· A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal...

Amathematicalframework forevolutionary

ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications

Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED

Conclusions

Extensions

A mathematical framework forevolutionary ecology

Yosef Cohen

University of Minnesota St. Paul, Minnesota


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Outline

Key references

Games vs ED

Formal definition

ApplicationsSingle-trait competitionTwo-traits competitionPredator prey

Point processED

Host pathogenPoint processED

Conclusions

Extensions


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Key references

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 adaptivespace. Journal of Applied Mathematics 2005:403–424.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Key references





ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Key references





ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Key references





ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Outline

Key references

Games vs ED

Formal definition


Point processED


Conclusions

Extensions


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

From evolutionary games to evolutionarydistributions

I We start with the case of a single population density,z and a single adaptive trait x.

I Thenz′ = f (z, x, t) .

I Next, we derive the strategy dynamics in some way

x′ = g (z, x, t)

I and solve for x (and sometimes for z also) to obtainstability or dynamics in a game theoretic context.


ecology

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x′ = g (z, x, t)



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I Then

z′ = f (z, x, t) .


x′ = g (z, x, t)



ecology

Yosef Cohen

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Games vs ED

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Extensions





x′ = g (z, x, t)



ecology

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Games vs ED

Formaldefinition

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Extensions





x′ = g (z, x, t)



ecology

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Extensions





x′ = g (z, x, t)



ecology

Yosef Cohen

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Extensions





x′ = g (z, x, t)



ecology

Yosef Cohen

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Games vs ED

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Extensions

Evolutionary Distributions (ED)

I Decompose f to components that reflect growth anddecline:

f (z, x, t) = β̃ (z, x, t)− µ (z, x, t) .

I There are good reasons to assume that β̃ is linear. Sowe write

β̃ (z, x, t) = βz(x, t).

I Assume random mutations on progeny with fractionη.

So ...


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Yosef Cohen

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Extensions



f (z, x, t) = β̃ (z, x, t)− µ (z, x, t) .


β̃ (z, x, t) = βz(x, t).


So ...


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Yosef Cohen

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f (z, x, t) = β̃ (z, x, t)− µ (z, x, t) .


β̃ (z, x, t) = βz(x, t).


So ...


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f (z, x, t) = β̃ (z, x, t)− µ (z, x, t) .


β̃ (z, x, t) = βz(x, t).


So ...


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Yosef Cohen

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Extensions



f (z, x, t) = β̃ (z, x, t)− µ (z, x, t) .


β̃ (z, x, t) = βz(x, t).


So ...


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Yosef Cohen

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Extensions



f (z, x, t) = β̃ (z, x, t)− µ (z, x, t) .


β̃ (z, x, t) = βz(x, t).


So ...


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Yosef Cohen

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Extensions


∂tz (x, t) = (1− η)βz (x, t) +12βη [z (x + ∆) + z (x−∆)]− µ (z, x, t) .

With Taylor series expansion of z around x, we obtainapproximately

∂tz = z +12∆2βη∂xxz − µ (z, x, t) .


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∂tz (x, t) = (1− η) βz (x, t) +12βη [z (x + ∆) + z (x−∆)]− µ (z, x, t) .


∂tz = z +12∆2βη∂xxz − µ (z, x, t) .


ecology

Yosef Cohen

Key references

Games vs ED

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Extensions




∂tz = z +12∆2βη∂xxz − µ (z, x, t) .


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

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Conclusions

Extensions




∂tz = z +12∆2βη∂xxz − µ (z, x, t) .


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

ED (continued)

For a single ED with m orthogonal adaptive traits, wehave

∂tz = z +12∆2β

m∑i=1

ηi∂xixiz − µ (z,x, t) .


ecology

Yosef Cohen

Key references

Games vs ED

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Conclusions

Extensions

ED (continued)

For a single ED with m orthogonal adaptive traits, wehave

∂tz = z +12∆2β

m∑i=1

ηi∂xixiz − µ (z,x, t) .


