Area Effects and Competition in Fragments

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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

  1. 1. ift-unesp Area Effects and Competition in FragmentsPresented at the BioForum-UFRN Roberto A. Kraenkel January 28th, 2010 Institute for Theoretical Physics - UNESP São Paulo
  2. 2. ift-unesp Area Effects and Competition in FragmentsPresented at the BioForum-UFRN Roberto A. Kraenkel January 28th, 2010 Institute for Theoretical Physics - UNESP São Paulo
  3. 3. ift-unespWhat are area effects
  4. 4. ift-unespWhat are area effects• Properties of populations may be different depending on the space available.
  5. 5. ift-unespWhat are area effects• Properties of populations may be different depending on the space available.• In particular, densities depend on area of the habitat.
  6. 6. ift-unespWhat 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. 7. ift-unespWhat 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. 8. ift-unespAREA EFFECTS
  9. 9. ift-unesp AREA EFFECTSArea effects come from:
  10. 10. ift-unesp AREA EFFECTS Area effects come from:diffusion through the boundaries
  11. 11. ift-unesp AREA EFFECTS Area effects come from:diffusion through the boundaries X
  12. 12. ift-unesp AREA EFFECTS Area effects come from:diffusion through the boundaries X growth inside the area
  13. 13. ift-unesp AREA EFFECTS Area effects come from:iffusion through the boundaries X growth inside the area
  14. 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. 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. 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. 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. 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. 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. 20. ift-unespHowever....
  21. 21. ift-unesp However....not all populations respond in the same way to area change
  22. 22. ift-unesp However....not all populations respond in the same way to area change Some respond positively to area increase .
  23. 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. 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. 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 tounderstand the situation that’s a minority....
  26. 26. ift-unespFisher, Kolmogorov and Skellam...
  27. 27. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with:
  28. 28. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion
  29. 29. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion growth
  30. 30. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion growth saturation
  31. 31. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion growth saturation
  32. 32. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion growth saturation diffusion
  33. 33. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion growth saturation diffusion growth
  34. 34. ift-unespFisher, Kolmogorov and Skellam...To model the situation we posit an equation with: Diffusion growth saturation diffusion growth saturation
  35. 35. ift-unesp
  36. 36. ift-unesp
  37. 37. ift-unespTake space to be one-dimensional
  38. 38. ift-unesp Take space to be one-dimensionalIf the habitat is infinite solutions look like:
  39. 39. ift-unesp Take space to be one-dimensionalIf the habitat is infinite solutions look like:
  40. 40. ift-unesp Take space to be one-dimensionalIf the habitat is infinite solutions look like:If the habitat is finite
  41. 41. ift-unesp Take space to be one-dimensionalIf the habitat is infinite solutions look like:If the habitat is finite
  42. 42. ift-unesp Take space to be one-dimensionalIf the habitat is infinite solutions look like:If the habitat is finite
  43. 43. ift-unesp Take space to be one-dimensionalIf 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. 44. ift-unesp Take space to be one-dimensionalIf the habitat is infinite solutions look like: How does the maximum density depend on the size of the patch?
  45. 45. ift-unesp
  46. 46. ift-unesp How does the maximumdensity depend on the size of the patch?
  47. 47. ift-unesp How does the maximum density depend on the size of the patch?maximal density L
  48. 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. 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. 50. ift-unesp
  51. 51. ift-unespAny truth in these conclusions?
  52. 52. ift-unespAny truth in these conclusions? Seems so....
  53. 53. ift-unespAny truth in these conclusions? Seems so.... PDBFF results for understorey birds in fragments of Amazonian forest (Ferraz et alli, Science 2007)
  54. 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. 55. Competition models in patches ift-unesp joint work with F.S. Azevedo
  56. 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. 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 becompetively stronger than species 2
  58. 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 becompetively stronger than species 2
  59. 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 becompetively stronger than species 2Recall: if space is unbouded ==> competitive exclusion
  60. 60. ift-unespCompetition models in patches
  61. 61. ift-unespCompetition models in patches
  62. 62. ift-unespCompetition models in patches Let the patch have area A and plot the maximal density against the area
  63. 63. ift-unespCompetition models in patches Let the patch have area A and plot the maximal density against the areamax. density ASpecies 1
  64. 64. ift-unespCompetition models in patches Let the patch have area A and plot the maximal density against the areamax. density max. density A ASpecies 1 Species 2
  65. 65. ift-unespCompetition models in patches Let the patch have area A and plot the maximal density against the area Population 2 grows with area decreasemax. density max. density A ASpecies 1 Species 2
  66. 66. ift-unespCompetition models in patches Let the patch have area A and plot the maximal density against the area Population 2 grows with area decreasemax. density max. density A A Competitive exclusion is not valid!Species 1 Species 2
  67. 67. ift-unespCompetition models in patches Let the patch have area A and plot the maximal density against the area Population 2 grows with area decreasemax. density max. density Strong coexistence A A Competitive exclusion is not valid!Species 1 Species 2
  68. 68. ift-unespLet us re-phrase it
  69. 69. ift-unesp Let us re-phrase itWe have coexistence mediated by habitat restriction
  70. 70. ift-unesp Let us re-phrase itWe have coexistence mediated by habitat restrictionIndeed: area reduction affects the stronger competitor
  71. 71. ift-unesp Let us re-phrase it We have coexistence mediated by habitat restriction Indeed: area reduction affects the stronger competitorOn a certain scale, the weaker competitor gets release of competitive pressure
  72. 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 pressureSo, what we found is competitive release induced by area effects.
  73. 73. ift-unespAny chance seeing this in nature?
  74. 74. ift-unesp Any chance seeing this in nature?PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
  75. 75. ift-unesp Any chance seeing this in nature?PDBFF data ( M. L.S.P. Jorge, Biol. Conser. v. 141(2008) 617.)
  76. 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. 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. 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. 79. ift-unespDiscussion
  80. 80. ift-unesp DiscussionTo fit a model doesn’t prove it is right...
  81. 81. ift-unesp Discussion To fit a model doesn’t prove it is right...We could imagine other models that reproduce the data
  82. 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. 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 isno competition” ( as competition for resources is indirect)
  84. 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 isno competition” ( as competition for resources is indirect) Better: consider consumer-resource models
  85. 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 isno competition” ( as competition for resources is indirect) Better: consider consumer-resource models ( we are working on it...)
  86. 86. ift-unespThank for your attention Roberto Kraenkel kraenkel@ift.unesp.br

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