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
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.
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.
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 A1− −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 B1− −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 = A1− 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
-