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Outline

Key references

Games vs ED

Formal definition


Point processED


Conclusions

Extensions


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

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Conclusions

Extensions

A formal definition of ED

Define the mth order mutation operator

mA := 1 + km∑

i=1

ηi∂xixi

where k := ∆2β/2.

zi ∈ R0+, i = 1, . . ., n is the distribution of the density oftypes with mi adaptive traits xi.

x = [x1, . . . ,xn].

Define the bounded open set X ⊂ RM (where M =∑ni=1 mi) with boundary ∂X . Then ...


ecology

Yosef Cohen

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Extensions



mA := 1 + km∑

i=1

ηi∂xixi



x = [x1, . . . ,xn].



ecology

Yosef Cohen

Key references

Games vs ED

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Extensions



mA := 1 + k

m∑i=1

ηi∂xixi



x = [x1, . . . ,xn].



ecology

Yosef Cohen

Key references

Games vs ED

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Extensions



mA := 1 + k

m∑i=1

ηi∂xixi



x = [x1, . . . ,xn].



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



mA := 1 + k

m∑i=1

ηi∂xixi



x = [x1, . . . ,xn].



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



mA := 1 + k

m∑i=1

ηi∂xixi



x = [x1, . . . ,xn].



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



mA := 1 + k

m∑i=1

ηi∂xixi



x = [x1, . . . ,xn].



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Definition An ED, zi (x, t), is the solution of the system

∂tzi (x, t) = βi miAzi (xi, t)− µi (z,x, t) ,

with the data

zi (x, 0) = z0 (x)

and

∂xizi (x, t)|x=∂X = 0, i = 1, . . . , n.


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Yosef Cohen

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Games vs ED

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Conclusions

Extensions



with the data

zi (x, 0) = z0 (x)

and

∂xizi (x, t)|x=∂X = 0, i = 1, . . . , n.


ecology

Yosef Cohen

Key references

Games vs ED

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Conclusions

Extensions



with the data

zi (x, 0) = z0 (x)

and

∂xizi (x, t)|x=∂X = 0, i = 1, . . . , n.


ecology

Yosef Cohen

Key references

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Conclusions

Extensions



with the data

zi (x, 0) = z0 (x)

and

∂xizi (x, t)|x=∂X = 0, i = 1, . . . , n.


ecology

Yosef Cohen

Key references

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Applications


Conclusions

Extensions



with the data

zi (x, 0) = z0 (x)

and

∂xizi (x, t)|x=∂X = 0, i = 1, . . . , n.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Outline

Key references

Games vs ED

Formal definition


Point processED


Conclusions

Extensions


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Applications

I With this framework, we can now port all pointprocess population ecology models.

I Here are some applications ....


ecology

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Conclusions

Extensions

Applications




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Applications




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

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Extensions

Competition - single trait without selection

The point process is

z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

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Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



z′ = rz − r

kz2.

The ED is

∂tz = rAz − r

kz2,

with data

z (x, 0) = 20 + sin (x) ,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0.

We obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

No selection

x

t

z

x

t


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Yosef Cohen

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Formaldefinition

Applications


Conclusions

Extensions

No selection

x

t

z

x

t


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Yosef Cohen

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Games vs ED

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Conclusions

Extensions

Competition - single trait with selection

Assume single trait adaptation to competition and bestadaptation to some value of carrying capacity. Then ...

α (x, ξ) = kα(1 + k exp

[−1

2

(x− ξ

σα

)2])

and

k (x) = km(1 + exp

[−1

2

(x− 5π/2

σk

)2])

and the ED is now ...


ecology

Yosef Cohen

Key references

Games vs ED

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Conclusions

Extensions



α (x, ξ) = kα(1 + k exp

[−1

2

(x− ξ

σα

)2])

and

k (x) = km(1 + exp

[−1

2

(x− 5π/2

σk

)2])



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



α (x, ξ) = kα(1 + k exp

[−1

2

(x− ξ

σα

)2])

and

k (x) = km(1 + exp

[−1

2

(x− 5π/2

σk

)2])



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



α (x, ξ) = kα(1 + k exp

[−1

2

(x− ξ

σα

)2])

and

k (x) = km(1 + exp

[−1

2

(x− 5π/2

σk

)2])



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



α (x, ξ) = kα(1 + k exp

[−1

2

(x− ξ

σα

)2])

and

k (x) = km(1 + exp

[−1

2

(x− 5π/2

σk

)2])



