SlideShare a Scribd company logo
Facilitation Systems
         Maria Luisa Jorge
       Marina Cenamo Salles
        Raquel A. F. Neves
         Sabastian Krieger
        Tomás Gallo Aquino
       Victoria Romeo Aznar


Southern Summer Mathematical Biology
       IFT – UNESP – São Paulo
             January 2012
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
Facilitation
  Species that
positively affects
another 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 gerardi

More tolerant to
high salinities
acts on
amelioration of
physical conditions:

shades the soil –
limits surface
evaporation and
accumulation of soil
salts.
Iva frutescens

Relatively intolerant to high soil salinities and
waterlogged soil conditions
S
     -               -
                 --
                    B
Juncus girardi
                  -      Iva frutescens
Bertness & Hacker, 1994
-

         A                      -                     B


Change of       Exponential   Saturation    Competition     Effect of
species A       term of       or logistic   effect of       salinity on
(facilitator)   population    term of       species B       species A
abundance       growth        population    (facilitated)   (facilitator)
over time                     growth        on species A
                                            (facilitator)
-
                               -                           B
               A


Change of      Exponential   Saturation        Competition     Effect of
species B      term of       or logistic       effect of       salinity on
(facilitated   population    term of           species A       species B
abundance      growth for    population        (facilitator)   (facilitated)
over time      species B     growth for        on species
                             species B         B
                                               (facilitated)
-

A                                      B


    Final      Initial  Effect of
    salinity   salinity abundance of
                        species A on
                        salinity
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
Looking for fixed points
Condition
:
  dA
      A f , B f =0              A=0∨B=1− A− F A e
                                                      − A
                                                             / c AB                            − A f
                                                                                                            Don't
  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 f
Ok, 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
How do fixed points varie with the
parameters?
                                 ' 1

                             ' 0 ∃ one A f ≠0
                                   − A
                        0'   e   f
                                        ∃ two A f ≠0
                           else don' t ∃ A f ≠0




We can have:
    No solution
    Two solutions
Some critical cases

             dA      dB
A=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 doesn't suffer too much                          survives
      stress, and doesn't suffer
      competition




     S B 1 ∃                          B f =0 is stable
                  one fixed
                                     B f =1−S B don ' t ∃
                  points


      Although B doesn't have
                                                            dies
      competition, the stress is hard
Some critical cases

             dB     dA             − A
B=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 doesn't suffer too much
                                                     survives
      stress, and doesn't 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
Species A
                             Species B (with facilitation)
                             Control for species B
                             Salinity




Temporal variation of both species and salinity when, at
equilibrium, both co-exist.
Species A
                              Species B (with facilitation)
                              Control for species B
                              Salinity




Temporal variation of both species and salinity when, at
equilibrium, both co-exist.
Species A
                              Species B (with facilitation)
                              Control for species B
                              Salinity




Temporal variation of both species and salinity when, at
         equilibrium, species B goes extinct.
Species A
                               Species B (with facilitation)
                               Control for species B




Variation of abundance of both species with respect to
                      salinity.
¿




Conditions of facilitation (treat - control > 0),
competition (treat - control < 0) and no difference (treat
- control = 0) with respect to a gradient of salinity and
competition of B on A. Beta ab: 0.68
Conditions of facilitation (treat - control > 0),
competition (treat - control < 0) and no difference (treat
- control = 0) with respect to a gradient of salinity and
competition of A on B. Beta ba: 1.2
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.
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
Other scenarios…
               Auto-


      S
               facilitation of
               the facilitated




  -     -
               Why does it



   -- -
               matter
               biologically?




