Introduction to Adaptive Dynamics

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.

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.

Fitness  Fitness is the long term population growth rate of a rare mutant strategy.  x is resident strategy, y is mutant strategy, E x 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).

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

 S x (x) = 0 and is the local fitness gradient.  D(x) > 0, y > x or 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, s x (y) > 0 implies s y (x) 0 implies s y (x) < 0, i.e. x cannot recover once mutant is common and x rare.

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.

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.

Pairwise Invadability Plots  These represent the spread of mutants in a given population.  Indicate the sign of s x (y) for all possible values of x and y.  Along main diagonal s x (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 s x (y)=0 and intersection of this with main diagonal corresponds to singular strategy.

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

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

 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.

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