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



α (x, ξ) = kα(1 + k exp

[−1

2

(x− ξ

σα

)2])

and

k (x) = km(1 + exp

[−1

2

(x− 5π/2

σk

)2])



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Competition - single trait with selection(continued)

∂tz = rAz − r

kmz (x, t)

∫ 9π/2

π/2α (x, ξ) z (ξ, t) dξ,

and data

z (x, 0) = 0.005,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0

and we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

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Extensions


∂tz = rAz − r

kmz (x, t)

∫ 9π/2

π/2α (x, ξ) z (ξ, t) dξ,

and data

z (x, 0) = 0.005,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0

and we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


∂tz = rAz − r

kmz (x, t)

∫ 9π/2

π/2α (x, ξ) z (ξ, t) dξ,

and data

z (x, 0) = 0.005,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0

and we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


∂tz = rAz − r

kmz (x, t)

∫ 9π/2

π/2α (x, ξ) z (ξ, t) dξ,

and data

z (x, 0) = 0.005,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0

and we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


∂tz = rAz − r

kmz (x, t)

∫ 9π/2

π/2α (x, ξ) z (ξ, t) dξ,

and data

z (x, 0) = 0.005,

∂xz (π/2, t) = ∂xz (9π/2, t) = 0

and we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Single trait selection for α and k

x

t

z

x


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Single trait selection for α and k

x

t

z

x


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Games vs ED

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Extensions

Two-traits competition

I x1 selected for carrying capacityI x2 selected for competitive abilityI The traits are orthogonal

Then ...


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Yosef Cohen

Key references

Games vs ED

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Extensions


I x1 selected for carrying capacity

I x2 selected for competitive abilityI The traits are orthogonal

Then ...


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Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


I x1 selected for carrying capacityI x2 selected for competitive ability

I The traits are orthogonalThen ...


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Yosef Cohen

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Games vs ED

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Applications


Conclusions

Extensions



Then ...


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Yosef Cohen

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Games vs ED

Formaldefinition

Applications


Conclusions

Extensions



Then ...


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Yosef Cohen

Key references

Games vs ED

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Applications


Conclusions

Extensions

Two-traits single ED

∂tz = r 2Az − r

k (x1)z

∫ 9π/2

π/2α (x2, ξ) z (x1, ξ, t) dξ,

and data

z (x, 0) = 20∂x1z (π/2, x2, t) = ∂x1z (9π/2, x2, t) = 0,

∂x2z (x1, π/2, t) = ∂x2z (x1, 9π/2, t) = 0,

we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

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Applications


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Extensions


∂tz = r 2Az − r

k (x1)z

∫ 9π/2

π/2α (x2, ξ) z (x1, ξ, t) dξ,

and data

z (x, 0) = 20∂x1z (π/2, x2, t) = ∂x1z (9π/2, x2, t) = 0,

∂x2z (x1, π/2, t) = ∂x2z (x1, 9π/2, t) = 0,

we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


∂tz = r 2Az − r

k (x1)z

∫ 9π/2

π/2α (x2, ξ) z (x1, ξ, t) dξ,

and data

z (x, 0) = 20∂x1z (π/2, x2, t) = ∂x1z (9π/2, x2, t) = 0,

∂x2z (x1, π/2, t) = ∂x2z (x1, 9π/2, t) = 0,

we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


∂tz = r 2Az − r

k (x1)z

∫ 9π/2

π/2α (x2, ξ) z (x1, ξ, t) dξ,

and data

z (x, 0) = 20∂x1z (π/2, x2, t) = ∂x1z (9π/2, x2, t) = 0,

∂x2z (x1, π/2, t) = ∂x2z (x1, 9π/2, t) = 0,

we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


∂tz = r 2Az − r

k (x1)z

∫ 9π/2

π/2α (x2, ξ) z (x1, ξ, t) dξ,

and data

z (x, 0) = 20∂x1z (π/2, x2, t) = ∂x1z (9π/2, x2, t) = 0,

∂x2z (x1, π/2, t) = ∂x2z (x1, 9π/2, t) = 0,

we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


x1

x2

z

x1


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Yosef Cohen

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Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


x1

x2

z

x1


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Yosef Cohen

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Games vs ED

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Applications


Conclusions

Extensions

Predator prey

Next, an application with regard to predator prey.

