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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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Facilitation SystemsMaria Luisa Jorge
Marina Cenamo SallesRaquel A. F. NevesSabastian Krieger
Tomás Gallo AquinoVictoria Romeo Aznar
Southern Summer Mathematical BiologyIFT – 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
Species that positively affects another species,
directly or indirectly.
Facilitation
Bruno et al. TREE 2003
Bru
no e
t al.,
2003
Bruno et al., 2003
A B
S- -
--
-
Salt Marshes
Physical conditions
- Waterlogged soils
- High soil salinities
Juncus gerardi
More tolerant to high salinitiesacts 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
B
S- -
--
-
Juncus girardi Iva frutescens
Bertness & Hacker, 1994
-
Change of species A (facilitator) abundance over time
Exponential term of population growth
Saturation or logistic term of population growth
Competition effect of species B (facilitated) on species A (facilitator)
Effect of salinity on species A (facilitator)
-
A B
-
Change of species B (facilitated abundance over time
Exponential term of population growth for species B
Saturation or logistic term of population growth for species B
Competition effect of species A (facilitator) on species B (facilitated)
Effect of salinity on species B (facilitated)
- BA
Final salinity
Effect of abundance of species A on salinity
Initial salinity
-
BA
Reducing the number of parameters. Nondimensionalization
dAdt
=r A A1−AK A
−bAB B−a AS 0 e− A
dBdt
=r B B1−BK B
−bBA A−aB S0 e− A
dA'dt
=A ' 1−A '−cAB B '−F A e− A'
dB 'dt
=rB ' 1−B '−cBA A '−F B e− A'
Looking for fixed points
dAdt
A f , B f =0
dBdt
A f , B f =0
Condition:
A=0∨B=1−A−F A e− A
/ cAB
B=0∨B=1−cBA A−F Be− A
− A f− e− A f=0
Don't have analytical solution !!
Ok, we look ...
A f are that
1− ' A f= ' e− A f
'0 ∃ one A f≠0else don ' t ∃ A f≠0
'1
We can have:
No solutionOne solution
=1−cAB , =1−c AB cBA , =F A−F B cAB
How do fixed points varie with the parameters?
'1
'0 ∃ one A f≠0
0 ' e− A f ∃ two A f≠0
else don' t ∃ A f≠0
We can have:
No solutionTwo solutions
Some critical cases
A=A f=0dAdt
=0dBdt
= B 1−B−S B=0 B f=0, B f=1−S B0
S B1 ∃ two fixed points
B f=0 is instableB f=1−S B is stable
If B doesn't suffer too much stress, and doesn't suffer competition
survives
S B1 ∃ one fixed points
B f=0 is stableB f=1−S B don ' t ∃
Although B doesn't have competition, the stress is hard
dies
Some critical cases
B=B f=0dBdt
=0dAdt
=A1−A−S A e− A
=0 A f=0
S A1 ∃ A f=0 is instable∃ A f ∈0,1 is stable
If A doesn't suffer too much stress, and doesn't suffer competition
survives
S A1 ∃A f=0 is stable
A f0 don ' t ∃ if is small∃ two A f ∈0,1 if is large.One stable , the other instable.
With a high self-facilitation, A can survive if it has already large numbers
dies
and some A f0
survives
Temporal variation of both species and salinity when, at equilibrium, both co-exist.
Species ASpecies B (with facilitation)Control for species BSalinity
Temporal variation of both species and salinity when, at equilibrium, both co-exist.
Species ASpecies B (with facilitation)Control for species BSalinity
Temporal variation of both species and salinity when, at equilibrium, species B goes extinct.
Species ASpecies B (with facilitation)Control for species BSalinity
Variation of abundance of both species with respect to salinity.
Species ASpecies B (with facilitation)Control for species B
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
With A 125 350 300
Bert
ness
& H
ack
er,
199
4
Other scenarios…
A B
S- -
--
--
Auto-facilitation of the facilitated
Why does it matter biologically?
Forest Clearing
A B
S- -
--
--
This is what we had in the previous model:
There are clear stable points
And this is what I ended up with:
UNTRUE!
Population at time t = 4000
Another snapshot time:
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 Competition/
facilitationEcological succession!
Successive oscilations
T = 400
T = 4000
Looking for the bifurcation
Converges fast to stability
Oscilating at very low frequency and high amplitude