 A      B
    -
Facilitation in Population Dynamics
Forest Clearing
S
-     -
  ---
A     B
   -
This is what we had in the previous model:




There are
clear stable
points
And this is what I ended up with:




UNTRUE!
Another snapshot time:




Population at
time t = 4000
Population at
time 400




                Competitive
                exclusion
Plain competition: A is excluded
Population at
time 400




                Competitive
                exclusion     Competition/
                              facilitation
B needs A, both are stable
Population at
time 400




                Competitive
                exclusion                   Successive    Competition/
                                            oscilations   facilitation
                              Ecological
                              succession!
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Facilitation in Population Dynamics
Looking for the bifurcation



            T = 400
                              T = 4000
Converges
                 fast to
                 stability




Oscilating at
very low
frequency and
high amplitude

More Related Content

More from Roberto Kraenkel

population dynamics of insects
population dynamics of insects population dynamics of insects
population dynamics of insects
Roberto Kraenkel
 
Tópicos de Biologia-Matemática
Tópicos de Biologia-MatemáticaTópicos de Biologia-Matemática
Tópicos de Biologia-Matemática
Roberto Kraenkel
 
Histerese, Bi-estabilidade e Desertificação
Histerese, Bi-estabilidade e DesertificaçãoHisterese, Bi-estabilidade e Desertificação
Histerese, Bi-estabilidade e Desertificação
Roberto Kraenkel
 
Ondas de Choque: Água, Luz e Condensados
Ondas de Choque: Água, Luz e CondensadosOndas de Choque: Água, Luz e Condensados
Ondas de Choque: Água, Luz e Condensados
Roberto Kraenkel
 
Estabilização dinâmica
Estabilização dinâmica Estabilização dinâmica
Estabilização dinâmica
Roberto Kraenkel
 
Aula quatro jornadas12_handout
Aula quatro jornadas12_handoutAula quatro jornadas12_handout
Aula quatro jornadas12_handout
Roberto Kraenkel
 
Tópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática IIITópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática III
Roberto Kraenkel
 
Tópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática IITópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática II
Roberto Kraenkel
 
Tópicos em Biologia Matemática I
Tópicos em Biologia Matemática ITópicos em Biologia Matemática I
Tópicos em Biologia Matemática I
Roberto Kraenkel
 
Intraguild mutualism
Intraguild mutualismIntraguild mutualism
Intraguild mutualism
Roberto Kraenkel
 
Indirect effects affects ecosystem dynamics
Indirect effects affects ecosystem dynamicsIndirect effects affects ecosystem dynamics
Indirect effects affects ecosystem dynamics
Roberto Kraenkel
 
Atraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas BiológicosAtraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas Biológicos
Roberto Kraenkel
 
Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado  Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado
Roberto Kraenkel
 
Mesopredadores
MesopredadoresMesopredadores
Mesopredadores
Roberto Kraenkel
 
Mutualismo
Mutualismo Mutualismo
Mutualismo
Roberto Kraenkel
 
Efeito Allee
Efeito AlleeEfeito Allee
Efeito Allee
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
 
Métodos Matemáticos em Biologia de POpulações
Métodos Matemáticos em Biologia de POpulações Métodos Matemáticos em Biologia de POpulações
Métodos Matemáticos em Biologia de POpulações
Roberto Kraenkel
 
Métodos Matemáticos em Biologia de Populações III
Métodos Matemáticos em Biologia de Populações IIIMétodos Matemáticos em Biologia de Populações III
Métodos Matemáticos em Biologia de Populações III
Roberto Kraenkel
 
Métodos Matemáticos em Biologia de Populações II
Métodos Matemáticos em Biologia de Populações IIMétodos Matemáticos em Biologia de Populações II
Métodos Matemáticos em Biologia de Populações II
Roberto Kraenkel
 

More from Roberto Kraenkel (20)

population dynamics of insects
population dynamics of insects population dynamics of insects
population dynamics of insects
 
Tópicos de Biologia-Matemática
Tópicos de Biologia-MatemáticaTópicos de Biologia-Matemática
Tópicos de Biologia-Matemática
 
Histerese, Bi-estabilidade e Desertificação
Histerese, Bi-estabilidade e DesertificaçãoHisterese, Bi-estabilidade e Desertificação
Histerese, Bi-estabilidade e Desertificação
 
