The document describes a model of the evolution of specialist versus generalist strategies in a spatially continuous landscape with two habitat types. The model examines the conditions under which a generalist strategy that uses both habitats equally is evolutionarily stable versus specialist strategies that focus on one habitat. It finds that a generalist strategy is always convergence stable, while it is evolutionarily stable under conditions of a concave trade-off between habitats and sufficient migration between habitats.

1. Evolution of specialist vs generalist strategies
in a continuous environment
Florence D´ebarre and Sylvain Gandon
CNRS UMR 5175, CEFE, Montpellier, France
August 28th 2009

2. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica...
Turin
Venaco
Montpellier
Paris
London
Rome
Madrid
© A. Coreau
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 2 / 16

3. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica... Species composition
Turin
Venaco
Montpellier
Paris
London
Rome
Madrid
Burnt
Scrub
Pines
Oaks
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 3 / 16

4. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica... Vegetation types
Turin
Venaco
Montpellier
Paris
London
Rome
Madrid
High VegetationLow Vegetation
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 4 / 16

5. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica... Vegetation types
Turin
Venaco
Montpellier
Paris
London
Rome
Madrid
High VegetationLow Vegetation
www.fatbirder.com
Sylvia undata
www.ebnitalia.it
Sylvia moltonii
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 4 / 16

6. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica...
High VegetationLow Vegetation
Low
Vegetation
High
Vegetation
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 5 / 16

7. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica...
High VegetationLow Vegetation
Low
Vegetation
High
Vegetation
High
Vegetation
Low
Vegetation
distance
migration
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 5 / 16

8. Introduction Model Results Conclusion More?
Introduction
A landscape in Venaco, Corsica...
Question
What are the conditions for the evolu-
tion of generalist vs specialist strate-
gies in a spatially continuous land-
scape ?
Low
Vegetation
High
Vegetation
High
Vegetation
Low
Vegetation
distance
migration
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 5 / 16

9. Introduction Model Results Conclusion More?
The environment
A linear environment with two habitats
Habitat AHabitat B Habitat B
Spatial location
0qS qSS S
S Total size of the environment
q Proportion of habitat A
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 6 / 16

10. Introduction Model Results Conclusion More?
The environment
A linear environment with two habitats
Habitat AHabitat B Habitat B
Spatial location
0qS qSS S
S Total size of the environment
q Proportion of habitat A
Migration kernel
Σ
distance
migration probability
0
σ Migration range
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 6 / 16

11. Introduction Model Results Conclusion More?
Trade-offs
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 7 / 16

12. Introduction Model Results Conclusion More?
Trade-offs
With a power trade-off
rA(s) = s
β
rB (s) = (1 − s)
β
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 7 / 16

13. Introduction Model Results Conclusion More?
Trade-offs
With a power trade-off
rA(s) = s
β
rB (s) = (1 − s)
β
Fitness in habitat A
FitnessinhabitatB
Β
0.25
0.5
0.75
1
1.5
2
3
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 7 / 16

14. Introduction Model Results Conclusion More?
Trade-offs
With a power trade-off
rA(s) = s
β
rB (s) = (1 − s)
β
Fitness in habitat A
FitnessinhabitatB
Β
0.25
0.5
0.75
1
1.5
2
3 With a Gaussian trade-off
ΣrΣr
Habitat B Habitat A
Trait value
Fitness
0Θ Θ
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 7 / 16

15. Introduction Model Results Conclusion More?
Trade-offs
With a power trade-off
rA(s) = s
β
rB (s) = (1 − s)
β
Fitness in habitat A
FitnessinhabitatB
Β
0.25
0.5
0.75
1
1.5
2
3 With a Gaussian trade-off
ΣrΣr
Habitat B Habitat A
Trait value
Fitness
0Θ Θ
Fitness in habitat A
FitnessinhabitatB
Θ
Σr
0.5
0.75
1
1.25
1.5
2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 7 / 16

16. Introduction Model Results Conclusion More?
The Model
Soft Selection
Local density regulation
Constant local densities
and NO habitat choice
. . . and spatially varying fitness ri (x)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 8 / 16

17. Introduction Model Results Conclusion More?
The Model
Soft Selection
Local density regulation
Constant local densities
and NO habitat choice
. . . and spatially varying fitness ri (x)
Migration
Diffusion approximation
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 8 / 16

18. Introduction Model Results Conclusion More?
The Model
Soft Selection
Local density regulation
Constant local densities
and NO habitat choice
. . . and spatially varying fitness ri (x)
Migration
Diffusion approximation
∂pi
∂t
(x, t) =
ri (x)
¯r(x, t)
− 1 pi (x, t) +
σ2
2
∂2
pi
∂x2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 8 / 16

