Co-evolution Co-evolution using using

adaptive dynamicsadaptive dynamics

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resident strain x

- at equilibrium

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resident strain x

mutant strain y

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resident strain x

mutant strain y

Fitness: sx(y) < 0

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resident strain x

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resident strain x

mutant strain y

Fitness: sx(y) > 0

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resident strain x

mutant strain y

Fitness: sx(y) > 0

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mutant strain y

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mutant strain y ↓

resident strain x

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• 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: 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

Singularities

• Points in evolution.

• In co-evolution, fitness gradient is a function of x1 and x2

• Solving ∂sx1(y1)/∂y1|y1=x1=x1*=0 gives x1*=x1*(x2)

• Likewise ∂sx2(y2)/∂y2|y2=x2=x2*=0 → x2*=x2*(x1)

Plotting the singular curves

• (x1**,x2**) =co-evolutionary singularity

Taylor Expansion

2**222

2

12

**22

**11

21

12

**22

**11

21

12

2**112

1

12

**11

**11

11

12

2**112

1

12

**22

2

1**11

1

1**11

1

1**11

)-( ∂

∂

2

1)-()-(

∂ ∂

∂

)-()-( ∂ ∂

∂)-(

∂

∂

2

1

)-()-( ∂

∂)-(

∂

∂

2

1

)-( ∂

∂)-(

∂

∂)-(

∂

∂)()(

**i

**i

**i

**i

**i

**i

**i

**i

**i

**11

xxx

sxxxy

xy

s

xxxxxx

sxy

y

s

xyxxyx

sxx

x

s

xxx

sxy

y

sxx

x

sxsys

xx

xx

xx

xxxxx

Evaluating at y1=x1

2**22

x

22

12

x21

12

x21

12

**22

**11

x

21

12

x11

12

x

21

12

2**11

**22

x2

1**11

x1

1

)-( ∂

∂

2

1

∂ ∂

∂

∂ ∂

∂)-()-(

∂

∂

∂ ∂

∂2

∂

∂)-(

2

1

)-( ∂

∂)-(

∂

∂0

**i

**i

**i

**i

**i

**i

**i

**i

xxx

s

xy

s

xx

sxxxx

y

s

yx

s

x

sxx

xxx

sxx

x

s

Fitness functions

)-()-( ∂ ∂

∂2

)-( ∂

∂)-(

∂

∂)-(

2

1)(

**22

**11

21

12

**112

1

12

**112

1

12

111

**i

**i

**i

1

xxxxxx

s

xyy

sxx

x

syxys

x

xx

x

)-()-( ∂ ∂

∂2

)-( ∂

∂)-(

∂

∂)-(

2

1)(

**22

**11

21

22

**222

2

22

**222

2

22

222

**i

**i

**i

2

xxxxxx

s

xyy

sxx

x

syxys

x

xx

x

ESS

• Co-evolutionary singularity ESS iff:

and0**

21

12

ixy

s0

**22

22

ixy

s

Convergence Stability

111

11

1 xy

y

sX

dt

dx

The canonical equation:

Convergence Stability

111

11

1 xy

y

sX

dt

dx

The canonical equation:

In co-evolution:

22

11

2

22

1

11

2

1

xy

xy

y

sX

y

sX

x

x

CS continued…

**

22

**11

22

22

22

22

2

21

22

2

21

12

121

12

21

12

1

2

1

2

2

xx

xx

x

s

y

sX

xx

sX

xx

sX

x

s

y

sX

x

x

•Simplifies to:

CS continued…

**

22

**11

22

22

22

22

2

21

22

2

21

12

121

12

21

12

1

2

1

2

2

xx

xx

x

s

y

sX

xx

sX

xx

sX

x

s

y

sX

x

x

•Simplifies to:

•Signs of the eigenvalues λ1 and λ2 determine the type of co-evolutionary singularity:

λ1, λ2 < 0 λ1, λ2 > 0 λ1 < 0, λ2 > 0

(vv)

Predator-prey example

XZkZqZrdt

dZ

XZkXqXrdt

dX

zxzz

xzxx

2

2

YZkYXYqrdt

dYyzyy

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 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

Fitness functions

• Fitness for prey:

• Giving:

dt

dY

Yys

Yx

1lim

0

XkkZqkgws

ZkkXqkfys

wxzwz

zyxyx

ESS & CS

• ESS: a < 0 and b > 0

• CS:

Derive conditions, on a and b, for various types of co-evolutionary singularity

3

14

2

1

6

16

1

3

54

2

5

b

a

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!