QUANTITATIVE LAWS June 13 -June 24

1. Impact of Environmental Noise
in communities of neutral species
Jordi Hidalgo – University of Padova

2. Samir Suweis, Amos Maritan
Tommaso Spanio, Miguel Á. Muñoz
J. Hidalgo, S. Suweis and A. Maritan,
Species coexistence in a neutral dynamics with environmental noise (submitted)
T. Spanio, J. Hidalgo and M. A. Muñoz,
Impact of environmental colored noise in population dynamics with extinction (in preparation)

3. Neutral Theory of Biodiversity
Many of the observed patterns can be explained by the
simple mechanisms of birth, death and migration,
assuming EQUIVALENCE AMONG SPECIES (NEUTRALITY)
(e.g. Species abundance distribution, Species Area Relationship, ...)
(from Cell.com)

4. However...
Role of temporal scales??
(we will avoid white noise approximation)
PRECIPITATIONS
TEMPERATURE
NUTRIENTS IN SOIL
ETC...
1) Most of the approaches based
on neutral assumptions do
not consider that ABIOTIC
CONDITIONS may change
2) Different species can have
different responses to such
fluctuations
Can we study the effect of environmental
variability in neutral dynamics without
compromising the NEUTRAL assumption?

5. Environmental variability has important consequences
in the functioning and organization of ecological systems
BET-HEDGING STRATEGIES
(see Ricardo's next talk!)
MECHANISMS FOR
SPECIES COEXISTENCE
(Chesson's storage effect)
PATTERNS IN ECOLOGICAL
SYSTEMS
Neutral Theory leads to successful predictions
for static patterns, BUT fails to estimate
several dynamical measures
(e.g. scaling of population fluctuations)

6. Some simple questions
● Coexistence of species or monodominance?
● Under which conditions a species get extinguished?
● How much time does it take to disappear?
We propose a SUPER top-down approach,
and start from simple models for single and multiple (NEUTRAL) species dynamics
IMPACT OF ENVIRONMENTAL NOISE??
TEMPORAL SCALES?

7. Colored noise
is not that bad...

8. Analytical tools
to deal with COLORED Noise
Tool #1: Unified Color Noise Approximation (UCNA)
2-variable Markovian problem 1-variable Markovian problem
Adiabatic approximation
(Exact when )
“Interpolation” for intermediate
white noise

9. Analytical tools
to deal with COLORED Noise
Tool #2: Dichotomic Markov Noise (DMN)
Many analytical tools with DMN: Stationary state distributions, mean first passage times, ...

10. Environmental noise in
single species dynamics

11. The Contact Process
DYNAMICS
A = ACTIVE (INFECTED, COLONIZED RESOURCES, ...)
I = INACTIVE (SUSCEPTIBLE, AVAILABLE RESOURCE, …) WE FOCUS ON THE
MEAN-FIELD CASE
(FULLY CONNECTED
NETWORK)
N Interacting “particles”

12. The Contact Process
Deterministic “mean-field” equation
Phase diagram
BUT!
Any FINITE system can eventually
reach the absorbing state...
How
Mean Extinction Times
(MET) scale with the system size N?

13. The Contact Process
Phase diagram Scaling of Mean Extinction Times
Dependence on the parameters?
Survival threshold?
How much time does activity take to disappear?
How probable is it?
IF RATES FLUCTUATE IN TIME...
● Just more parameters?
● Do we have to change the way in which we
understand Absorbing Vs Active phase?
● Role of time-scales?

14. Contact Process with
environmental Noise
+
UCNA
white noise

15. Contact Process with
environmental Noise
SIMULATIONS “AVOIDING”
THE ABSORBINGSTATE!
(e.g. recovering after absorbing...)
Physically real?
What if the system can fall into
the absorbing state? (FINITE SYSTEMS)

16. Environmental noise smears the usual phase diagram
into a more variegated one
in which the absorbing behavior
slowly blends into the active one

17. Environmental noise
in communities of NEUTRAL Species

18. Some recent works on
the impact of Environmental Variability
in neutral communities
● Model for multiple bacterial colonies
growing under limited resources
(logistic model)
● Species respond differently to the
environmental conditions
(neutral on average)
Environmental variability FAVORS one of the species to monodominate
● Neutral Theory of Biodiversity
(Voter model with speciation)
● Species respond differently to the
environment (neutral on average)
Environmental variability increases
the TOTAL number of species for
highly variable environments
(CHESSON STORAGE EFFECT)

19. Voter model with 2 species

20. Voter model with 2 species
COEXISTENCE?? The system can always reach MONODOMINANCE...
NO PARAMETERS, NO PHASE DIAGRAM...
Cox, J., 1989, Ann. Probab. 17 (4), 1333.
What is the impact of
environmental noise
on species coexistence?
NEUTRAL ASSUMPTION

21. Voter model with 2 species + Env. Noise
Neutrality is kept on
TEMPORAL AVERAGE
EXPLICIT TIME-DEPENDENCY?
PROBLEM: DETAILS MATTER!!!
Environment affects mortality: Environment affects fitness:

22. Linear fitness case For constant environments
(mean-field)
EXTINCTION TIMES
In agreement with:
(Environment modulates
GROWTH RATES
linearly)

23. Relative fitness case For constant environments
(mean-field)
EXTINCTION TIMES
In agreement with:
(Environment modulates
species fitness,
but effective fitness is re-
scaled by the total one)

24. CHESSON'S
STORAGE EFFECT
Different responses to environmental fluctuations
depending on the species abundance:
Small populations are slowly decimated during unfavorable conditions
and rapidly grow during favorable periods (in relation to its abundance),
(The opposite occurs for large populations)
Effective drift that propitiates species to coexist

25. Take home message
Further questions:
● Observables that we should measure in order to identify
the “correct dependence” with environmental fluctuations?
● Interplay between demographic & environmental noise?
● Environmental variability ↔ Spatial patterns?
(soon on Arxiv!)
T. Spanio, J. Hidalgo and M. A. Muñoz,
Impact of environmental colored noise in population dynamics with extinction
J. Hidalgo, S. Suweis and A. Maritan,
Species coexistence in a neutral dynamics with environmental noise (in Review)
● Environmental variability changes the way in which we understand phase
transitions in the context of population dynamics.
● In neutral dynamics, environmental variability can favor/disfavor species
coexistence depending on how the environment modulates species fitness.
● The interplay between the timescales of the dynamics and the environment
plays a fundamental role (white noise approximation are not always valid)
THANKS
FOR YOUR ATTENTION!