19. Introduction Model Results Conclusion More?
The Model
Soft Selection
Local density regulation
Constant local densities
and NO habitat choice
. . . and spatially varying fitness ri (x)
Migration
Diffusion approximation
∂pi
∂t
(x, t) =
ri (x)
¯r(x, t)
− 1 pi (x, t) +
σ2
2
∂2
pi
∂x2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 8 / 16

20. Introduction Model Results Conclusion More?
The Model
Soft Selection
Local density regulation
Constant local densities
and NO habitat choice
. . . and spatially varying fitness ri (x)
Migration
Diffusion approximation
∂pi
∂t
(x, t) =
ri (x)
¯r(x, t)
− 1 pi (x, t) +
σ2
2
∂2
pi
∂x2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 8 / 16

21. Introduction Model Results Conclusion More?
The Model
Soft Selection
Local density regulation
Constant local densities
and NO habitat choice
. . . and spatially varying fitness ri (x)
Migration
Diffusion approximation
∂pi
∂t
(x, t) =
ri (x)
¯r(x, t)
− 1 pi (x, t) +
σ2
2
∂2
pi
∂x2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 8 / 16

22. Introduction Model Results Conclusion More?
Methods
Adaptive Dynamics
Assumptions
Rare mutations (decoupling of ecological and evolutionary time scales)
Mutations of small phenotypic effect
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 9 / 16

23. Introduction Model Results Conclusion More?
Pairwise Invasibility Plots (PIP)
Does a mutant with trait sm invade a population fixed for trait sr?
+
+
-
-
mutant'strait(sm)
resident's trait (sr)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 10 / 16

24. Introduction Model Results Conclusion More?
Pairwise Invasibility Plots (PIP)
Does a mutant with trait sm invade a population fixed for trait sr?
+
+
-
-
mutant'strait(sm)
resident's trait (sr)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 10 / 16

25. Introduction Model Results Conclusion More?
Pairwise Invasibility Plots (PIP)
Does a mutant with trait sm invade a population fixed for trait sr?
+
+
-
-
mutant'strait(sm)
resident's trait (sr)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 10 / 16

26. Introduction Model Results Conclusion More?
Pairwise Invasibility Plots (PIP)
+
+
-
-
+
+
-
-
-
-
+
+
-
-
+
+
A B
C D
CS
not CS
ES not ES
mutant
resident
Evolutionary Stability (ES)Convergence
Stability (CS)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 10 / 16

27. Introduction Model Results Conclusion More?
Evolution of the adaptation trait s
Gaussian Trade-off
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 11 / 16

28. Introduction Model Results Conclusion More?
Evolution of the adaptation trait s
Gaussian Trade-off
Singular Strategy
Generalist
s∗
= q θ + (1 − q) (−θ)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 11 / 16

29. Introduction Model Results Conclusion More?
Evolution of the adaptation trait s
Gaussian Trade-off
Singular Strategy
Generalist
s∗
= q θ + (1 − q) (−θ)
Convergence stability
Always
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 11 / 16

30. Introduction Model Results Conclusion More?
Evolution of the adaptation trait s
Gaussian Trade-off
Singular Strategy
Generalist
s∗
= q θ + (1 − q) (−θ)
Convergence stability
Always
Evolutionary stability
Concave trade-off
θ
σr
< 1
2
√
q (1−q)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 11 / 16

31. Introduction Model Results Conclusion More?
Evolution of the adaptation trait s
Gaussian Trade-off
Singular Strategy
Generalist
s∗
= q θ + (1 − q) (−θ)
Convergence stability
Always
Evolutionary stability
Concave trade-off
θ
σr
< 1
2
√
q (1−q)
Enough migration
(σ/S)2
2 >
q (1−q)
3
8 d q (1−q) (θ/σr )2
1−4 q (1−q) (θ/σr )2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 11 / 16

32. Introduction Model Results Conclusion More?
Evolution of the adaptation trait s
Gaussian Trade-off
Singular Strategy
Generalist
s∗
= q θ + (1 − q) (−θ)
Convergence stability
Always
Evolutionary stability
Concave trade-off
θ
σr
< 1
2
√
q (1−q)
Enough migration
(σ/S)2
2 >
q (1−q)
3
8 d q (1−q) (θ/σr )2
1−4 q (1−q) (θ/σr )2
Power Trade-off
Singular Strategy
Generalist
s∗
= q
Convergence stability
Always
Evolutionary stability
Concave trade-off
β < 1
Enough migration
(σ/S)2
2 >
q (1−q)
3
2 d β
1−β
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 11 / 16

33. Introduction Model Results Conclusion More?
Illustration with the Gaussian trade-off
θ
σr
= 0.75
Trait value
Fitness
0Θ Θ
θ
σr
= 1.25
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 12 / 16

34. Introduction Model Results Conclusion More?
Illustration with the Gaussian trade-off
θ
σr
= 0.75
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
θ
σr
= 1.25
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 12 / 16

