Area Effects and Competition in Fragments

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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?

ift-unespHow does the maximum

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)

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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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...)

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Thank for your attention

Roberto Kraenkelkraenkel@ift.unesp.br