We start with the point process and then move on to ED...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey

Next, an application with regard to predator prey.

We start with the point process and then move on to ED...


ecology

Yosef Cohen

Key references

Games vs ED

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Applications


Conclusions

Extensions

Predator prey - point process

Let

z1 preyz2 predator

z′1 = rz1 −r

kz21 −

az1

b + cz1z2,

z′2 = daz1

b + cz1z2 − µz2

2 .

With certain parameter values we obtain ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

z1 preyz2 predator

z′1 = rz1 −r

kz21 −

az1

b + cz1z2,

z′2 = daz1

b + cz1z2 − µz2

2 .



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

z1 prey

z2 predator

z′1 = rz1 −r

kz21 −

az1

b + cz1z2,

z′2 = daz1

b + cz1z2 − µz2

2 .



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

z1 preyz2 predator

z′1 = rz1 −r

kz21 −

az1

b + cz1z2,

z′2 = daz1

b + cz1z2 − µz2

2 .



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

z1 preyz2 predator

z′1 = rz1 −r

kz21 −

az1

b + cz1z2,

z′2 = daz1

b + cz1z2 − µz2

2 .



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

z1 preyz2 predator

z′1 = rz1 −r

kz21 −

az1

b + cz1z2,

z′2 = daz1

b + cz1z2 − µz2

2 .



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Limit cycle

0 200 400 600 8001000t

010203040506070

z

prey�thin,predator�thick

0 10 20 30 40 50 60 70z1�t�

2345678

z 2�t�

limit cycle


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Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Limit cycle

0 200 400 600 8001000t

010203040506070

zprey�thin,predator�thick

0 10 20 30 40 50 60 70z1�t�

2345678

z 2�t�

limit cycle


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey ED

I z1 evolves on x1

I z2 evolves on x2

I Predation is at its maximum when x1 = x2 withsome phenotypic plasticity σ

I Then ...

α (x1, x2) = exp

[−1

2

(x1 − x2

σ

)2]

.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey ED

I z1 evolves on x1

I z2 evolves on x2


I Then ...

α (x1, x2) = exp

[−1

2

(x1 − x2

σ

)2]

.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey ED

I z1 evolves on x1

I z2 evolves on x2


I Then ...

α (x1, x2) = exp

[−1

2

(x1 − x2

σ

)2]

.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey ED

I z1 evolves on x1

I z2 evolves on x2


I Then ...

α (x1, x2) = exp

[−1

2

(x1 − x2

σ

)2]

.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey ED

I z1 evolves on x1

I z2 evolves on x2


I Then ...

α (x1, x2) = exp

[−1

2

(x1 − x2

σ

)2]

.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Predator prey ED

I z1 evolves on x1

I z2 evolves on x2


I Then ...

α (x1, x2) = exp

[−1

2

(x1 − x2

σ

)2]

.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The mutation operators

Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.

Then the point process becomes ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Let

zi ≡ zi (x1, x2, t) ,

Az1 := z1 +12∆2η1∂x1x1z1

and

Az2 := z2 +12∆2η2∂x2x2z1.



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Two ED two traits

∂tz1 = rAz1 −r

kz21 − α (x)

az1

b + cz1z2,

∂tz2 = dα (x)az1

b + cz1Az2 − µz2

2 ,

with initial conditions

z1 (x, 0) = 10,

z2 (x, 0) = 1

and boundary conditions

∂x1z1 (π/2, x2, t) = ∂x1z1 (9π/2, x2, t) = 0,

∂x2z2 (x1, π/2, t) = ∂x2z2 (x1, 9π/2, t) = 0.

Now ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Two ED two traits

∂tz1 = rAz1 −r

kz21 − α (x)

az1

b + cz1z2,

∂tz2 = dα (x)az1

b + cz1Az2 − µz2

2 ,


z1 (x, 0) = 10,

z2 (x, 0) = 1


∂x1z1 (π/2, x2, t) = ∂x1z1 (9π/2, x2, t) = 0,

∂x2z2 (x1, π/2, t) = ∂x2z2 (x1, 9π/2, t) = 0.

