· A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal...
Transcript of · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal...
![Page 1: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/1.jpg)
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
![Page 2: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/2.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Outline
Key references
Games vs ED
Formal definition
ApplicationsSingle-trait competitionTwo-traits competitionPredator prey
Point processED
Host pathogenPoint processED
Conclusions
Extensions
![Page 3: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/3.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 4: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/4.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 5: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/5.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 6: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/6.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 7: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/7.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Outline
Key references
Games vs ED
Formal definition
ApplicationsSingle-trait competitionTwo-traits competitionPredator prey
Point processED
Host pathogenPoint processED
Conclusions
Extensions
![Page 8: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/8.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 9: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/9.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 10: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/10.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 Then
z′ = 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.
![Page 11: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/11.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 12: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/12.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 13: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/13.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 14: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/14.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 15: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/15.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 16: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/16.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 17: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/17.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 18: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/18.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 19: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/19.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 20: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/20.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 21: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/21.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Evolutionary Distributions (ED)
∂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) .
![Page 22: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/22.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Evolutionary Distributions (ED)
∂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) .
![Page 23: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/23.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Evolutionary Distributions (ED)
∂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) .
![Page 24: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/24.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Evolutionary Distributions (ED)
∂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) .
![Page 25: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/25.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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) .
![Page 26: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/26.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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) .
![Page 27: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/27.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Outline
Key references
Games vs ED
Formal definition
ApplicationsSingle-trait competitionTwo-traits competitionPredator prey
Point processED
Host pathogenPoint processED
Conclusions
Extensions
![Page 28: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/28.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 29: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/29.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 30: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/30.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
A formal definition of ED
Define the mth order mutation operator
mA := 1 + k
m∑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 ...
![Page 31: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/31.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
A formal definition of ED
Define the mth order mutation operator
mA := 1 + k
m∑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 ...
![Page 32: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/32.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
A formal definition of ED
Define the mth order mutation operator
mA := 1 + k
m∑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 ...
![Page 33: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/33.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
A formal definition of ED
Define the mth order mutation operator
mA := 1 + k
m∑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 ...
![Page 34: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/34.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
A formal definition of ED
Define the mth order mutation operator
mA := 1 + k
m∑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 ...
![Page 35: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/35.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 36: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/36.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 37: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/37.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 38: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/38.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 39: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/39.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 40: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/40.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Outline
Key references
Games vs ED
Formal definition
ApplicationsSingle-trait competitionTwo-traits competitionPredator prey
Point processED
Host pathogenPoint processED
Conclusions
Extensions
![Page 41: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/41.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Applications
I With this framework, we can now port all pointprocess population ecology models.
I Here are some applications ....
![Page 42: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/42.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Applications
I With this framework, we can now port all pointprocess population ecology models.
I Here are some applications ....
![Page 43: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/43.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Applications
I With this framework, we can now port all pointprocess population ecology models.
I Here are some applications ....
![Page 44: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/44.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 45: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/45.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 46: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/46.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 47: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/47.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 48: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/48.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 49: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/49.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 50: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/50.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 51: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/51.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
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 ...
![Page 52: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/52.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
No selection
x
t
z
x
t
![Page 53: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/53.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
No selection
x
t
z
x
t
![Page 54: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/54.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 55: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/55.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 56: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/56.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 57: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/57.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 58: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/58.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 59: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/59.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 60: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/60.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 61: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/61.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 62: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/62.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 63: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/63.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 64: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/64.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 65: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/65.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Single trait selection for α and k
x
t
z
x
![Page 66: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/66.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Single trait selection for α and k
x
t
z
x
![Page 67: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/67.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits competition
I x1 selected for carrying capacityI x2 selected for competitive abilityI The traits are orthogonal
Then ...
![Page 68: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/68.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits competition
I x1 selected for carrying capacity
I x2 selected for competitive abilityI The traits are orthogonal
Then ...
![Page 69: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/69.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits competition
I x1 selected for carrying capacityI x2 selected for competitive ability
I The traits are orthogonalThen ...
![Page 70: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/70.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits competition
I x1 selected for carrying capacityI x2 selected for competitive abilityI The traits are orthogonal
Then ...
![Page 71: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/71.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits competition
I x1 selected for carrying capacityI x2 selected for competitive abilityI The traits are orthogonal
Then ...
![Page 72: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/72.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 73: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/73.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 74: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/74.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 75: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/75.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 76: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/76.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 77: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/77.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits single ED
x1
x2
z
x1
![Page 78: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/78.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Two-traits single ED
x1
x2
z
x1
![Page 79: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/79.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Predator prey
Next, an application with regard to predator prey.
We start with the point process and then move on to ED...
![Page 80: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/80.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Predator prey
Next, an application with regard to predator prey.
We start with the point process and then move on to ED...
![Page 81: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/81.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 82: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/82.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 83: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/83.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Predator prey - point process
Let
z1 prey
z2 predator
z′1 = rz1 −r
kz21 −
az1
b + cz1z2,
z′2 = daz1
b + cz1z2 − µz2
2 .
With certain parameter values we obtain ...
![Page 84: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/84.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 85: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/85.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 86: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/86.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 87: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/87.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 88: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/88.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 89: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/89.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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]
.
![Page 90: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/90.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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]
.
![Page 91: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/91.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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]
.
