1. Co-evolution using adaptive dynamics

2. Flashback to last week resident strain x - at equilibrium

3. Flashback to last week resident strain x mutant strain y

4. Flashback to last week resident strain x mutant strain y Fitness: s x (y) < 0

5. Flashback to last week resident strain x

6. Flashback to last week resident strain x mutant strain y Fitness: s x (y) > 0

7. Flashback to last week resident strain x mutant strain y Fitness: s x (y) > 0

8. Flashback to last week mutant strain y

9. Flashback to last week mutant strain y ↓ resident strain x

10. Flashback to last week This continues…

11. Assumptions Assumptions of adaptive dynamics: –Population settles to a (point) equilibrium before mutations. –All individuals are identical and denoted by strategy, eg. x. Additional assumptions: –In co-evolution, only one mutation at any time.

12. Introduction to Co-evolution Two evolving strains: x 1 and x 2 Fitness functions: s x 1 (y 1 ) = s 1 (x 1,x 2,y 1 ) s x 2 (y 2 ) = s 2 (x 2,x 1,y 2 ) Fitness gradients ∂s x i (y i )/∂y i|yi=xi for i=1,2

13. Singularities Points in evolution. In co-evolution, fitness gradient is a function of x 1 and x 2 Solving ∂s x 1 (y 1 )/∂y 1|y 1 =x 1 =x 1 * =0 gives x 1 *=x 1 *(x 2 ) Likewise ∂s x 2 (y 2 )/∂y 2|y2=x2=x2* =0 → x 2 *=x 2 *(x 1 )

14. Plotting the singular curves ( x 1 **,x 2 ** ) =co-evolutionary singularity

15. Taylor Expansion

16. Evaluating at y 1 =x 1

17. Fitness functions

18. ESS Co-evolutionary singularity ESS iff: and

19. Convergence Stability The canonical equation:

20. Convergence Stability The canonical equation: In co-evolution:

21. CS continued… Simplifies to:

22. CS continued… Simplifies to: Signs of the eigenvalues λ 1 and λ 2 determine the type of co-evolutionary singularity: λ 1, λ 2 0 λ 1 0 (vv)

23. Predator-prey example Dynamics of the resident prey ( x ) and predator ( z ): A mutation in the prey ( y):

24. Trade-off Between intrinsic growth rates ( r ) and predation rates ( k ). Split k xz into k x k z Trade-offs: –r x = f(k x ) where f(k x ) = a(k x -1) 2 + k x + 1 –r z = g(k z ) where g(k z ) = b(k z -1) 2 + k z - 0.2

25. Fitness functions Fitness for prey: Giving:

26. ESS & CS ESS: a 0 CS: Derive conditions, on a and b, for various types of co-evolutionary singularity

27. Types of singularity

28. Running simulations

29. Simulations cont… Prey branching

30. Simulations cont… Predator branching

31. Simulations cont… Both prey and predator branching

32. The problem… Should be branching, branching

33. Solutions?? Two singularities in close proximity. Look more “locally” about each one. Develop a more global theory!