Now ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Two ED two traits

∂tz1 = rAz1 −r

kz21 − α (x)

az1

b + cz1z2,

∂tz2 = dα (x)az1

b + cz1Az2 − µz2

2 ,


z1 (x, 0) = 10,

z2 (x, 0) = 1


∂x1z1 (π/2, x2, t) = ∂x1z1 (9π/2, x2, t) = 0,

∂x2z2 (x1, π/2, t) = ∂x2z2 (x1, 9π/2, t) = 0.

Now ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Two ED two traits

∂tz1 = rAz1 −r

kz21 − α (x)

az1

b + cz1z2,

∂tz2 = dα (x)az1

b + cz1Az2 − µz2

2 ,


z1 (x, 0) = 10,

z2 (x, 0) = 1


∂x1z1 (π/2, x2, t) = ∂x1z1 (9π/2, x2, t) = 0,

∂x2z2 (x1, π/2, t) = ∂x2z2 (x1, 9π/2, t) = 0.

Now ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Two ED two traits

∂tz1 = rAz1 −r

kz21 − α (x)

az1

b + cz1z2,

∂tz2 = dα (x)az1

b + cz1Az2 − µz2

2 ,


z1 (x, 0) = 10,

z2 (x, 0) = 1


∂x1z1 (π/2, x2, t) = ∂x1z1 (9π/2, x2, t) = 0,

∂x2z2 (x1, π/2, t) = ∂x2z2 (x1, 9π/2, t) = 0.

Now ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Two ED two traits

∂tz1 = rAz1 −r

kz21 − α (x)

az1

b + cz1z2,

∂tz2 = dα (x)az1

b + cz1Az2 − µz2

2 ,


z1 (x, 0) = 10,

z2 (x, 0) = 1


∂x1z1 (π/2, x2, t) = ∂x1z1 (9π/2, x2, t) = 0,

∂x2z2 (x1, π/2, t) = ∂x2z2 (x1, 9π/2, t) = 0.

Now ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Phenotypic plasticity σ = π/3

Prey

x1

x2

z1

x1

Predator

x1

x2

z2

x1


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Phenotypic plasticity σ = π/3

Prey

x1

x2

z1

x1

Predator

x1

x2

z2

x1


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Phenotypic plasticity σ = π

Prey

x1

x2

z1

x1

Predator

x1

x2

z2

x1


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Phenotypic plasticity σ = π

Prey

x1

x2

z1

x1

Predator

x1

x2

z2

x1


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen

The model is from ...

Anderson and May (1980, equations 3,5 and 6; 1981).


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The point process

I z1 - density of hostI z2 - density of infected hostI z3 - density of pathogens

We have ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The point process

I z1 - density of host

I z2 - density of infected hostI z3 - density of pathogens

We have ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The point process

I z1 - density of hostI z2 - density of infected host

I z3 - density of pathogens

We have ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The point process


We have ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The point process


We have ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

The point process (continued)

z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.

The relevant parameters are

α̃ additional death rate due to infectionµ̃ death rate of infective stages


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.


α̃ additional death rate due to infection

µ̃ death rate of infective stages


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


z′1 = (a− b) z1 − α̃z2,

z′2 = νz3 (z1 − z2)− (α̃ + b + γ) z2,

z′3 = λz2 − (µ̃ + νz1) z3.




ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED

I x1 - adaptive trait that affects death rate of hostsdue to infection (α)

I x2 - adaptive trait that affects pathogen death rate ofinfective stages (µ)

I At some value of x1 the value of α is at its minimumI At some value of x2 the value of µ is at its minimum

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED




Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED




Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED



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 ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED




Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED




Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Coevolution

α (x1) = α̃

(1− 0.1 exp

[−1

2

(x1 − 5π/2

σα

)2])

,

µ (x2) = µ̃

(1 + 0.1 exp

[−1

2

(x2 − 5π/2

σµ

)2])

.

