The document discusses how the number of traits under selection can influence evolutionary and coevolutionary processes. It reviews previous frameworks that have studied this question using adaptive dynamics and quantitative genetics approaches. Generally previous work has found that branching is easier and evolutionary/coevolutionary equilibria are less stable when more traits are under selection. The presentation aims to further explore how the dimensionality of trait space impacts evolutionary dynamics.

1. (Co)evolution in multiple dimensions:
How does the number of traits under selection influence
evolutionary and coevolutionary processes?
Florence Débarre
@flodebarre
Modelling Biological Evolution 2015, Leicester
1 / 17

2. (Co)evolution in multiple dimensions:
How does the number of traits under selection influence
evolutionary and coevolutionary processes?
Florence Débarre
@flodebarre
with
Scott
Nuismer
and
Michael
Doebeli
Modelling Biological Evolution 2015, Leicester
1 / 17

3. (c)https://www.flickr.com/photos/andreaskay/15409471645
2 / 17

4. (c)C.Marley,ChrysinaPrismCM-12
2 / 17

5. (c)C.Marley,ChrysinaPrismCM-12
2 / 17

6. (c)C.Marley,ChrysinaPrismCM-12
2 / 17

7. (c)C.Marley,ChrysinaPrismCM-12
How does the number of traits under
selection influence the stability
of (co)evolutionary equilibria?
2 / 17

8. Previously on
the topic
3 / 17

9. (2010)
∂φ(x)
∂t
= r φ(x) 1 −
∫ α (x, y) φ(y)dy
K (x)
.
4 / 17

10. (2010)
∂φ(x)
∂t
= r φ(x) 1 −
∫ α (x, y) φ(y)dy
K (x)
.
Adaptive dynamics framework
4 / 17

11. (2010)
∂φ(x)
∂t
= r φ(x) 1 −
∫ α (x, y) φ(y)dy
K (x)
.
Adaptive dynamics framework
Trait 1
Trait2
Kα
4 / 17

12. (2010)
∂φ(x)
∂t
= r φ(x) 1 −
∫ α (x, y) φ(y)dy
K (x)
.
Adaptive dynamics framework
Trait 1
Trait2
Kα
4 / 17

13. (2010)
∂φ(x)
∂t
= r φ(x) 1 −
∫ α (x, y) φ(y)dy
K (x)
.
Adaptive dynamics framework
Trait 1
Trait2
Kα
4 / 17

14. (2010)
∂φ(x)
∂t
= r φ(x) 1 −
∫ α (x, y) φ(y)dy
K (x)
.
Adaptive dynamics framework
Trait 1
Trait2
Kα
Branching is easier in
high-dimensional phenotype spaces.
4 / 17

15. (2012)
f(x, y) =
i
exp γi (xi − yi)2
5 / 17

16. (2012)
f(x, y) =
i
exp γi (xi − yi)2
Quantitative genetics framework
5 / 17

17. (2012)
f(x, y) =
i
exp γi (xi − yi)2
Quantitative genetics framework
Trait 1
Trait2
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5 / 17

18. (2012)
f(x, y) =
i
exp γi (xi − yi)2
Quantitative genetics framework
Trait 1
Trait2
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5 / 17

19. (2012)
f(x, y) =
i
exp γi (xi − yi)2
Quantitative genetics framework
Trait 1
Trait2
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Escape is easier in high-dimensional
phenotype spaces.
5 / 17

20. Are they so different?
6 / 17

21. Are they so different?
Single species Two antogonistic species
6 / 17

22. Are they so different?
Single species Two antogonistic species
Adaptive dynamics Quantitative genetics
6 / 17

23. Are they so different?
Single species Two antogonistic species
Adaptive dynamics Quantitative genetics
Evolutionary stability Convergence stability
6 / 17

24. Are they so different?
Single species Two antogonistic species
Adaptive dynamics Quantitative genetics
Evolutionary stability Convergence stability
Equilibria are destabilized in higher dimensions
6 / 17

25. Now, our work
7 / 17

26. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
8 / 17

27. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
depends on all φj, Nj, and t
8 / 17

28. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
depends on all φj, Nj, and t
A moment-based approach
For all species i, follow changes in
Mean traits ¯xij, and
Variances and covariances Gi,kl
8 / 17

29. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
depends on all φj, Nj, and t
A moment-based approach
For all species i, follow changes in
Mean traits ¯xij, and
Variances and covariances Gi,kl
Crucial assumptions
Small trait variances
8 / 17

30. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
depends on all φj, Nj, and t
A moment-based approach
For all species i, follow changes in
Mean traits ¯xij, and
Variances and covariances Gi,kl
Crucial assumptions
Small trait variances
wi(xi) = wi(¯xi) + (xi − ¯xi)T· Dwi(¯xi) +
1
2
(xi − ¯xi)T· D2wi(¯xi) ·(xi − ¯xi)+O(δ3
).
8 / 17

31. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
depends on all φj, Nj, and t
A moment-based approach
For all species i, follow changes in
Mean traits ¯xij, and
Variances and covariances Gi,kl
Crucial assumptions
Small trait variances
wi(xi) = wi(¯xi) + (xi − ¯xi)T· Dwi(¯xi) +
1
2
(xi − ¯xi)T· D2wi(¯xi) ·(xi − ¯xi)+O(δ3
).
Gradient vector (β) Hessian matrix
8 / 17

32. Methods
∆φi(xi; t) = wi(xi) −1 φi(xi; t).
depends on all φj, Nj, and t
A moment-based approach
For all species i, follow changes in
Mean traits ¯xij, and
Variances and covariances Gi,kl
Crucial assumptions
Small trait variances
wi(xi) = wi(¯xi) + (xi − ¯xi)T· Dwi(¯xi) +
1
2
(xi − ¯xi)T· D2wi(¯xi) ·(xi − ¯xi)+O(δ3
).
Gradient vector (β) Hessian matrix
Multivariate Gaussian distributions (¯xi, Gi)
8 / 17

33. Dynamics and stabilities
Changes in mean traits
9 / 17

34. Dynamics and stabilities
Changes in mean traits
∆¯xi = Gi · Dwi(¯xi) + O(δ4
).
9 / 17

35. Dynamics and stabilities
Changes in mean traits
∆¯xi = Gi · Dwi(¯xi) + O(δ4
).
Changes in trait covariances
∆Gi = Gi · D2
wi(¯xi) · Gi + O(δ5
).
9 / 17

36. Dynamics and stabilities
Changes in mean traits
∆¯xi = Gi · Dwi(¯xi) + O(δ4
).
Equilibria: x∗
Changes in trait covariances
∆Gi = Gi · D2
wi(¯xi) · Gi + O(δ5
).
9 / 17

37. Dynamics and stabilities
Changes in mean traits
∆¯xi = Gi · Dwi(¯xi) + O(δ4
).
Equilibria: x∗
Convergence stability: Eigenvalues of Jacobian J, with blocks
Jih = Gi · Cih,
Cih,jl =
∂(Dwi)j
∂¯xhl ¯x=x∗
Changes in trait covariances
∆Gi = Gi · D2
wi(¯xi) · Gi + O(δ5
).
9 / 17

38. Dynamics and stabilities
Changes in mean traits
∆¯xi = Gi · Dwi(¯xi) + O(δ4
).
Equilibria: x∗
Convergence stability: Eigenvalues of Jacobian J, with blocks
Jih = Gi · Cih,
Cih,jl =
∂(Dwi)j
∂¯xhl ¯x=x∗
Changes in trait covariances
∆Gi = Gi · D2
wi(¯xi) · Gi + O(δ5
).
Evolutionary stability: Eigenvalues of D2wi(¯xi). (Leimar 2009)
9 / 17

39. Stabilities
Convergence stability (CS) Evolutionary stability (ES)
10 / 17

40. Stabilities
Convergence stability (CS)
Changes in the mean traits
Attractor/Repellor
Trait 1
Trait2
q
Evolutionary stability (ES)
10 / 17

41. Stabilities
Convergence stability (CS)
Changes in the mean traits
Attractor/Repellor
Trait 1
Trait2
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Evolutionary stability (ES)
10 / 17

42. Stabilities
Convergence stability (CS)
Changes in the mean traits
Attractor/Repellor
Trait 1
Trait2
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Evolutionary stability (ES)
10 / 17

43. Stabilities
Convergence stability (CS)
Changes in the mean traits
Attractor/Repellor
Trait 1
Trait2
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Evolutionary stability (ES)
Changes in the trait variances
Stabilizing/Diversifying selection
Trait 1
Trait2
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10 / 17

44. Stabilities
Convergence stability (CS)
Changes in the mean traits
Attractor/Repellor
Trait 1
Trait2
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Evolutionary stability (ES)
Changes in the trait variances
Stabilizing/Diversifying selection
Trait 1
Trait2
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10 / 17

45. Stabilities
Convergence stability (CS)
Changes in the mean traits
Attractor/Repellor
Trait 1
Trait2
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Evolutionary stability (ES)
Changes in the trait variances
Stabilizing/Diversifying selection
Trait 1
Trait2
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10 / 17

46. Convergence stability
Jacobian J, Jih = Gi · Cih
Evolutionary stability
Hessian matrices D2
wi
11 / 17

47. Three effects of dimensionality
Convergence stability
Jacobian J, Jih = Gi · Cih
Evolutionary stability
Hessian matrices D2
wi
11 / 17

48. Three effects of dimensionality
Convergence stability
Jacobian J, Jih = Gi · Cih
Evolutionary stability
Hessian matrices D2
wi
Combinatorial effect
Trait 1
Trait2
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11 / 17

49. Three effects of dimensionality
Convergence stability
Jacobian J, Jih = Gi · Cih
Evolutionary stability
Hessian matrices D2
wi
Combinatorial effect
Trait 1
Trait2
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CS , ES .
11 / 17

50. Three effects of dimensionality
Convergence stability
Jacobian J, Jih = Gi · Cih
Evolutionary stability
Hessian matrices D2
wi
Combinatorial effect
Trait 1
Trait2
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