1. Introduction to Adaptive Dynamics

2. Definition • Adaptive dynamics looks at the long term effects of small mutations on a system. • If mutant invades monomorphic population, we can tell if invasion is successful. • Can be applied to various ecological settings. • Give conditions for each possible evolutionary outcome.

3. Why AD? • Community Dynamics (having known number of strains); using the Jacobian method we obtain information about start and end points but not how we get from one to other. • Adaptive Dynamics (an infinite number of strains) gives us information about start and end points and also the path it takes.

4. Fitness • Fitness is the long term population growth rate of a rare mutant strategy. • x is resident strategy, y is mutant strategy, Ex is environment with x at equilibrium, ρ is smooth function of strategy and environmental parameters (i.e. good environment, ρ +ve, population grows, poor environment, ρ –ve, population decreases).

5. Sx(y) > 0 mutant population may increase. • Sx(y) < 0 mutant population will die out. • Small mutations implies x and y are similar so linear approx of fitness is

6. Sx(x) = 0 and is the local fitness gradient. • D(x) > 0, y > x or D(x) < 0, y < x then y can invade x. • D(x) = 0 at evolutionary singular strategy, x*. • D(x) tells us what direction population evolves in so with y near x, sx(y) > 0 implies sy(x) < 0, i.e. x cannot recover once mutant is common and x rare.

7. Properties of x* • ESS is evolutionary trap, i.e. once established in a population no further evolutionary change is possible. • CS is evolutionary attractor, i.e. any nearby mutant strategy evolves towards the evolutionary singular strategy.

8. Evolutionary Outcomes • CS and ESS – Evolutionary attractor. • CS not ESS – Evolutionary Branching Point. • ESS not CS – Garden of Eden Point. • Neither ESS nor CS – Evolutionary Repellor.

9. Pairwise Invadability Plots • These represent the spread of mutants in a given population. • Indicate the sign of sx(y) for all possible values of x and y. • Along main diagonal sx(y) is zero. • +ve above and -ve below indicates positive fitness gradient. • -ve above and +ve below indicates negative fitness gradient. • Contains another line where sx(y)=0 and intersection of this with main diagonal corresponds to singular strategy.

11. Example • Different methods to find invading eigenvalue: • Jacobian Method • Invasion Analysis • Reading off model (model dependent) • N* and Y* are the steady state values.

12. Substituting for Y* & N* and introducing trade-off r = f(β), and β=x, r = f(x) and βm=y, rm = f(y) • Using the above we find • Setting this equal to zero gives a solution for x*.

13. Condition for ESS • Condition for CS • Combinations of above inequalities will give either Attractors, Repellors, Branching points or Garden of Eden point as discussed earlier.

14. We use a concave trade-off. In this case it is neither ESS or CS so we get a repellor: