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Facilitation Systems         Maria Luisa Jorge       Marina Cenamo Salles        Raquel A. F. Neves         Sabastian Krie...
Presentation Outline• Facilitation: definition and conceptual  models• Facilitation vs. competition in a gradient  of envi...
Facilitation  Species thatpositively affectsanother species,   directly or   indirectly.
Bruno et al. TREE
Bruno et al., 2003
Bruno et al., 2003
S-       -  --A    B   -
Salt Marshes          Physical conditions          - Waterlogged soils          - High soil salinities
Juncus gerardiMore tolerant tohigh salinitiesacts onamelioration ofphysical conditions:shades the soil –limits surfaceevap...
Iva frutescensRelatively intolerant to high soil salinities andwaterlogged soil conditions
S     -               -                 --                    BJuncus girardi                  -      Iva frutescens
Bertness & Hacker, 1994
-         A                      -                     BChange of       Exponential   Saturation    Competition     Effect...
-                               -                           B               AChange of      Exponential   Saturation      ...
-A                                      B    Final      Initial  Effect of    salinity   salinity abundance of            ...
Reducing the number of parameters.Nondimensionalization        dA           A                   − A           =r A A1−  ...
Looking for fixed pointsCondition:  dA      A f , B f =0              A=0∨B=1− A− F A e                                ...
How do fixed points varie with theparameters?                                  1                              0 ∃ one ...
Some critical cases             dA      dBA=A f =0       =0     = B 1− B−S B =0        B f =0, B f =1−S B 0         ...
Some critical cases             dB     dA             − AB=B f =0       =0  = A1− A−S A e =0              A f =0 and ...
Species A                             Species B (with facilitation)                             Control for species B     ...
Species A                              Species B (with facilitation)                              Control for species B   ...
Species A                              Species B (with facilitation)                              Control for species B   ...
Species A                               Species B (with facilitation)                               Control for species BV...
¿Conditions of facilitation (treat - control > 0),competition (treat - control < 0) and no difference (treat- control = 0)...
Conditions of facilitation (treat - control > 0),competition (treat - control < 0) and no difference (treat- control = 0) ...
Conditions of facilitation (treat - control < 0), competition(treat - control > 0) and no difference (treat - control = 0)...
Real world example sustaining our model!   Sp. A     Stress level 0   Stress level 1   Stress level 2 Without B        900...
Other scenarios…               Auto-      S               facilitation of               the facilitated  -     -          ...
Forest Clearing
S-     -  ---A     B   -
This is what we had in the previous model:There areclear stablepoints
And this is what I ended up with:UNTRUE!
Another snapshot time:Population attime t = 4000
Population attime 400                Competitive                exclusion
Plain competition: A is excluded
Population attime 400                Competitive                exclusion     Competition/                              fa...
B needs A, both are stable
Population attime 400                Competitive                exclusion                   Successive    Competition/    ...
Looking for the bifurcation            T = 400                              T = 4000
Converges                 fast to                 stabilityOscilating atvery lowfrequency andhigh amplitude
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
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Facilitation in Population Dynamics

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This presentation is the result of a student group project, during the Southern-Summer School on Mathematical-Biology, hold in São Paulo, January 2012, http://www.ictp-saifr.org/mathbio

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Transcript of "Facilitation in Population Dynamics"

