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.

4.
Bruno et al. TREE

5.
Bruno et al., 2003

6.
Bruno et al., 2003

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

12.
Bertness & Hacker, 1994

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
-

30.
Forest Clearing

31.
S
- -
---
A B
-

32.
This is what we had in the previous model:
There are
clear stable
points

33.
And this is what I ended up with:
UNTRUE!

34.
Another snapshot time:
Population at
time t = 4000

35.
Population at
time 400
Competitive
exclusion

36.
Plain competition: A is excluded

37.
Population at
time 400
Competitive
exclusion Competition/
facilitation

38.
B needs A, both are stable

39.
Population at
time 400
Competitive
exclusion Successive Competition/
oscilations facilitation
Ecological
succession!

40.
Looking for the bifurcation
T = 400
T = 4000

41.
Converges
fast to
stability
Oscilating at
very low
frequency and
high amplitude