Download presentation
Presentation is loading. Please wait.
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, 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).
5
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
6
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.
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 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.
11
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.
12
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*.
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:
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.