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

Flashback to last week resident strain x

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

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

Flashback to last week mutant strain y

Flashback to last week mutant strain y ↓ resident strain x

Flashback to last week This continues…

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.

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

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 )

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

Taylor Expansion

Evaluating at y 1 =x 1

Fitness functions

ESS Co-evolutionary singularity ESS iff: and

Convergence Stability The canonical equation:

Convergence Stability The canonical equation: In co-evolution:

CS continued… Simplifies to:

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)

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

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

Fitness functions Fitness for prey: Giving:

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

Types of singularity

Running simulations

Simulations cont… Prey branching

Simulations cont… Predator branching

Simulations cont… Both prey and predator branching

The problem… Should be branching, branching

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