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Presented at the BioFprum in Natal, RN, Brazil, in 2010. Joint work with and F. Azevedo and D.J. Pamplona da Silva.
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Area Effects and Competition in Fragments
Roberto A. KraenkelInstitute for Theoretical Physics - UNESP
São Paulo
Presented at the BioForum-UFRNJanuary 28th, 2010
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Area Effects and Competition in Fragments
Roberto A. KraenkelInstitute for Theoretical Physics - UNESP
São Paulo
Presented at the BioForum-UFRNJanuary 28th, 2010
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What are area effects
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What are area effects
• Properties of populations may be different depending on the space available.
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What are area effects
• Properties of populations may be different depending on the space available.
• In particular, densities depend on area of the habitat.
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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.
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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?
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AREA EFFECTS
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AREA EFFECTS
Area effects come from:
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AREA EFFECTS
Area effects come from:
diffusion through the boundaries
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
These depend on perimeter of the fragment
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
These depend on perimeter of the fragment
These depend on the area of the fragment
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
These depend on perimeter of the fragment
These depend on the area of the fragment
Say the typical radius of the patch is R
Area grows with R^2Perimeter grows with R
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
These depend on perimeter of the fragment
These depend on the area of the fragment
Say the typical radius of the patch is R
Area grows with R^2Perimeter grows with R
Large Areas -- dominated by growth
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
These depend on perimeter of the fragment
These depend on the area of the fragment
Say the typical radius of the patch is R
Area grows with R^2Perimeter grows with R
Large Areas -- dominated by growth
Small Areas -- dominated by diffusion
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AREA EFFECTS
Area effects come from:
diffusion through the boundariesX
growth inside the area
These depend on perimeter of the fragment
These depend on the area of the fragment
The larger the area, the larger the density of the population
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However....
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However....
not all populations respond in the same way to area change
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However....
not all populations respond in the same way to area change
Some respond positively to area increase .
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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!
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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....
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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....We will build models to understand the situation
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Fisher, Kolmogorov and Skellam...
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion growth
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion growth
saturation
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion growth
saturation
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion growth
saturation
diffusion
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion growth
saturation
diffusion growth
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Fisher, Kolmogorov and Skellam...
To model the situation we posit an equation with:
Diffusion growth
saturation
diffusion growth saturation
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Take space to be one-dimensional
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
If the habitat is finite
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
If the habitat is finite
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
If the habitat is finite
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
If the habitat is finite
Notice: the height of the plateau depends on the size of the patch.
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If the habitat is infinite solutions look like:
Take space to be one-dimensional
How does the maximum density depend on the size
of the patch?
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ift-unespHow does the maximum
density depend on the size of the patch?
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density depend on the size of the patch?
maximal density
L
ift-unespHow does the maximum
density depend on the size of the patch?
maximal density
L
There is a minimal length ( area) for a
population to exist on a patch!
(Skellam 1951)
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density depend on the size of the patch?
maximal density
L
There is a minimal length ( area) for a
population to exist on a patch!
(Skellam 1951)
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Any truth in these conclusions?
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Any truth in these conclusions?
Seems so....
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Any truth in these conclusions?
Seems so....
PDBFF results for understorey birds in
fragments of Amazonian forest
(Ferraz et alli, Science 2007)
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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
ift-unespCompetition models in patchesjoint work with F.S. Azevedo
ift-unespCompetition models in patchesjoint 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.
ift-unespCompetition models in patchesjoint work with F.S. Azevedo
Let species 1 be competively stronger
than species 2
This is written in non-dimensional variables. This is not essential. What it means: both species diffuse, grow, saturate and compete.
ift-unespCompetition models in patchesjoint work with F.S. Azevedo
Let species 1 be competively stronger
than species 2
This is written in non-dimensional variables. This is not essential. What it means: both species diffuse, grow, saturate and compete.
ift-unespCompetition models in patchesjoint work with F.S. Azevedo
Let species 1 be competively stronger
than species 2
Recall: if space is unbouded ==> competitive exclusion
This is written in non-dimensional variables. This is not essential. What it means: both species diffuse, grow, saturate and compete.
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Competition models in patches
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Competition models in patches
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Competition models in patches
Let the patch have area A and plot the maximal density against the
area
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Competition models in patches
Let the patch have area A and plot the maximal density against the
area
Species 1
A
max. density
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Competition models in patches
Let the patch have area A and plot the maximal density against the
area
Species 1 Species 2
AA
max. densitymax. density
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Competition models in patches
Let the patch have area A and plot the maximal density against the
area
Species 1 Species 2
Population 2 grows with area
decrease
AA
max. densitymax. density
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Competition models in patches
Let the patch have area A and plot the maximal density against the
area
Species 1 Species 2Competitive exclusion is not valid!
Population 2 grows with area
decrease
AA
max. densitymax. density
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Competition models in patches
Let the patch have area A and plot the maximal density against the
area
Species 1 Species 2Competitive exclusion is not valid!
Strong coexistence
Population 2 grows with area
decrease
AA
max. densitymax. density
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Let us re-phrase it
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Let us re-phrase it
We have coexistence mediated by habitat restriction
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Let us re-phrase it
We have coexistence mediated by habitat restriction
Indeed: area reduction affects the stronger competitor
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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
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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.
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Any chance seeing this in nature?
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Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
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Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
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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)
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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)
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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)
Fitted Model
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Discussion
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Discussion
To fit a model doesn’t prove it is right...
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Discussion
To fit a model doesn’t prove it is right...
We could imagine other models that reproduce the data
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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...
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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)
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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
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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...)