SlideShare a Scribd company logo
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
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
ift-unesp


What are area effects
ift-unesp


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


Area effects come from:
ift-unesp
       AREA EFFECTS


    Area effects come from:

diffusion through the boundaries
ift-unesp
       AREA EFFECTS


    Area effects come from:

diffusion through the boundaries
                X
ift-unesp
       AREA EFFECTS


    Area effects come from:

diffusion through the boundaries
                X
      growth inside the area
ift-unesp
                     AREA EFFECTS


  Area effects come from:

iffusion through the boundaries
               X
     growth inside the area
ift-unesp
                     AREA EFFECTS


  Area effects come from:

iffusion through the boundaries   These depend on perimeter of the fragment

               X
     growth inside the area
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
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
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
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
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
ift-unesp


However....
ift-unesp


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


               However....
not all populations respond in the same way to area change



    Some respond positively to area increase .
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!
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....
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....
ift-unesp



Fisher, Kolmogorov and Skellam...
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                     Diffusion
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                     Diffusion
                     growth
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                     Diffusion
                     growth
                    saturation
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                     Diffusion
                     growth
                    saturation
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                         Diffusion
                         growth
                        saturation



                diffusion
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                         Diffusion
                         growth
                        saturation



                diffusion     growth
ift-unesp



Fisher, Kolmogorov and Skellam...

To model the situation we posit an equation with:

                         Diffusion
                         growth
                        saturation



                diffusion     growth   saturation
ift-unesp
ift-unesp
ift-unesp




Take space to be one-dimensional
ift-unesp




          Take space to be one-dimensional

If the habitat is infinite
   solutions look like:
ift-unesp




          Take space to be one-dimensional

If the habitat is infinite
   solutions look like:
ift-unesp




          Take space to be one-dimensional

If the habitat is infinite
   solutions look like:




If the habitat is finite
ift-unesp




          Take space to be one-dimensional

If the habitat is infinite
   solutions look like:




If the habitat is finite
ift-unesp




          Take space to be one-dimensional

If the habitat is infinite
   solutions look like:




If the habitat is finite
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
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?
ift-unesp
ift-unesp
 How does the maximum
density depend on the size
       of the patch?
ift-unesp
                   How does the maximum
                  density depend on the size
                         of the patch?


maximal density




                               L
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
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
ift-unesp
ift-unesp

Any truth in these conclusions?
ift-unesp

Any truth in these conclusions?

         Seems so....
ift-unesp

Any truth in these conclusions?

         Seems so....




                     PDBFF results for
                    understorey birds in
                       fragments of
                     Amazonian forest
                   (Ferraz et alli, Science 2007)
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
Competition models in patches                     ift-unesp

                   joint work with F.S. Azevedo
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.
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
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
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
ift-unesp

Competition models in patches
ift-unesp

Competition models in patches
ift-unesp

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

Let us re-phrase it
ift-unesp

       Let us re-phrase it

We have coexistence mediated by habitat restriction
ift-unesp

        Let us re-phrase it

We have coexistence mediated by habitat restriction

Indeed: area reduction affects the stronger competitor
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
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.
ift-unesp
Any chance seeing this in nature?
ift-unesp
          Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
ift-unesp
          Any chance seeing this in nature?
PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
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)
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)
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)
ift-unesp

Discussion
ift-unesp

            Discussion

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

                Discussion

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




Thank for your attention


                Roberto Kraenkel
              kraenkel@ift.unesp.br

More Related Content

Viewers also liked

Tópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática IIITópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática III
Roberto Kraenkel
 
Resonances and bi-modality in a model for cholera
Resonances and bi-modality in a model for choleraResonances and bi-modality in a model for cholera
Resonances and bi-modality in a model for cholera
Roberto Kraenkel
 
Aula quatro jornadas12_handout
Aula quatro jornadas12_handoutAula quatro jornadas12_handout
Aula quatro jornadas12_handout
Roberto Kraenkel
 
Intraguild mutualism
Intraguild mutualismIntraguild mutualism
Intraguild mutualism
Roberto Kraenkel
 
Modelos Matemáticos para Epidemias
Modelos Matemáticos para EpidemiasModelos Matemáticos para Epidemias
Modelos Matemáticos para Epidemias
Roberto Kraenkel
 
Mesopredadores
MesopredadoresMesopredadores
Mesopredadores
Roberto Kraenkel
 
Mutualismo
MutualismoMutualismo
Mutualismo
claufil
 
Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado  Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado
Roberto Kraenkel
 
Atraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas BiológicosAtraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas Biológicos
Roberto Kraenkel
 
Mutualismo
Mutualismo Mutualismo
Mutualismo
Roberto Kraenkel
 
Tópicos em Biologia Matemática I
Tópicos em Biologia Matemática ITópicos em Biologia Matemática I
Tópicos em Biologia Matemática I
Roberto Kraenkel
 
Métodos Matemáticos em Biologia de Populações V
Métodos Matemáticos em Biologia de Populações VMétodos Matemáticos em Biologia de Populações V
Métodos Matemáticos em Biologia de Populações V
Roberto Kraenkel
 
Tópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática IITópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática II
Roberto Kraenkel
 
Tópicos de Biologia-Matemática
Tópicos de Biologia-MatemáticaTópicos de Biologia-Matemática
Tópicos de Biologia-Matemática
Roberto Kraenkel
 
population dynamics of insects
population dynamics of insects population dynamics of insects
population dynamics of insects
Roberto Kraenkel
 
Efeito Allee
Efeito AlleeEfeito Allee
Efeito Allee
Roberto Kraenkel
 
Relações ecológicas
Relações ecológicasRelações ecológicas
Relações ecológicas
GMarques99
 

Viewers also liked (17)

Tópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática IIITópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática III
 
Resonances and bi-modality in a model for cholera
Resonances and bi-modality in a model for choleraResonances and bi-modality in a model for cholera
Resonances and bi-modality in a model for cholera
 
Aula quatro jornadas12_handout
Aula quatro jornadas12_handoutAula quatro jornadas12_handout
Aula quatro jornadas12_handout
 
Intraguild mutualism
Intraguild mutualismIntraguild mutualism
Intraguild mutualism
 
Modelos Matemáticos para Epidemias
Modelos Matemáticos para EpidemiasModelos Matemáticos para Epidemias
Modelos Matemáticos para Epidemias
 
Mesopredadores
MesopredadoresMesopredadores
Mesopredadores
 
Mutualismo
MutualismoMutualismo
Mutualismo
 
Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado  Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado
 
Atraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas BiológicosAtraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas Biológicos
 
Mutualismo
Mutualismo Mutualismo
Mutualismo
 
Tópicos em Biologia Matemática I
Tópicos em Biologia Matemática ITópicos em Biologia Matemática I
Tópicos em Biologia Matemática I
 
Métodos Matemáticos em Biologia de Populações V
Métodos Matemáticos em Biologia de Populações VMétodos Matemáticos em Biologia de Populações V
Métodos Matemáticos em Biologia de Populações V
 
Tópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática IITópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática II
 
Tópicos de Biologia-Matemática
Tópicos de Biologia-MatemáticaTópicos de Biologia-Matemática
Tópicos de Biologia-Matemática
 
population dynamics of insects
population dynamics of insects population dynamics of insects
population dynamics of insects
 
Efeito Allee
Efeito AlleeEfeito Allee
Efeito Allee
 
Relações ecológicas
Relações ecológicasRelações ecológicas
Relações ecológicas
 

Area Effects and Competition in Fragments

  • 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
  • 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?
  • 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
  • 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....
  • 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
  • 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?
  • 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
  • 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
  • 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
  • 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)
  • 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