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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What are area effects

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• Properties of populations may be different depending on the space available.

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• In particular, densities depend on area of the habitat.

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• Common sense says that densities are larger for larger areas.

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• 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


diffusion through the boundaries

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AREA EFFECTS


diffusion through the boundariesX

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AREA EFFECTS



growth inside the area

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AREA EFFECTS




These depend on perimeter of the fragment

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AREA EFFECTS





These depend on the area of the fragment

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AREA EFFECTS






Say the typical radius of the patch is R

Area grows with R^2Perimeter grows with R

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AREA EFFECTS








Large Areas -- dominated by growth

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AREA EFFECTS








Large Areas -- dominated by growth

Small Areas -- dominated by diffusion

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AREA EFFECTS






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....


Some respond positively to area increase .

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However....



Other just respond in the opposite way: as larger the area, smaller the population!

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However....




that’s a minority....

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However....




that’s a minority....We will build models to understand the situation

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Fisher, Kolmogorov and Skellam...

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To model the situation we posit an equation with:

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Diffusion

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Diffusion growth

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Diffusion growth

saturation

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Diffusion growth

saturation

diffusion

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Diffusion growth

saturation

diffusion growth

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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:


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If the habitat is finite

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If the habitat is finite

Notice: the height of the plateau depends on the size of the patch.

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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?



maximal density

L



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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Seems so....

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Seems so....

PDBFF results for understorey birds in

fragments of Amazonian forest

(Ferraz et alli, Science 2007)

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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


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



Let species 1 be competively stronger

than species 2

Recall: if space is unbouded ==> competitive exclusion


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Competition models in patches

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Let the patch have area A and plot the maximal density against the

area

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area

Species 1

A

max. density

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area

Species 1 Species 2

AA

max. densitymax. density

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area

Species 1 Species 2

Population 2 grows with area

decrease

AA


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area

Species 1 Species 2Competitive exclusion is not valid!


decrease

AA


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area

Species 1 Species 2Competitive exclusion is not valid!

Strong coexistence


decrease

AA


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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


Indeed: area reduction affects the stronger competitor

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Let us re-phrase it



On a certain scale, the weaker competitor gets release of competitive pressure

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Let us re-phrase it



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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PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)

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acouchies (Myoprocta sp., Dasyproctidae, Rodentia)

agoutis (Dasyprocta sp., Dasyproctidae, Rodentia)

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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


We could imagine other models that reproduce the data

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Discussion



Ours is possibly the simplest...

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Discussion




On a theoretical perspective we could argue that “there is no competition” ( as competition for resources is indirect)

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Discussion





Better: consider consumer-resource models

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Discussion





Better: consider consumer-resource models( we are working on it...)

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Thank for your attention

Roberto [email protected]

Area Effects and Competition in Fragments

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