Ondas de Choque: Água, Luz e Condensados
Ondas de Choque: Água, Luz e CondensadosOndas de Choque: Água, Luz e Condensados
Ondas de Choque: Água, Luz e Condensados
 
Estabilização dinâmica
Estabilização dinâmica Estabilização dinâmica
Estabilização dinâmica
 
Aula quatro jornadas12_handout
Aula quatro jornadas12_handoutAula quatro jornadas12_handout
Aula quatro jornadas12_handout
 
Tópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática IIITópicos de Biologia-Matemática III
Tópicos de Biologia-Matemática III
 
Tópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática IITópicos em Biologia-Matemática II
Tópicos em Biologia-Matemática II
 
Tópicos em Biologia Matemática I
Tópicos em Biologia Matemática ITópicos em Biologia Matemática I
Tópicos em Biologia Matemática I
 
Intraguild mutualism
Intraguild mutualismIntraguild mutualism
Intraguild mutualism
 
Indirect effects affects ecosystem dynamics
Indirect effects affects ecosystem dynamicsIndirect effects affects ecosystem dynamics
Indirect effects affects ecosystem dynamics
 
Atraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas BiológicosAtraso temporal em Sistemas Biológicos
Atraso temporal em Sistemas Biológicos
 
Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado  Epidemias Em Um Mundo Interligado
Epidemias Em Um Mundo Interligado
 
Mesopredadores
MesopredadoresMesopredadores
Mesopredadores
 
Mutualismo
Mutualismo Mutualismo
Mutualismo
 
Efeito Allee
Efeito AlleeEfeito Allee
Efeito Allee
 
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
 
Métodos Matemáticos em Biologia de POpulações
Métodos Matemáticos em Biologia de POpulações Métodos Matemáticos em Biologia de POpulações
Métodos Matemáticos em Biologia de POpulações
 
Métodos Matemáticos em Biologia de Populações III
Métodos Matemáticos em Biologia de Populações IIIMétodos Matemáticos em Biologia de Populações III
Métodos Matemáticos em Biologia de Populações III
 
Métodos Matemáticos em Biologia de Populações II
Métodos Matemáticos em Biologia de Populações IIMétodos Matemáticos em Biologia de Populações II
Métodos Matemáticos em Biologia de Populações II
 

Recently uploaded

NAEYC Code of Ethical Conduct Resource Book
NAEYC Code of Ethical Conduct Resource BookNAEYC Code of Ethical Conduct Resource Book
NAEYC Code of Ethical Conduct Resource Book
lakitawilson
 
CTD Punjab Police Past Papers MCQs PPSC PDF
CTD Punjab Police Past Papers MCQs PPSC PDFCTD Punjab Police Past Papers MCQs PPSC PDF
CTD Punjab Police Past Papers MCQs PPSC PDF
hammadmughal76316
 
SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
SEQUNCES  Lecture_Notes_Unit4_chapter11_sequenceSEQUNCES  Lecture_Notes_Unit4_chapter11_sequence
SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
Murugan Solaiyappan
 
(T.L.E.) Agriculture: Essentials of Gardening
(T.L.E.) Agriculture: Essentials of Gardening(T.L.E.) Agriculture: Essentials of Gardening
(T.L.E.) Agriculture: Essentials of Gardening
MJDuyan
 
NC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline PartnershipNC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline Partnership
Mebane Rash
 
Webinar Innovative assessments for SOcial Emotional Skills
Webinar Innovative assessments for SOcial Emotional SkillsWebinar Innovative assessments for SOcial Emotional Skills
Webinar Innovative assessments for SOcial Emotional Skills
EduSkills OECD
 
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
MysoreMuleSoftMeetup
 
Imagination in Computer Science Research
Imagination in Computer Science ResearchImagination in Computer Science Research
Imagination in Computer Science Research
Abhik Roychoudhury
 
