Facilitation in Population Dynamics

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