Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Evolutionary distributions
Definitions of
ED
A mathematical framework for evolutionary ecology
Applications
Yosef Cohen
University of Minnesota St. Paul, Minnesota
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Outline
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of ED
Applications
Competition
Single-trait competition
Two-traits competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual parasitism
Conclusions
Extensions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Key references
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Key references
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Cohen, Y. 2003. Distributed evolutionary games.
Evolutionary Ecology Research 5:1-14.
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Key references
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Cohen, Y. 2003. Distributed evolutionary games.
Evolutionary Ecology Research 5:1-14.
Cohen, Y. 2003. Distributed predator prey coevolution.
Evolutionary Ecology Research 5: 819-834.
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Key references
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Cohen, Y. 2003. Distributed evolutionary games.
Evolutionary Ecology Research 5:1-14.
Cohen, Y. 2003. Distributed predator prey coevolution.
Evolutionary Ecology Research 5: 819-834.
Cohen Y. 2005 Evolutionary distributions in adaptive
space. Journal of Applied Mathematics 2005:
403–424.
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
We start with the case of a single population density,
z and a single adaptive trait x.
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
We start with the case of a single population density,
z and a single adaptive trait x.
I
Then
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
We start with the case of a single population density,
z and a single adaptive trait x.
I
Then
z 0 = f (z, x, t) .
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
We start with the case of a single population density,
z and a single adaptive trait x.
I
Then
z 0 = f (z, x, t) .
I
Next, we derive the strategy dynamics in some way
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
We start with the case of a single population density,
z and a single adaptive trait x.
I
Then
z 0 = f (z, x, t) .
I
Next, we derive the strategy dynamics in some way
x0 = g (z, x, t)
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
From evolutionary games to evolutionary
distributions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
We start with the case of a single population density,
z and a single adaptive trait x.
I
Then
z 0 = f (z, x, t) .
I
Next, we derive the strategy dynamics in some way
x0 = g (z, x, t)
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
I
and solve for x (and sometimes for z also) to obtain
stability or dynamics in a game theoretic context.
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
I
Decompose f to components that reflect growth and
decline:
f (z, x, t) = βe (z, x, t) − µ (z, x, t) .
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
I
Decompose f to components that reflect growth and
decline:
f (z, x, t) = βe (z, x, t) − µ (z, x, t) .
I
There are good reasons to assume that βe is linear. So
we write
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
I
Decompose f to components that reflect growth and
decline:
f (z, x, t) = βe (z, x, t) − µ (z, x, t) .
I
There are good reasons to assume that βe is linear. So
we write
βe (z, x, t) = βz(x, t).
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
I
Decompose f to components that reflect growth and
decline:
f (z, x, t) = βe (z, x, t) − µ (z, x, t) .
I
There are good reasons to assume that βe is linear. So
we write
βe (z, x, t) = βz(x, t).
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
I
Assume random mutations on progeny with fraction
η.
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
I
Decompose f to components that reflect growth and
decline:
f (z, x, t) = βe (z, x, t) − µ (z, x, t) .
I
There are good reasons to assume that βe is linear. So
we write
βe (z, x, t) = βz(x, t).
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
I
Assume random mutations on progeny with fraction
η.
So ...
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
∂t z (x, t) = (1 − η) βz (x, t) +
1
βη [z (x + ∆) + z (x − ∆)] − µ (z, x, t) .
2
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
∂t z (x, t) = (1 − η) βz (x, t) +
1
βη [z (x + ∆) + z (x − ∆)] − µ (z, x, t) .
2
With Taylor series expansion of z around x, we obtain
approximately
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary Distributions (ED)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
∂t z (x, t) = (1 − η) βz (x, t) +
1
βη [z (x + ∆) + z (x − ∆)] − µ (z, x, t) .
2
With Taylor series expansion of z around x, we obtain
approximately
1
∂t z = z + ∆2 βη∂xx z − µ (z, x, t) .
2
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
ED (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
For a single ED with m orthogonal adaptive traits, we
have
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
ED (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
For a single ED with m orthogonal adaptive traits, we
have
m
X
1
ηi ∂xi xi z − µ (z, x, t) .
