Presented at the BioFprum in Natal, RN, Brazil, in 2010. Joint work with and F. Azevedo and D.J. Pamplona da Silva.

1. ift-unesp
Area Effects and Competition
in Fragments
Presented at the BioForum-UFRN Roberto A. Kraenkel
January 28th, 2010 Institute for Theoretical Physics - UNESP
São Paulo

2. ift-unesp
Area Effects and Competition
in Fragments
Presented at the BioForum-UFRN Roberto A. Kraenkel
January 28th, 2010 Institute for Theoretical Physics - UNESP
São Paulo

3. ift-unesp
What are area effects

4. ift-unesp
What are area effects
• Properties of populations may be different
depending on the space available.

5. ift-unesp
What are area effects
• Properties of populations may be different
depending on the space available.
• In particular, densities depend on area of
the habitat.

6. ift-unesp
What are area effects
• Properties of populations may be different
depending on the space available.
• In particular, densities depend on area of
the habitat.
• Common sense says that densities are
larger for larger areas.

7. ift-unesp
What are area effects
• Properties of populations may be different
depending on the space available.
• In particular, densities depend on area of
the habitat.
• Common sense says that densities are
larger for larger areas.
• Why?

8. ift-unesp
AREA EFFECTS

9. ift-unesp
AREA EFFECTS
Area effects come from:

10. ift-unesp
AREA EFFECTS
Area effects come from:
diffusion through the boundaries

11. ift-unesp
AREA EFFECTS
Area effects come from:
diffusion through the boundaries
X

12. ift-unesp
AREA EFFECTS
Area effects come from:
diffusion through the boundaries
X
growth inside the area

13. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries
X
growth inside the area

14. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries These depend on perimeter of the fragment
X
growth inside the area

15. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries These depend on perimeter of the fragment
X
growth inside the area These depend on the area of the fragment

16. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries These depend on perimeter of the fragment
X
growth inside the area These depend on the area of the fragment
Say the typical radius of the
patch is R
Area grows with R^2
Perimeter grows with R

17. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries These depend on perimeter of the fragment
X
growth inside the area These depend on the area of the fragment
Say the typical radius of the Large Areas -- dominated by
patch is R growth
Area grows with R^2
Perimeter grows with R

18. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries These depend on perimeter of the fragment
X
growth inside the area These depend on the area of the fragment
Say the typical radius of the Large Areas -- dominated by
patch is R growth
Area grows with R^2 Small Areas -- dominated by
Perimeter grows with R diffusion

19. ift-unesp
AREA EFFECTS
Area effects come from:
iffusion through the boundaries These depend on perimeter of the fragment
X
growth inside the area These depend on the area of the fragment
The larger the area, the larger the density of the population

20. ift-unesp
However....

21. ift-unesp
However....
not all populations respond in the same way to area change

22. ift-unesp
However....
not all populations respond in the same way to area change
Some respond positively to area increase .

23. ift-unesp
However....
not all populations respond in the same way to area change
Some respond positively to area increase .
Other just respond in the opposite way: as larger the
area, smaller the population!

24. ift-unesp
However....
not all populations respond in the same way to area change
Some respond positively to area increase .
Other just respond in the opposite way: as larger the
area, smaller the population!
that’s a minority....

25. ift-unesp
However....
not all populations respond in the same way to area change
Some respond positively to area increase .
Other just respond in the opposite way: as larger the
area, smaller the population!
We will build models to
understand the situation
that’s a minority....

26. ift-unesp
Fisher, Kolmogorov and Skellam...

27. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:

28. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion

29. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
growth

30. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
growth
saturation

31. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
growth
saturation

32. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
growth
saturation
diffusion

33. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
growth
saturation
diffusion growth

34. ift-unesp
Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
growth
saturation
diffusion growth saturation

35. ift-unesp

36. ift-unesp

37. ift-unesp
Take space to be one-dimensional

38. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:

39. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:

40. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:
If the habitat is finite

41. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:
If the habitat is finite

42. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:
If the habitat is finite

43. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:
Notice: the height of the plateau
depends on the size of the patch.
If the habitat is finite

44. ift-unesp
Take space to be one-dimensional
If the habitat is infinite
solutions look like:
How does the maximum
density depend on the size
of the patch?

45. ift-unesp

46. ift-unesp
How does the maximum
density depend on the size
of the patch?

47. ift-unesp
How does the maximum
density depend on the size
of the patch?
maximal density
L

48. ift-unesp
How does the maximum
density depend on the size
of the patch?
maximal density
There is a minimal
length ( area) for a
population to exist on
a patch!
(Skellam 1951)
L

49. ift-unesp
How does the maximum
density depend on the size
of the patch?
maximal density
There is a minimal
length ( area) for a
population to exist on
a patch!
(Skellam 1951)
L

50. ift-unesp

51. ift-unesp
Any truth in these conclusions?

52. ift-unesp
Any truth in these conclusions?
Seems so....

