + m) and C(r) of the degree of investment.
Now we have invasion _tnessS(r;m) = B(r + m)
Give an expression the selection gradient. Find an expression for the
singular strategies in terms of the functions B and C and conditions
for evolutionary stability of thesesingular strategies.
4. Now consider quadratic costs and bene_ts.
B(x) = ax2 + bx and C(x) = cx2 + dx
Show that the non-zero singular strategy is unique and _nd its value.
Find conditions forevolutionary stability of the singular strategy.
5
5. There are _ve qualitatively di_erent evolutionary outcomes with quadratic
costs: always no investment, always full investment, either fullinvestment
or no investment depending on initial conditions, intermediate
investment by all, or high investment by some, low investment by others.
Sketch pairwise invasibility plots for each of these...