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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. - PowerPoint PPT Presentation
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Co-evolution Co-evolution using using
adaptive dynamicsadaptive 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: sx(y) < 0
Flashback to last week
resident strain x
Flashback to last week
resident strain x
mutant strain y
Fitness: sx(y) > 0
Flashback to last week
resident strain x
mutant strain y
Fitness: sx(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: 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!