  1. 1. Facilitation Systems Maria Luisa Jorge Marina Cenamo Salles Raquel A. F. Neves Sabastian Krieger Tomás Gallo Aquino Victoria Romeo AznarSouthern Summer Mathematical Biology IFT – UNESP – São Paulo January 2012
  2. 2. Presentation Outline• Facilitation: definition and conceptual models• Facilitation vs. competition in a gradient of environmental stress• The simplest model• A real case our simple model• Results of the model• Other possible (and more complicated) scenarios
  3. 3. Facilitation Species thatpositively affectsanother species, directly or indirectly.
  4. 4. Bruno et al. TREE
  5. 5. Bruno et al., 2003
  6. 6. Bruno et al., 2003
  7. 7. S- - --A B -
  8. 8. Salt Marshes Physical conditions - Waterlogged soils - High soil salinities
  9. 9. Juncus gerardiMore tolerant tohigh salinitiesacts onamelioration ofphysical conditions:shades the soil –limits surfaceevaporation andaccumulation of soilsalts.
  10. 10. Iva frutescensRelatively intolerant to high soil salinities andwaterlogged soil conditions
  11. 11. S - - -- BJuncus girardi - Iva frutescens
  12. 12. Bertness & Hacker, 1994
  13. 13. - A - BChange of Exponential Saturation Competition Effect ofspecies A term of or logistic effect of salinity on(facilitator) population term of species B species Aabundance growth population (facilitated) (facilitator)over time growth on species A (facilitator)
  14. 14. - - B AChange of Exponential Saturation Competition Effect ofspecies B term of or logistic effect of salinity on(facilitated population term of species A species Babundance growth for population (facilitator) (facilitated)over time species B growth for on species species B B (facilitated)
  15. 15. -A B Final Initial Effect of salinity salinity abundance of species A on salinity
  16. 16. Reducing the number of parameters.Nondimensionalization dA A − A =r A A1− −b AB B−a A S 0 e  dt KA dA − A =A 1−A −c AB B − F A e  dt dB B − A =r B B1− −b BA A−a B S 0 e  dt KB dB − A =rB 1−B −c BA A − F B e  dt
  17. 17. Looking for fixed pointsCondition: dA  A f , B f =0 A=0∨B=1− A− F A e − A / c AB − A f Dont dt Af are that − A f − e =0 have dB B=0∨ B=1−c BA A− F B e − A dt  A f , B f =0 analytical solution !! =1−c AB , =1−c AB c BA , = F A −F B c AB − A fOk, we look ... 1− A f = e  1  0 ∃ one A f ≠0 else don t ∃ A f ≠0 We can have: No solution One solution
  18. 18. How do fixed points varie with theparameters?  1  0 ∃ one A f ≠0 − A 0   e f ∃ two A f ≠0 else don t ∃ A f ≠0We can have: No solution Two solutions
  19. 19. Some critical cases dA dBA=A f =0  =0  = B 1− B−S B =0 B f =0, B f =1−S B 0 dt dt S B 1 ∃ B f =0 is instable two fixed B f =1−S B is stable points If B doesnt suffer too much survives stress, and doesnt suffer competition S B 1 ∃ B f =0 is stable one fixed B f =1−S B don t ∃ points Although B doesnt have dies competition, the stress is hard
  20. 20. Some critical cases dB dA − AB=B f =0  =0  = A1− A−S A e =0 A f =0 and some A f 0 dt dt S A 1 ∃ A f =0 is instable ∃ A f ∈0,1 is stable If A doesnt suffer too much survives stress, and doesnt suffer competition A f =0 is stable S A 1 ∃ A f 0 don t ∃ if  is small ∃ two A f ∈0,1 if  is large. One stable , the other instable. dies With a high self-facilitation, A can survive if it has already large numbers survives
  21. 21. Species A Species B (with facilitation) Control for species B SalinityTemporal variation of both species and salinity when, atequilibrium, both co-exist.
  22. 22. Species A Species B (with facilitation) Control for species B SalinityTemporal variation of both species and salinity when, atequilibrium, both co-exist.
  23. 23. Species A Species B (with facilitation) Control for species B SalinityTemporal variation of both species and salinity when, at equilibrium, species B goes extinct.
  24. 24. Species A Species B (with facilitation) Control for species BVariation of abundance of both species with respect to salinity.
  25. 25. ¿Conditions of facilitation (treat - control > 0),competition (treat - control < 0) and no difference (treat- control = 0) with respect to a gradient of salinity andcompetition of B on A. Beta ab: 0.68
  26. 26. Conditions of facilitation (treat - control > 0),competition (treat - control < 0) and no difference (treat- control = 0) with respect to a gradient of salinity andcompetition of A on B. Beta ba: 1.2
  27. 27. Conditions of facilitation (treat - control < 0), competition(treat - control > 0) and no difference (treat - control = 0) with respect to a gradient of competition of A-B and B-A.
  28. 28. Real world example sustaining our model! Sp. A Stress level 0 Stress level 1 Stress level 2 Without B 900 300 300 With B 500 700 600 Sp. B Competition Facilitation Facilitation ( - -) (+ +) (+ +) Without A 750 200 150,2 Bertness & Hacker, 1994 With A 125 350 300
  29. 29. Other scenarios… Auto- S facilitation of the facilitated - - Why does it -- - matter biologically? A B -
  30. 30. Forest Clearing
  31. 31. S- - ---A B -
  32. 32. This is what we had in the previous model:There areclear stablepoints
  33. 33. And this is what I ended up with:UNTRUE!
  34. 34. Another snapshot time:Population attime t = 4000
  35. 35. Population attime 400 Competitive exclusion
  36. 36. Plain competition: A is excluded
  37. 37. Population attime 400 Competitive exclusion Competition/ facilitation
  38. 38. B needs A, both are stable
  39. 39. Population attime 400 Competitive exclusion Successive Competition/ oscilations facilitation Ecological succession!
  40. 40. Looking for the bifurcation T = 400 T = 4000
  41. 41. Converges fast to stabilityOscilating atvery lowfrequency andhigh amplitude
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