∂t z = z + ∆2 β
2
i=1
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Outline
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of ED
Applications
Competition
Single-trait competition
Two-traits competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual parasitism
Conclusions
Extensions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Define the mth order mutation operator
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Define the mth order mutation operator
Introduction
mA
:= 1 + k
m
X
i=1
Key references
Games vs ED
ηi ∂xi xi
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Define the mth order mutation operator
Introduction
mA
:= 1 + k
m
X
i=1
Key references
Games vs ED
ηi ∂xi xi
Definitions of
ED
Applications
where k :=
∆2 β/2.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Define the mth order mutation operator
Introduction
mA
:= 1 + k
m
X
Key references
Games vs ED
ηi ∂xi xi
i=1
Definitions of
ED
Applications
where k :=
∆2 β/2.
zi ∈ R0+ , i = 1, . . ., n is the distribution of the density of
types with mi adaptive traits xi .
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Define the mth order mutation operator
Introduction
mA
:= 1 + k
m
X
Key references
Games vs ED
ηi ∂xi xi
i=1
Definitions of
ED
Applications
where k :=
∆2 β/2.
zi ∈ R0+ , i = 1, . . ., n is the distribution of the density of
types with mi adaptive traits xi .
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
x = [x1 , . . . , xn ].
Extensions
Definitions of ED
Evolutionary
distributions
Yosef Cohen
Define the mth order mutation operator
Introduction
mA
:= 1 + k
m
X
Key references
Games vs ED
ηi ∂xi xi
i=1
Definitions of
ED
Applications
where k :=
∆2 β/2.
zi ∈ R0+ , i = 1, . . ., n is the distribution of the density of
types with mi adaptive traits xi .
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
x = [x1 , . . . , xn ].
Define
the bounded open set X ⊂ RM (where M =
Pn
i=1 mi ) with boundary ∂X . Then ...
Extensions
Evolutionary
distributions
Definition
A linear ED, zi (x, t), is the solution of the system
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A linear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A linear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
(where F is some functional) with the data
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A linear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
(where F is some functional) with the data
zi (x, 0) = z0 (x)
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A linear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
(where F is some functional) with the data
zi (x, 0) = z0 (x)
and (Neumann boundary conditions)
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
∂xi zi (x, t)|x=∂X = 0, i = 1, . . . , n.
Extensions
Evolutionary
distributions
Definition
A nonlinear ED, zi (x, t), is the solution of the system
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A nonlinear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (z, x, t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A nonlinear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (z, x, t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
(where F is some functional) with the data
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A nonlinear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (z, x, t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
(where F is some functional) with the data
zi (x, 0) = z0 (x)
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Evolutionary
distributions
Yosef Cohen
Definition
A nonlinear ED, zi (x, t), is the solution of the system
∂t zi (x, t) = βi (z, x, t)
mi Azi (xi , t)
− µi (z, F (x) , t) ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
(where F is some functional) with the data
zi (x, 0) = z0 (x)
and (Neumann boundary conditions)
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
∂xi zi (x, t)|x=∂X = 0, i = 1, . . . , n.
Extensions
Outline
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of ED
Applications
Competition
Single-trait competition
Two-traits competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual parasitism
Conclusions
Extensions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Applications
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Applications
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
With this framework, we can now port all point
process population ecology models.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Applications
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
With this framework, we can now port all point
process population ecology models.
I
Here are some applications ....
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
r
z 0 = rz − z 2 .
k
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
r
z 0 = rz − z 2 .
k
Key references
Games vs ED
Definitions of
ED
Applications
The linear ED is
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
r
z 0 = rz − z 2 .
k
Key references
Games vs ED
Definitions of
ED
Applications
The linear ED is
r
∂t z = rAz − z 2 ,
k
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
r
z 0 = rz − z 2 .
k
Key references
Games vs ED
Definitions of
ED
Applications
The linear ED is
r
∂t z = rAz − z 2 ,
k
with data
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
r
z 0 = rz − z 2 .
k
Key references
Games vs ED
Definitions of
ED
Applications
The linear ED is
r
∂t z = rAz − z 2 ,
k
with data
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
z (x, 0) = 20 + sin (x) ,
∂x z (π/2, t) = ∂x z (9π/2, t) = 0,
Extensions
Competition - single trait without selection
Evolutionary
distributions
Yosef Cohen
The point process is
Introduction
r
z 0 = rz − z 2 .
k
Key references
Games vs ED
Definitions of
ED
Applications
The linear ED is
r
∂t z = rAz − z 2 ,
k
with data
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
z (x, 0) = 20 + sin (x) ,
∂x z (π/2, t) = ∂x z (9π/2, t) = 0,
we obtain ...