![Page 92: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/92.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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]
.
![Page 93: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/93.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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]
.
![Page 94: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/94.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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]
.
![Page 95: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/95.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 96: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/96.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 97: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/97.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 98: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/98.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 99: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/99.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 100: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/100.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 101: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/101.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 102: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/102.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 103: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/103.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 104: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/104.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 105: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/105.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 106: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/106.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 107: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/107.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 108: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/108.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Phenotypic plasticity σ = π/3
Prey
x1
x2
z1
x1
Predator
x1
x2
z2
x1
![Page 109: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/109.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Phenotypic plasticity σ = π/3
Prey
x1
x2
z1
x1
Predator
x1
x2
z2
x1
![Page 110: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/110.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Phenotypic plasticity σ = π
Prey
x1
x2
z1
x1
Predator
x1
x2
z2
x1
![Page 111: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/111.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Phenotypic plasticity σ = π
Prey
x1
x2
z1
x1
Predator
x1
x2
z2
x1
![Page 112: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/112.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Host pathogen
The model is from ...
Anderson and May (1980, equations 3,5 and 6; 1981).
![Page 113: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/113.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Host pathogen
The model is from ...
Anderson and May (1980, equations 3,5 and 6; 1981).
![Page 114: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/114.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Host pathogen
The model is from ...
Anderson and May (1980, equations 3,5 and 6; 1981).
![Page 115: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/115.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
The point process
I z1 - density of hostI z2 - density of infected hostI z3 - density of pathogens
We have ...
![Page 116: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/116.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
The point process
I z1 - density of host
I z2 - density of infected hostI z3 - density of pathogens
We have ...
![Page 117: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/117.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
The point process
I z1 - density of hostI z2 - density of infected host
I z3 - density of pathogens
We have ...
![Page 118: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/118.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
The point process
I z1 - density of hostI z2 - density of infected hostI z3 - density of pathogens
We have ...
![Page 119: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/119.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
The point process
I z1 - density of hostI z2 - density of infected hostI z3 - density of pathogens
We have ...
![Page 120: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/120.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 121: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/121.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 122: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/122.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 123: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/123.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 124: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/124.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 125: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/125.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 infection
µ̃ death rate of infective stages
![Page 126: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/126.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 127: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/127.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 128: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/128.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 129: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/129.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 130: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/130.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 minimum
I At some value of x2 the value of µ is at its minimum
Then ...
![Page 131: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/131.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 132: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/132.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 133: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/133.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 134: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/134.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 135: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/135.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 136: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/136.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 137: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/137.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 138: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/138.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 139: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/139.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 140: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/140.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 141: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/141.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 142: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/142.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 143: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/143.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 144: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/144.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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.
![Page 145: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/145.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Anticipated effect of α and µ
Host
x1
x2
�Α
x2Pahogen
x1
x2
�Μ
x2
![Page 146: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/146.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Anticipated effect of α and µ
Host
x1
x2
�Α
x2Pahogen
x1
x2
�Μ
x2
![Page 147: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/147.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Stable surfaces of ED
Host
x1
x2
z1
x2Pathogen
x1
x2
z3
x2
The rise and fall ...
![Page 148: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/148.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Stable surfaces of ED
Host
x1
x2
z1
x2Pathogen
x1
x2
z3
x2
The rise and fall ...
![Page 149: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/149.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Stable surfaces of ED
Host
x1
x2
z1
x2Pathogen
x1
x2
z3
x2
The rise and fall ...
![Page 150: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/150.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Outline
Key references
Games vs ED
Formal definition
ApplicationsSingle-trait competitionTwo-traits competitionPredator prey
Point processED
Host pathogenPoint processED
Conclusions
Extensions
![Page 151: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/151.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 152: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/152.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 153: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/153.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 154: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/154.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 155: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/155.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 156: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/156.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 157: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/157.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 158: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/158.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 159: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/159.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 160: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/160.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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 ...
![Page 161: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/161.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 162: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/162.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
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
![Page 163: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/163.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Outline
Key references
Games vs ED
Formal definition
ApplicationsSingle-trait competitionTwo-traits competitionPredator prey
Point processED
Host pathogenPoint processED
Conclusions
Extensions
![Page 164: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/164.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Extensions
I ED and learningI Mating systemsI Sexual reproductionI Thanks for you attention
![Page 165: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/165.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Extensions
I ED and learning
I Mating systemsI Sexual reproductionI Thanks for you attention
![Page 166: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/166.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Extensions
I ED and learningI Mating systems
I Sexual reproductionI Thanks for you attention
![Page 167: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/167.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Extensions
I ED and learningI Mating systemsI Sexual reproduction
I Thanks for you attention
![Page 168: · A mathematical framework for evolutionary ecology Yosef Cohen Key references Games vs ED Formal definition Applications Single-trait competition Two-traits competition Predator](https://reader034.fdocuments.us/reader034/viewer/2022042315/5f03ee8e7e708231d40b7b46/html5/thumbnails/168.jpg)
Amathematicalframework forevolutionary
ecology
Yosef Cohen
Key references
Games vs ED
Formaldefinition
Applications
Single-traitcompetitionTwo-traitscompetitionPredator preyPoint processEDHost pathogenPoint processED
Conclusions
Extensions
Extensions
I ED and learningI Mating systemsI Sexual reproductionI Thanks for you attention