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Co-evolution using adaptive dynamics

Co-evolution using adaptive dynamics. Flashback to last week. resident strain x - at equilibrium. Flashback to last week. resident strain x mutant strain y. Flashback to last week. resident strain x mutant strain y Fitness: s x (y) < 0.

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Co-evolution using adaptive dynamics

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  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: sx(y) < 0

  5. Flashback to last week resident strain x

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

  7. Flashback to last week resident strain x mutant strain y Fitness: sx(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: x1 and x2 • Fitness functions: sx1(y1) = s1(x1,x2,y1) sx2(y2) = s2(x2,x1,y2) • Fitness gradients ∂sxi(yi)/∂yi|yi=xi for i=1,2

  13. Singularities • Points in evolution. • In co-evolution, fitness gradient is a function of x1 and x2 • Solving ∂sx1(y1)/∂y1|y1=x1=x1*=0gives x1*=x1*(x2) • Likewise ∂sx2(y2)/∂y2|y2=x2=x2*=0→x2*=x2*(x1)

  14. Plotting the singular curves • (x1**,x2**) =co-evolutionary singularity

  15. Taylor Expansion

  16. Evaluating at y1=x1

  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 λ1 andλ2 determine the type of co-evolutionary singularity: • λ1, λ2 < 0 λ1, λ2 > 0 λ1 < 0, λ2 > 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 kxz into kxkz • Trade-offs: • rx = f(kx) where f(kx) = a(kx-1)2 + kx + 1 • rz = g(kz) where g(kz) = b(kz-1)2 + kz - 0.2

  25. Fitness functions • Fitness for prey: • Giving:

  26. ESS & CS • ESS: a < 0 and b > 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!

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