Extensions
No selection
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
EDFramework-2d.nb
No selection
Evolutionary
distributions
Yosef Cohen
30
t
Introduction
Key references
Games vs ED
20
Definitions of
ED
10
Applications
0
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
100
80
z 60
40
Conclusions
20
Extensions
5
x
10
Competition - single trait with selection
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
Evolutionary
distributions
Yosef Cohen
Introduction
Assume single trait adaptation to competition and best
adaptation to some value of carrying capacity. Then ...
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
Evolutionary
distributions
Yosef Cohen
Introduction
Assume single trait adaptation to competition and best
adaptation to some value of carrying capacity. Then ...
"
#!
1 x−ξ 2
α (x, ξ) = kα 1 + k exp −
2
σα
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
Evolutionary
distributions
Yosef Cohen
Introduction
Assume single trait adaptation to competition and best
adaptation to some value of carrying capacity. Then ...
"
#!
1 x−ξ 2
α (x, ξ) = kα 1 + k exp −
2
σα
and
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
Evolutionary
distributions
Yosef Cohen
Introduction
Assume single trait adaptation to competition and best
adaptation to some value of carrying capacity. Then ...
"
#!
1 x−ξ 2
α (x, ξ) = kα 1 + k exp −
2
σα
and
"
k (x) = km
1
1 + exp −
2
x − 5π/2
σk
2 #!
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
Evolutionary
distributions
Yosef Cohen
Introduction
Assume single trait adaptation to competition and best
adaptation to some value of carrying capacity. Then ...
"
#!
1 x−ξ 2
α (x, ξ) = kα 1 + k exp −
2
σα
and
"
k (x) = km
1
1 + exp −
2
x − 5π/2
σk
2 #!
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
and the linear ED is now ...
Competition - single trait with selection
(continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
(continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
rAz − rF (x) z (x, t) ,
Z 9π/2
1
F (x) :=
α (x, ξ) z (ξ, t) dξ
k (x) π/2
∂t z
and data
=
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
(continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
rAz − rF (x) z (x, t) ,
Z 9π/2
1
F (x) :=
α (x, ξ) z (ξ, t) dξ
k (x) π/2
∂t z
=
and data
z (x, 0) = 0.005,
∂x z (π/2, t) = ∂x z (9π/2, t) = 0
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Competition - single trait with selection
(continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
rAz − rF (x) z (x, t) ,
Z 9π/2
1
F (x) :=
α (x, ξ) z (ξ, t) dξ
k (x) π/2
∂t z
=
and data
z (x, 0) = 0.005,
∂x z (π/2, t) = ∂x z (9π/2, t) = 0
and we obtain ...
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Single trait selection for α and k
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
EDFramework-2d.nb
Single trait selection for α and k
Evolutionary
distributions
Yosef Cohen
0 0.2
t
0.4
Introduction
0.6
Key references
Games vs ED
0.8
1
8
Definitions of
ED
Applications
6
4 z
2
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
0
5
10
x
Extensions
Two-traits competition
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits competition
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
x1 selected for carrying capacity
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits competition
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
x1 selected for carrying capacity
I
x2 selected for competitive ability
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits competition
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
x1 selected for carrying capacity
I
x2 selected for competitive ability
I
The traits are orthogonal
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits competition
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
x1 selected for carrying capacity
I
x2 selected for competitive ability
I
The traits are orthogonal
Then ...
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
r 2 Az − rF (x) z,
Z 9π/2
1
α (x2 , ξ) z (x1 , ξ, t) dξ,
F (x) :=
k (x1 ) π/2
∂t z
=
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
r 2 Az − rF (x) z,
Z 9π/2
1
α (x2 , ξ) z (x1 , ξ, t) dξ,
F (x) :=
k (x1 ) π/2
∂t z
and data
=
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
r 2 Az − rF (x) z,
Z 9π/2
1
α (x2 , ξ) z (x1 , ξ, t) dξ,
F (x) :=
k (x1 ) π/2
∂t z
=
and data
z (x, 0) = 20
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
∂x1 z (π/2, x2 , t) = ∂x1 z (9π/2, x2 , t) = 0,
Conclusions
∂x2 z (x1 , π/2, t) = ∂x2 z (x1 , 9π/2, t) = 0,
Extensions
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
r 2 Az − rF (x) z,
Z 9π/2
1
α (x2 , ξ) z (x1 , ξ, t) dξ,
F (x) :=
k (x1 ) π/2
∂t z
=
and data
z (x, 0) = 20
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
∂x1 z (π/2, x2 , t) = ∂x1 z (9π/2, x2 , t) = 0,
Conclusions
∂x2 z (x1 , π/2, t) = ∂x2 z (x1 , 9π/2, t) = 0,
Extensions
we obtain ...