35. Introduction Model Results Conclusion More?
Illustration with the Gaussian trade-off
θ
σr
= 0.75
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
1 ESS
2 Branching
Proportion of habitat A
Migration
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
θ
σr
= 1.25
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 12 / 16

36. Introduction Model Results Conclusion More?
Illustration with the Gaussian trade-off
θ
σr
= 0.75
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
1 ESS
2 Branching
Proportion of habitat A
Migration
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
θ
σr
= 1.25
Trait value
Fitness
0Θ Θ
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 12 / 16

37. Introduction Model Results Conclusion More?
Illustration with the Gaussian trade-off
θ
σr
= 0.75
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
1 ESS
2 Branching
Proportion of habitat A
Migration
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
θ
σr
= 1.25
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 12 / 16

38. Introduction Model Results Conclusion More?
Illustration with the Gaussian trade-off
θ
σr
= 0.75
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
1 ESS
2 Branching
Proportion of habitat A
Migration
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
θ
σr
= 1.25
Trait value
Fitness
0Θ Θ Fitness in habitat A
FitnessinhabitatB
1 1
2 2
3 Branching
Proportion of habitat A
Migration
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 12 / 16

39. Introduction Model Results Conclusion More?
Two-patch models
(or metapopulations)
High
Vegetation
Low
Vegetation
distance
migration
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 13 / 16

40. Introduction Model Results Conclusion More?
Two-patch models
(or metapopulations)
Continuous-time soft selection model with limited migration
Habitat A Habitat B
1-q q
μ
μ
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 13 / 16

41. Introduction Model Results Conclusion More?
Two-patch models: evolution
With a Gaussian Trade-off
Continuous environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
(σ/S)2
2
>
q (1 − q)
3
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
Two-patch environment
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 14 / 16

42. Introduction Model Results Conclusion More?
Two-patch models: evolution
With a Gaussian Trade-off
Continuous environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
(σ/S)2
2
>
q (1 − q)
3
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
Two-patch environment
Singular Strategy
s∗ = θ (2 q − 1)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 14 / 16

43. Introduction Model Results Conclusion More?
Two-patch models: evolution
With a Gaussian Trade-off
Continuous environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
(σ/S)2
2
>
q (1 − q)
3
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
Two-patch environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 14 / 16

44. Introduction Model Results Conclusion More?
Two-patch models: evolution
With a Gaussian Trade-off
Continuous environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
(σ/S)2
2
>
q (1 − q)
3
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
Two-patch environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 14 / 16

45. Introduction Model Results Conclusion More?
Two-patch models: evolution
With a Gaussian Trade-off
Continuous environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
(σ/S)2
2
>
q (1 − q)
3
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
Two-patch environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
µ >
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 14 / 16

46. Introduction Model Results Conclusion More?
Two-patch models: evolution
With a Gaussian Trade-off
Continuous environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
(σ/S)2
2
>
q (1 − q)
3
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
Two-patch environment
Singular Strategy
s∗ = θ (2 q − 1)
Convergence stability
Always
Evolutionary stability
θ
σr
<
1
2 q (1 − q)
µ >
8 d q (1 − q) (θ/σr )2
1 − 4 q (1 − q) (θ/σr )2
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 14 / 16

47. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
Evolution of generalists when . . .
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

48. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
Evolution of generalists when . . .
Concave trade-off
close habitat optima
weak trade-off
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

49. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
Evolution of generalists when . . .
Concave trade-off
close habitat optima
weak trade-off
Strong migration
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

50. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
Evolution of generalists when . . .
Concave trade-off
close habitat optima
weak trade-off
Strong migration
Evolution with small mutations
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

51. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
Evolution of generalists when . . .
Concave trade-off
close habitat optima
weak trade-off
Strong migration
Evolution with small mutations
Same singular strategies
Same condition for convergence stability
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

52. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
Evolution of generalists when . . .
Concave trade-off
close habitat optima
weak trade-off
Strong migration
Evolution with small mutations
Same singular strategies
Same condition for convergence stability
Almost same condition for evolutionary stability
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

53. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
. . . But with large mutations
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

54. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
. . . But with large mutations
Two patches Never more than two species coexisting
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

55. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
. . . But with large mutations
Two patches Never more than two species coexisting
Continuous Sometimes more than two species coexisting
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

56. Introduction Model Results Conclusion More?
Conclusion: Spatially continuous vs. patchy
. . . But with large mutations
Two patches Never more than two species coexisting
Continuous Sometimes more than two species coexisting
Spatial location
SA
SB
G
0qS qSS S
0.
0.2
0.4
0.6
0.8
1.
Spatial location
Proportion
time
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 15 / 16