Let

Az1 = z1 +12∆2η1∂x1x1 ,

Az3 = z3 +12∆2η2∂x2x2 .

Then ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED

∂tz1 = aAz1 − bz1 − α (x1) z2,

∂tz2 = νAz3 (z1 − z2)− (α (x1) + b + γ) z2,

∂tz3 = λz2 − (µ (x2) + νz1) z3,

with data

zi (x, 0) = 1000zi (π/2, x2, t) = zi (9π/2, x2, t)zi (x1, π/2, t) = zi (x2, 9π/2, t)

i = 1, 2, 3.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED

∂tz1 = aAz1 − bz1 − α (x1) z2,

∂tz2 = νAz3 (z1 − z2)− (α (x1) + b + γ) z2,

∂tz3 = λz2 − (µ (x2) + νz1) z3,

with data


i = 1, 2, 3.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED

∂tz1 = aAz1 − bz1 − α (x1) z2,

∂tz2 = νAz3 (z1 − z2)− (α (x1) + b + γ) z2,

∂tz3 = λz2 − (µ (x2) + νz1) z3,

with data


i = 1, 2, 3.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED

∂tz1 = aAz1 − bz1 − α (x1) z2,

∂tz2 = νAz3 (z1 − z2)− (α (x1) + b + γ) z2,

∂tz3 = λz2 − (µ (x2) + νz1) z3,

with data


i = 1, 2, 3.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Host pathogen ED

∂tz1 = aAz1 − bz1 − α (x1) z2,

∂tz2 = νAz3 (z1 − z2)− (α (x1) + b + γ) z2,

∂tz3 = λz2 − (µ (x2) + νz1) z3,

with data


i = 1, 2, 3.


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Anticipated effect of α and µ

Host

x1

x2

�Α

x2Pahogen

x1

x2

�Μ

x2


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Anticipated effect of α and µ

Host

x1

x2

�Α

x2Pahogen

x1

x2

�Μ

x2


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Stable surfaces of ED

Host

x1

x2

z1

x2Pathogen

x1

x2

z3

x2

The rise and fall ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Host

x1

x2

z1

x2Pathogen

x1

x2

z3

x2



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions


Host

x1

x2

z1

x2Pathogen

x1

x2

z3

x2



ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Outline

Key references

Games vs ED

Formal definition


Point processED


Conclusions

Extensions


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Conclusions

I With ED, it is clear how one can obtain ESS atminimum fitness

I For organisms with small number of genes, we canhope to map the power set of genes to phenotypictraits

I Then population genetics problems become algebraicproblems

I For smooth games (not matrix games) ED bypassesgames


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Conclusions






ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Conclusions






ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Conclusions






ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Conclusions






ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Conclusions (continued)

I A stable ED surface (homogeneous or not) is an ESSin the context of point processes

I Because of stability of non-homogeneous surfaces,fitness of phenotypes can have any value

I ED are functions in Sobolev space; they need not besmooth; they even need not be continuous

Example ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions





Example ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions





Example ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions





Example ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions





Example ...


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Mutual parasitism

05

10x

010

2030

40t

0

10

20

30

40

z1

05

10x

010

2030t

05

10x

010

2030

40t

0

20

40

z2

05

10x

010

2030t


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Mutual parasitism

05

10x

010

2030

40t

0

10

20

30

40

z1

05

10x

010

2030t

05

10x

010

2030

40t

0

20

40

z2

05

10x

010

2030t


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Outline

Key references

Games vs ED

Formal definition


Point processED


Conclusions

Extensions


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Extensions

I ED and learningI Mating systemsI Sexual reproductionI Thanks for you attention


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Extensions

I ED and learning

I Mating systemsI Sexual reproductionI Thanks for you attention


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Extensions

I ED and learningI Mating systems

I Sexual reproductionI Thanks for you attention


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

Applications


Conclusions

Extensions

Extensions

I ED and learningI Mating systemsI Sexual reproduction

I Thanks for you attention


ecology

Yosef Cohen

Key references

Games vs ED

Formaldefinition

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Conclusions

Extensions

Extensions

I ED and learningI Mating systemsI Sexual reproductionI Thanks for you attention

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· A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal...

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