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
EDFramework-2d.nb
Two-traits single ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
6
4
z
2
5
x2
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
5
10
10
x1
Predator prey
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Next, an application with regard to predator prey.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Next, an application with regard to predator prey.
We start with the point process and then move on to ED
...
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey - point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey - point process
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey - point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Let
z1 prey
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey - point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Let
z1 prey
z2 predator
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey - point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Let
z1 prey
z2 predator
r
az1
z10 = rz1 − z12 −
z2 ,
k
b + cz1
az1
z20 = d
z2 − µz22 .
b + cz1
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey - point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Let
z1 prey
z2 predator
r
az1
z10 = rz1 − z12 −
z2 ,
k
b + cz1
az1
z20 = d
z2 − µz22 .
b + cz1
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
With certain parameter values we obtain ...
Limit cycle
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Limit cycle
Evolutionary
distributions
Yosef Cohen
z
preythin,predatorthick
70
60
50
40
30
20
10
0
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
0 200 400 600 8001000
t
Extensions
Predator prey ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
z1 evolves on x1
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
z1 evolves on x1
I
z2 evolves on x2
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
z1 evolves on x1
I
z2 evolves on x2
I
Predation is at its maximum when x1 = x2 with
some phenotypic plasticity σ
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
z1 evolves on x1
I
z2 evolves on x2
I
Predation is at its maximum when x1 = x2 with
some phenotypic plasticity σ
I
Then ...
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Predator prey ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
z1 evolves on x1
I
z2 evolves on x2
I
Predation is at its maximum when x1 = x2 with
some phenotypic plasticity σ
I
Then ...
Applications
"
1
α (x1 , x2 ) = exp −
2
x1 − x2
σ
2 #
.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
zi ≡ zi (x1 , x2 , t) ,
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
zi ≡ zi (x1 , x2 , t) ,
Definitions of
ED
Applications
1
Az1 := z1 + ∆2 η1 ∂x1 x1 z1
2
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
zi ≡ zi (x1 , x2 , t) ,
Definitions of
ED
Applications
1
Az1 := z1 + ∆2 η1 ∂x1 x1 z1
2
and
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
zi ≡ zi (x1 , x2 , t) ,
Definitions of
ED
Applications
1
Az1 := z1 + ∆2 η1 ∂x1 x1 z1
2
and
1
Az2 := z2 + ∆2 η2 ∂x2 x2 z1 .
2
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The mutation operators
Evolutionary
distributions
Yosef Cohen
Introduction
Let
Key references
Games vs ED
zi ≡ zi (x1 , x2 , t) ,
Definitions of
ED
Applications
1
Az1 := z1 + ∆2 η1 ∂x1 x1 z1
2
and
1
Az2 := z2 + ∆2 η2 ∂x2 x2 z1 .
2
Then the point process becomes ...
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two ED two traits
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two ED two traits
Evolutionary
distributions
Yosef Cohen
az1
r
z2 ,
∂t z1 = rAz1 − z12 − α (x)
k
b + cz1
az1
∂t z2 = dα (x)
Az2 − µz22 ,
b + cz1
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two ED two traits
Evolutionary
distributions
Yosef Cohen
az1
r
z2 ,
∂t z1 = rAz1 − z12 − α (x)
k
b + cz1
az1
∂t z2 = dα (x)
Az2 − µz22 ,
b + cz1
with initial conditions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two ED two traits
Evolutionary
distributions
Yosef Cohen
az1
r
z2 ,
∂t z1 = rAz1 − z12 − α (x)
k
b + cz1
az1
∂t z2 = dα (x)
Az2 − µz22 ,
b + cz1
with initial conditions
z1 (x, 0) = 10,
z2 (x, 0) = 1
and boundary conditions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Two ED two traits
Evolutionary
distributions
Yosef Cohen
az1
r
z2 ,
∂t z1 = rAz1 − z12 − α (x)
k
b + cz1
az1
∂t z2 = dα (x)
Az2 − µz22 ,
b + cz1
with initial conditions
z1 (x, 0) = 10,
z2 (x, 0) = 1
and boundary conditions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
∂x1 z1 (π/2, x2 , t) = ∂x1 z1 (9π/2, x2 , t) = 0,
∂x2 z2 (x1 , π/2, t) = ∂x2 z2 (x1 , 9π/2, t) = 0.
