Impact of Environmental Noisein communities of neutral species

Jordi Hidalgo – University of Padova

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)

Neutral Theory of Biodiversity

Many of the observed patterns can be explained by thesimple mechanisms of birth, death and migration,

assuming EQUIVALENCE AMONG SPECIES (NEUTRALITY)

(e.g. Species abundance distribution, Species Area Relationship, ...)

(from Cell.com)

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 donot 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?

Environmental variability has important consequencesin 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 ECOLOGICALSYSTEMS

Neutral Theory leads to successful predictionsfor static patterns, BUT fails to estimate

several dynamical measures(e.g. scaling of population fluctuations)

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?

Colored noiseis not that bad...

Analytical toolsto 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

Analytical toolsto deal with COLORED Noise

Tool #2: Dichotomic Markov Noise (DMN)

Many analytical tools with DMN: Stationary state distributions, mean first passage times, ...

Environmental noise insingle species dynamics

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”

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?

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?

Contact Process with environmental Noise

+

UCNA

white noise

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)

Environmental noise smears the usual phase diagraminto a more variegated one

in which the absorbing behaviorslowly blends into the active one

Environmental noise in communities of NEUTRAL Species

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 increasesthe TOTAL number of species for highly variable environments(CHESSON STORAGE EFFECT)

Voter model with 2 species

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

Voter model with 2 species + