Coevolution using adaptive dynamics

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Co-evolution using adaptive dynamics
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Flashback to last week

• resident strain x
• - at equilibrium

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Flashback to last week

• resident strain x
• mutant strain y

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Flashback to last week

• resident strain x
• mutant strain y
• Fitness sx(y) lt 0

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Flashback to last week

• resident strain x

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Flashback to last week

• resident strain x
• mutant strain y
• Fitness sx(y) gt 0

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Flashback to last week

• resident strain x
• mutant strain y
• Fitness sx(y) gt 0

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Flashback to last week

• mutant strain y

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Flashback to last week

• mutant strain y ?
• resident strain x

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Flashback to last week

• This continues

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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.

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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)/?yiyixi for i1,2

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Singularities

• Points in evolution.
• In co-evolution, fitness gradient is a function
  of x1 and x2
• Solving ?sx1(y1)/?y1y1x1x10 gives
  x1x1(x2)
• Likewise ?sx2(y2)/?y2y2x2x20 ?
  x2x2(x1)

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Plotting the singular curves

• (x1,x2) co-evolutionary singularity

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Taylor Expansion
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Evaluating at y1x1
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Fitness functions
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ESS

• Co-evolutionary singularity ESS iff
• and

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Convergence Stability
The canonical equation
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Convergence Stability
The canonical equation In co-evolution
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CS continued

• Simplifies to

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CS continued

• Simplifies to
• Signs of the eigenvalues ?1 and ?2 determine the
  type of co-evolutionary singularity
• ?1, ?2 lt 0
  ?1, ?2 gt 0 ?1 lt 0, ?2 gt 0 (vv)

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

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Fitness functions

• Fitness for prey
• Giving

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ESS CS

• ESS a lt 0 and b gt 0
• CS
• Derive conditions, on a and b, for various types
  of co-evolutionary singularity

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Types of singularity
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Running simulations
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Simulations cont
Prey branching
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Simulations cont
Predator branching
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Simulations cont
Both prey and predator branching
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The problem

• Should be branching, branching

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Solutions??

• Two singularities in close proximity.
• Look more locally about each one.
• Develop a more global theory!