Two ED two traits
Evolutionary
distributions
Yosef Cohen
az1
r
z2 ,
∂t z1 = rAz1 − z12 − α (x)
k
b + cz1
az1
∂t z2 = dα (x)
Az2 − µz22 ,
b + cz1
with initial conditions
z1 (x, 0) = 10,
z2 (x, 0) = 1
and boundary conditions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
∂x1 z1 (π/2, x2 , t) = ∂x1 z1 (9π/2, x2 , t) = 0,
∂x2 z2 (x1 , π/2, t) = ∂x2 z2 (x1 , 9π/2, t) = 0.
Now ...
Phenotypic plasticity σ = π/3
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Phenotypic plasticity σ = π/3
Evolutionary
distributions
Yosef Cohen
EDFramework-2d.nb
1
Introduction
Key references
Games vs ED
Prey
Definitions of
ED
Predator
Applications
10
100
7.5
99.5
z1
5 z2
2.5
99
5
x2
5
5
10
10
x1
x2
0
5
10
10
x1
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Phenotypic plasticity σ = π
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Phenotypic plasticity σ = π
Evolutionary
distributions
Yosef Cohen
EDFramework-2d.nb
Introduction
Key references
Games vs ED
Definitions of
ED
Predator
Prey
Applications
99.8
8
99.7
z1
99.6
6
99.5
5
x2
99.4
5
10
10
x1
5
x2
5
10
10
x1
z2
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
1
Host pathogen
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
The model is from ...
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
The model is from ...
Anderson and May (1980, equations 3,5 and 6; 1981).
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
z1 - density of host
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
z1 - density of host
I
z2 - density of infected host
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
z1 - density of host
I
z2 - density of infected host
I
z3 - density of pathogens
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
z1 - density of host
I
z2 - density of infected host
I
z3 - density of pathogens
We have ...
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
z10 = (a − b) z1 − α
e z2 ,
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
z10 = (a − b) z1 − α
e z2 ,
z20
= νz3 (z1 − z2 ) − (e
α + b + γ) z2 ,
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
z10 = (a − b) z1 − α
e z2 ,
z20
= νz3 (z1 − z2 ) − (e
α + b + γ) z2 ,
z30 = λz2 − (e
µ + νz1 ) z3 .
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
z10 = (a − b) z1 − α
e z2 ,
z20
= νz3 (z1 − z2 ) − (e
α + b + γ) z2 ,
z30 = λz2 − (e
µ + νz1 ) z3 .
The relevant parameters are
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
z10 = (a − b) z1 − α
e z2 ,
z20
= νz3 (z1 − z2 ) − (e
α + b + γ) z2 ,
z30 = λz2 − (e
µ + νz1 ) z3 .
The relevant parameters are
α
e additional death rate due to infection
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The point process (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
z10 = (a − b) z1 − α
e z2 ,
z20
= νz3 (z1 − z2 ) − (e
α + b + γ) z2 ,
z30 = λz2 − (e
µ + νz1 ) z3 .
The relevant parameters are
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
α
e additional death rate due to infection
Conclusions
µ
e death rate of infective stages
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
x1 - adaptive trait that affects death rate of hosts
due to infection (α)
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
I
x1 - adaptive trait that affects death rate of hosts
due to infection (α)
x2 - adaptive trait that affects pathogen death rate of
infective stages (µ)
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
x1 - adaptive trait that affects death rate of hosts
due to infection (α)
I
x2 - adaptive trait that affects pathogen death rate of
infective stages (µ)
I
At some value of x1 the value of α is at its minimum
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
x1 - adaptive trait that affects death rate of hosts
due to infection (α)
I
x2 - adaptive trait that affects pathogen death rate of
infective stages (µ)
I
At some value of x1 the value of α is at its minimum
I
At some value of x2 the value of µ is at its minimum
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
x1 - adaptive trait that affects death rate of hosts
due to infection (α)
I
x2 - adaptive trait that affects pathogen death rate of
infective stages (µ)
I
At some value of x1 the value of α is at its minimum
I
At some value of x2 the value of µ is at its minimum
Then ...