How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17
Celine George
 
How to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POSHow to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POS
Celine George
 
formative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.Vformative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.V
DrRavindrakshirsagar1
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
heathfieldcps1
 
Individual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docxIndividual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docx
monicaaringo1
 
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 SlidesWhat is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
Celine George
 
Year-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 DashboardYear-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 Dashboard
Celine George
 
C# Interview Questions PDF By ScholarHat.pdf
C# Interview Questions PDF By ScholarHat.pdfC# Interview Questions PDF By ScholarHat.pdf
C# Interview Questions PDF By ScholarHat.pdf
Scholarhat
 
The Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdfThe Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdf
luzmilaglez334
 
Edukasyong Pantahanan at Pangkabuhayan 1: Personal Hygiene
Edukasyong Pantahanan at  Pangkabuhayan 1: Personal HygieneEdukasyong Pantahanan at  Pangkabuhayan 1: Personal Hygiene
Edukasyong Pantahanan at Pangkabuhayan 1: Personal Hygiene
MJDuyan
 
How To Create a Transient Model in Odoo 17
How To Create a Transient Model in Odoo 17How To Create a Transient Model in Odoo 17
How To Create a Transient Model in Odoo 17
Celine George
 
Principles of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptxPrinciples of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptx
ibtesaam huma
 

Recently uploaded (20)

NAEYC Code of Ethical Conduct Resource Book
NAEYC Code of Ethical Conduct Resource BookNAEYC Code of Ethical Conduct Resource Book
NAEYC Code of Ethical Conduct Resource Book
 
CTD Punjab Police Past Papers MCQs PPSC PDF
CTD Punjab Police Past Papers MCQs PPSC PDFCTD Punjab Police Past Papers MCQs PPSC PDF
CTD Punjab Police Past Papers MCQs PPSC PDF
 
SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
SEQUNCES  Lecture_Notes_Unit4_chapter11_sequenceSEQUNCES  Lecture_Notes_Unit4_chapter11_sequence
SEQUNCES Lecture_Notes_Unit4_chapter11_sequence
 
(T.L.E.) Agriculture: Essentials of Gardening
(T.L.E.) Agriculture: Essentials of Gardening(T.L.E.) Agriculture: Essentials of Gardening
(T.L.E.) Agriculture: Essentials of Gardening
 
NC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline PartnershipNC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline Partnership
 
Webinar Innovative assessments for SOcial Emotional Skills
Webinar Innovative assessments for SOcial Emotional SkillsWebinar Innovative assessments for SOcial Emotional Skills
Webinar Innovative assessments for SOcial Emotional Skills
 
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
 
Imagination in Computer Science Research
Imagination in Computer Science ResearchImagination in Computer Science Research
Imagination in Computer Science Research
 
How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17How To Update One2many Field From OnChange of Field in Odoo 17
How To Update One2many Field From OnChange of Field in Odoo 17
 
How to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POSHow to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POS
 
formative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.Vformative Evaluation By Dr.Kshirsagar R.V
formative Evaluation By Dr.Kshirsagar R.V
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
 
Individual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docxIndividual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docx
 
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 SlidesWhat is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
 
Year-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 DashboardYear-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 Dashboard
 
C# Interview Questions PDF By ScholarHat.pdf
C# Interview Questions PDF By ScholarHat.pdfC# Interview Questions PDF By ScholarHat.pdf
C# Interview Questions PDF By ScholarHat.pdf
 
The Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdfThe Cruelty of Animal Testing in the Industry.pdf
The Cruelty of Animal Testing in the Industry.pdf
 
Edukasyong Pantahanan at Pangkabuhayan 1: Personal Hygiene
Edukasyong Pantahanan at  Pangkabuhayan 1: Personal HygieneEdukasyong Pantahanan at  Pangkabuhayan 1: Personal Hygiene
Edukasyong Pantahanan at Pangkabuhayan 1: Personal Hygiene
 