57. Introduction Model Results Conclusion More?
Acknowledgments
Virginie Ravign´e
Thomas Lenormand
Minus van Baalen
Patrice David
Oph´elie Ronce
Fran¸cois Rousset
Audrey Coreau
Pierre-Andr´e Crochet
Aur´elie Cailleau
Federico Manna
Nicolas Rode
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 16 / 16

58. Introduction Model Results Conclusion More?
Acknowledgments
Virginie Ravign´e
Thomas Lenormand
Minus van Baalen
Patrice David
Oph´elie Ronce
Fran¸cois Rousset
Audrey Coreau
Pierre-Andr´e Crochet
Aur´elie Cailleau
Federico Manna
Nicolas Rode
and you for your attention !
F. D´ebarre and S. Gandon Evolution of specialist vs generalist strategies in a continuous environment 28-08-2009 16 / 16

59. Introduction Model Results Conclusion More?
Invasion condition
Does a mutant with trait sm invade a population fixed for trait sr?
Case where sm > sr : upper triangle in the PIP

60. Introduction Model Results Conclusion More?
Invasion condition
Does a mutant with trait sm invade a population fixed for trait sr?
Case where sm > sr : upper triangle in the PIP
Yes, when
a >
1
wA
arctan −
wB
wA
tanh (b
√
−wB )

61. Introduction Model Results Conclusion More?
Invasion condition
Does a mutant with trait sm invade a population fixed for trait sr?
Case where sm > sr : upper triangle in the PIP
Yes, when
a >
1
wA
arctan −
wB
wA
tanh (b
√
−wB )
Spatial Parameters
a =
qS
√
2
σ
; b =
(1 − q)S
√
2
σ

62. Introduction Model Results Conclusion More?
Invasion condition
Does a mutant with trait sm invade a population fixed for trait sr?
Case where sm > sr : upper triangle in the PIP
Yes, when
a >
1
wA
arctan −
wB
wA
tanh (b
√
−wB )
Spatial Parameters
a =
qS
√
2
σ
; b =
(1 − q)S
√
2
σ
Invasion fitness in absence of migration
wA =
rA(sm) − rA(sr)
rA(sr)
; wB =
rB (sm) − rB (sr)
rB (sr)

63. Introduction Model Results Conclusion More?
Habitats and fitness
With a general trade-off

64. Introduction Model Results Conclusion More?
Habitats and fitness
With a general trade-off
fitness in B = u( fitness in A )

65. Introduction Model Results Conclusion More?
Habitats and fitness
With a general trade-off
fitness in B = u( fitness in A )
rB (s) = u(rA(s))

66. Introduction Model Results Conclusion More?
Habitats and fitness
With a general trade-off
fitness in B = u( fitness in A )
rB (s) = u(rA(s))
u is a decreasing function
u is derivable (at least) twice

67. Introduction Model Results Conclusion More?
Evolution of s
With a general trade-off rB = u(rA)
Singular Strategy
rA(s∗
) u (rA(s∗
))
u(rA(s∗))
= −
q
1 − q
Convergence stability
u (rA(s∗
)) <
q u(rA(s∗
))
(1 − q)2 rA(s∗)2
Evolutionary stability
u (rA(s∗
)) < 0
(σ/S)2
2
>
q (1 − q)
3
2 q d u(rA(s∗
))
−u (rA(s∗)) rA(s∗)2 (1 − q)2

68. Introduction Model Results Conclusion More?
Two-patch models
(or metapopulations)
Continuous-time soft selection model with limited migration
Habitat A Habitat B
1-q q
μ
μ

69. Introduction Model Results Conclusion More?
Two-patch models
(or metapopulations)
Continuous-time soft selection model with limited migration
Habitat A Habitat B
1-q q
μ
μ
dpA
i
d t
= pA
i
rA(si )
¯rA
− 1 + µ (1 − q) (pB
i − pA
i )
dpB
i
d t
= pB
i
rB (si )
¯rB
− 1 + µ q (pA
i − pB
i )

70. Introduction Model Results Conclusion More?
Two-patch models
(or metapopulations)
Continuous-time soft selection model with limited migration
Habitat A Habitat B
1-q q
μ
μ
dpA
i
d t
= pA
i
rA(si )
¯rA
− 1 + µ (1 − q) (pB
i − pA
i )
dpB
i
d t
= pB
i
rB (si )
¯rB
− 1 + µ q (pA
i − pB
i )

71. Introduction Model Results Conclusion More?
Two-patch models
(or metapopulations)
Continuous-time soft selection model with limited migration
Habitat A Habitat B
1-q q
μ
μ
dpA
i
d t
= pA
i
rA(si )
¯rA
− 1 + µ (1 − q) (pB
i − pA
i )
dpB
i
d t
= pB
i
rB (si )
¯rB
− 1 + µ q (pA
i − pB
i )