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Coevolution
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Coevolution
Evolutionary
distributions
Yosef Cohen
"
1
α (x1 ) = α
e 1 − 0.1 exp −
2
x1 − 5π/2
σα
2 #!
Introduction
,
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Coevolution
Evolutionary
distributions
Yosef Cohen
"
1
α (x1 ) = α
e 1 − 0.1 exp −
2
x1 − 5π/2
σα
2 #!
x2 − 5π/2
σµ
2 #!
Introduction
,
Key references
Games vs ED
Definitions of
ED
Applications
"
1
µ (x2 ) = µ
e 1 + 0.1 exp −
2
.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Coevolution
Evolutionary
distributions
Yosef Cohen
"
1
α (x1 ) = α
e 1 − 0.1 exp −
2
x1 − 5π/2
σα
2 #!
x2 − 5π/2
σµ
2 #!
Introduction
,
Key references
Games vs ED
Definitions of
ED
Applications
"
1
µ (x2 ) = µ
e 1 + 0.1 exp −
2
Let
.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Coevolution
Evolutionary
distributions
Yosef Cohen
"
1
α (x1 ) = α
e 1 − 0.1 exp −
2
x1 − 5π/2
σα
2 #!
x2 − 5π/2
σµ
2 #!
Introduction
,
Key references
Games vs ED
Definitions of
ED
Applications
"
1
µ (x2 ) = µ
e 1 + 0.1 exp −
2
Let
1
Az1 = z1 + ∆2 η1 ∂x1 x1 ,
2
.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Coevolution
Evolutionary
distributions
Yosef Cohen
"
1
α (x1 ) = α
e 1 − 0.1 exp −
2
x1 − 5π/2
σα
2 #!
x2 − 5π/2
σµ
2 #!
Introduction
,
Key references
Games vs ED
Definitions of
ED
Applications
"
1
µ (x2 ) = µ
e 1 + 0.1 exp −
2
Let
1
Az1 = z1 + ∆2 η1 ∂x1 x1 ,
2
.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
1
Az3 = z3 + ∆2 η2 ∂x2 x2 .
2
Coevolution
Evolutionary
distributions
Yosef Cohen
"
1
α (x1 ) = α
e 1 − 0.1 exp −
2
x1 − 5π/2
σα
2 #!
x2 − 5π/2
σµ
2 #!
Introduction
,
Key references
Games vs ED
Definitions of
ED
Applications
"
1
µ (x2 ) = µ
e 1 + 0.1 exp −
2
Let
1
Az1 = z1 + ∆2 η1 ∂x1 x1 ,
2
.
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
1
Az3 = z3 + ∆2 η2 ∂x2 x2 .
2
Then ...
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
∂t z1 = aAz1 − bz1 − α (x1 ) z2 ,
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
∂t z1 = aAz1 − bz1 − α (x1 ) z2 ,
Introduction
Key references
Games vs ED
Definitions of
ED
∂t z2 = νAz3 (z1 − z2 ) − (α (x1 ) + b + γ) z2 ,
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
∂t z1 = aAz1 − bz1 − α (x1 ) z2 ,
Introduction
Key references
Games vs ED
Definitions of
ED
∂t z2 = νAz3 (z1 − z2 ) − (α (x1 ) + b + γ) z2 ,
∂t z3 = λz2 − (µ (x2 ) + νz1 ) z3 ,
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Host pathogen ED
Evolutionary
distributions
Yosef Cohen
∂t z1 = aAz1 − bz1 − α (x1 ) z2 ,
Introduction
Key references
Games vs ED
Definitions of
ED
∂t z2 = νAz3 (z1 − z2 ) − (α (x1 ) + b + γ) z2 ,
∂t z3 = λz2 − (µ (x2 ) + νz1 ) z3 ,
with data
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
zi (x, 0) = 1000
zi (π/2, x2 , t) = zi (9π/2, x2 , t)
zi (x1 , π/2, t) = zi (x2 , 9π/2, t)
i = 1, 2, 3.