How To Create a Transient Model in Odoo 17
How To Create a Transient Model in Odoo 17How To Create a Transient Model in Odoo 17
How To Create a Transient Model in Odoo 17
 
Principles of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptxPrinciples of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptx
 

Facilitation in Population Dynamics

  • 1. Facilitation Systems Maria Luisa Jorge Marina Cenamo Salles Raquel A. F. Neves Sabastian Krieger Tomás Gallo Aquino Victoria Romeo Aznar Southern Summer Mathematical Biology IFT – UNESP – São Paulo January 2012
  • 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. Facilitation Species that positively affects another species, directly or indirectly.
  • 7. S - - -- A B -
  • 8. Salt Marshes Physical conditions - Waterlogged soils - High soil salinities
  • 9. Juncus gerardi More tolerant to high salinities acts on amelioration of physical conditions: shades the soil – limits surface evaporation and accumulation of soil salts.
  • 10. Iva frutescens Relatively intolerant to high soil salinities and waterlogged soil conditions
  • 11. S - - -- B Juncus girardi - Iva frutescens
  • 13. - A - B Change of Exponential Saturation Competition Effect of species A term of or logistic effect of salinity on (facilitator) population term of species B species A abundance growth population (facilitated) (facilitator) over time growth on species A (facilitator)
  • 14. - - B A Change of Exponential Saturation Competition Effect of species B term of or logistic effect of salinity on (facilitated population term of species A species B abundance growth for population (facilitator) (facilitated) over time species B growth for on species species B B (facilitated)
  • 15. - A B Final Initial Effect of salinity salinity abundance of species A on salinity
  • 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. Looking for fixed points Condition : dA  A f , B f =0 A=0∨B=1− A− F A e − A / c AB − A f Don't 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 f Ok, 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. How do fixed points varie with the parameters?  ' 1  ' 0 ∃ one A f ≠0 − A 0'   e f ∃ two A f ≠0 else don' t ∃ A f ≠0 We can have: No solution Two solutions
  • 19. Some critical cases dA dB A=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 doesn't suffer too much survives stress, and doesn't suffer competition S B 1 ∃ B f =0 is stable one fixed B f =1−S B don ' t ∃ points Although B doesn't have dies competition, the stress is hard
  • 20. Some critical cases dB dA − A B=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 doesn't suffer too much survives stress, and doesn't 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. Species A Species B (with facilitation) Control for species B Salinity Temporal variation of both species and salinity when, at equilibrium, both co-exist.
  • 22. Species A Species B (with facilitation) Control for species B Salinity Temporal variation of both species and salinity when, at equilibrium, both co-exist.
  • 23. Species A Species B (with facilitation) Control for species B Salinity Temporal variation of both species and salinity when, at equilibrium, species B goes extinct.
  • 24. Species A Species B (with facilitation) Control for species B Variation of abundance of both species with respect to salinity.
  • 25. ¿ Conditions of facilitation (treat - control > 0), competition (treat - control < 0) and no difference (treat - control = 0) with respect to a gradient of salinity and competition of B on A. Beta ab: 0.68
  • 26. Conditions of facilitation (treat - control > 0), competition (treat - control < 0) and no difference (treat - control = 0) with respect to a gradient of salinity and competition of A on B. Beta ba: 1.2
  • 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. 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. Other scenarios… Auto- S facilitation of the facilitated - - Why does it -- - matter biologically? A B -
  • 32. S - - --- A B -
  • 33. This is what we had in the previous model: There are clear stable points
  • 34. And this is what I ended up with: UNTRUE!
  • 36. Population at time 400 Competitive exclusion
  • 37. Plain competition: A is excluded
  • 38. Population at time 400 Competitive exclusion Competition/ facilitation
  • 39. B needs A, both are stable
  • 40. Population at time 400 Competitive exclusion Successive Competition/ oscilations facilitation Ecological succession!
  • 47. Looking for the bifurcation T = 400 T = 4000
  • 48. Converges fast to stability Oscilating at very low frequency and high amplitude