53. ift-unesp
Any truth in these conclusions?
Seems so....
PDBFF results for
understorey birds in
fragments of
Amazonian forest
(Ferraz et alli, Science 2007)

54. ift-unesp
Any truth in these conclusions?
Seems so....
PDBFF results for
understorey birds in
fragments of
Amazonian forest
(Ferraz et alli, Science 2007)
Main hypothesis: non-interacting populations

55. Competition models in patches ift-unesp
joint work with F.S. Azevedo

56. Competition models in patches ift-unesp
joint work with F.S. Azevedo
This is written in non-dimensional variables. This is not essential. What it
means: both species diffuse, grow, saturate and compete.

57. Competition models in patches ift-unesp
joint work with F.S. Azevedo
This is written in non-dimensional variables. This is not essential. What it
means: both species diffuse, grow, saturate and compete.
Let species 1 be
competively stronger
than species 2

58. Competition models in patches ift-unesp
joint work with F.S. Azevedo
This is written in non-dimensional variables. This is not essential. What it
means: both species diffuse, grow, saturate and compete.
Let species 1 be
competively stronger
than species 2

59. Competition models in patches ift-unesp
joint work with F.S. Azevedo
This is written in non-dimensional variables. This is not essential. What it
means: both species diffuse, grow, saturate and compete.
Let species 1 be
competively stronger
than species 2
Recall: if space is unbouded ==> competitive
exclusion

60. ift-unesp
Competition models in patches

61. ift-unesp
Competition models in patches

62. ift-unesp
Competition models in patches
Let the patch have area A and plot
the maximal density against the
area

63. ift-unesp
Competition models in patches
Let the patch have area A and plot
the maximal density against the
area
max. density
A
Species 1

64. ift-unesp
Competition models in patches
Let the patch have area A and plot
the maximal density against the
area
max. density max. density
A
A
Species 1 Species 2

65. ift-unesp
Competition models in patches
Let the patch have area A and plot
the maximal density against the
area
Population 2
grows with area
decrease
max. density max. density
A
A
Species 1 Species 2

66. ift-unesp
Competition models in patches
Let the patch have area A and plot
the maximal density against the
area
Population 2
grows with area
decrease
max. density max. density
A
A
Competitive exclusion is not valid!
Species 1 Species 2

67. ift-unesp
Competition models in patches
Let the patch have area A and plot
the maximal density against the
area
Population 2
grows with area
decrease
max. density max. density
Strong
coexistence
A
A
Competitive exclusion is not valid!
Species 1 Species 2

68. ift-unesp
Let us re-phrase it

69. ift-unesp
Let us re-phrase it
We have coexistence mediated by habitat restriction

70. ift-unesp
Let us re-phrase it
We have coexistence mediated by habitat restriction
Indeed: area reduction affects the stronger competitor

71. ift-unesp
Let us re-phrase it
We have coexistence mediated by habitat restriction
Indeed: area reduction affects the stronger competitor
On a certain scale, the weaker competitor gets release of
competitive pressure

72. ift-unesp
Let us re-phrase it
We have coexistence mediated by habitat restriction
Indeed: area reduction affects the stronger competitor
On a certain scale, the weaker competitor gets release of
competitive pressure
So, what we found is competitive release induced by area effects.

73. ift-unesp
Any chance seeing this in nature?

74. ift-unesp
Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)

75. ift-unesp
Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)

76. ift-unesp
Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
acouchies
(Myoprocta sp.,
Dasyproctidae,
Rodentia)
agoutis
(Dasyprocta sp.,
Dasyproctidae,
Rodentia)

77. ift-unesp
Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
acouchies
(Myoprocta sp.,
Dasyproctidae,
Rodentia)
agoutis
(Dasyprocta sp.,
Dasyproctidae,
Rodentia)

78. ift-unesp
Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
acouchies
(Myoprocta sp.,
Dasyproctidae,
Rodentia)
Fitted Model
agoutis
(Dasyprocta sp.,
Dasyproctidae,
Rodentia)

79. ift-unesp
Discussion

80. ift-unesp
Discussion
To fit a model doesn’t prove it is right...

81. ift-unesp
Discussion
To fit a model doesn’t prove it is right...
We could imagine other models that reproduce the
data

82. ift-unesp
Discussion
To fit a model doesn’t prove it is right...
We could imagine other models that reproduce the
data
Ours is possibly the simplest...

83. ift-unesp
Discussion
To fit a model doesn’t prove it is right...
We could imagine other models that reproduce the
data
Ours is possibly the simplest...
On a theoretical perspective we could argue that “there is
no competition” ( as competition for resources is indirect)

84. ift-unesp
Discussion
To fit a model doesn’t prove it is right...
We could imagine other models that reproduce the
data
Ours is possibly the simplest...
On a theoretical perspective we could argue that “there is
no competition” ( as competition for resources is indirect)
Better: consider consumer-resource models

85. ift-unesp
Discussion
To fit a model doesn’t prove it is right...
We could imagine other models that reproduce the
data
Ours is possibly the simplest...
On a theoretical perspective we could argue that “there is
no competition” ( as competition for resources is indirect)
Better: consider consumer-resource models
( we are working on it...)

86. ift-unesp
Thank for your attention
Roberto Kraenkel
kraenkel@ift.unesp.br