Extensions
Anticipated effect of α and µ
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Anticipated effect of α and µ
Evolutionary
distributions
Yosef Cohen
EDFramework-2d.nb
1
Introduction
Key references
Games vs ED
Definitions of
ED
Host x
10 2
x1 10
5
Pahogen x
10 2
5
x1 10
5
-5
-5.2
-5.4
-a
Applications
5
-0.02
-0.0205
-0.021 -m
-0.0215
-0.022
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Stable surfaces of ED
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Stable surfaces of ED
Evolutionary
distributions
Yosef Cohen
Introduction
Host
Key references
Games vs ED
Pathogen x
2
x2
x1
Definitions of
ED
x1
Applications
z1
z3
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Stable surfaces of ED
Evolutionary
distributions
Yosef Cohen
Introduction
Host
Key references
Games vs ED
Pathogen x
2
x2
x1
Definitions of
ED
x1
Applications
z1
z3
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
The rise and fall ...
Mutual parasitism
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Mutual parasitism
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
The results here are from Cohen (2005).
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Mutual parasitism
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
The results here are from Cohen (2005).
I
We have a system of two ED. On each ED, similar
phenotypes (on x) benefit from each other, different
phenotypes harm each other.
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Mutual parasitism
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
The results here are from Cohen (2005).
I
We have a system of two ED. On each ED, similar
phenotypes (on x) benefit from each other, different
phenotypes harm each other.
I
With certain parameter values, under the assumption
of homogeneous stable surfaces we obtain two
eigenvalues, one negative and one positive.
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Mutual parasitism
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
The results here are from Cohen (2005).
I
We have a system of two ED. On each ED, similar
phenotypes (on x) benefit from each other, different
phenotypes harm each other.
I
With certain parameter values, under the assumption
of homogeneous stable surfaces we obtain two
eigenvalues, one negative and one positive.
The dynamics look like this ...
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Mutual parasitism (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Mutual parasitism (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
t 30
20
40
t 30
20
10
Definitions of
ED
40
Applications
10
0
40
0
30
40
z1 20
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
z2
20
10
0
0
0
0
5
x
10
5
x
Conclusions
10
Extensions
Outline
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of ED
Applications
Competition
Single-trait competition
Two-traits competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual parasitism
Conclusions
Extensions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
With ED, it is clear how one can obtain ESS at
minimum fitness
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
I
With ED, it is clear how one can obtain ESS at
minimum fitness
For organisms with small number of genes, we can
hope to map the power set of genes to phenotypic
traits
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
With ED, it is clear how one can obtain ESS at
minimum fitness
I
For organisms with small number of genes, we can
hope to map the power set of genes to phenotypic
traits
I
Then population genetics problems become algebraic
problems
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
I
With ED, it is clear how one can obtain ESS at
minimum fitness
I
For organisms with small number of genes, we can
hope to map the power set of genes to phenotypic
traits
I
Then population genetics problems become algebraic
problems
I
For smooth games (not matrix games) ED bypasses
games
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
A stable ED surface (homogeneous or not) is an ESS
in the context of point processes
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
A stable ED surface (homogeneous or not) is an ESS
in the context of point processes
I
Because of stability of non-homogeneous surfaces,
fitness of phenotypes can have any value
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Conclusions (continued)
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
I
A stable ED surface (homogeneous or not) is an ESS
in the context of point processes
I
Because of stability of non-homogeneous surfaces,
fitness of phenotypes can have any value
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Outline
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of ED
Applications
Competition
Single-trait competition
Two-traits competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual parasitism
Conclusions
Extensions
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Extensions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Extensions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
ED and learning
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Extensions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
ED and learning
I
Mating systems
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Extensions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
ED and learning
I
Mating systems
I
Sexual reproduction
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Extensions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
ED and learning
I
Mating systems
I
Sexual reproduction
I
Small population and ED in probability space
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions
Extensions
Evolutionary
distributions
Yosef Cohen
Introduction
Key references
Games vs ED
Definitions of
ED
I
ED and learning
I
Mating systems
I
Sexual reproduction
I
Small population and ED in probability space
I
Thanks for you attention
Applications
Single-trait
competition
Competition
Two-traits
competition
Predator prey
Point process
ED
Host pathogen
Point process
ED
Mutual
parasitism
Conclusions
Extensions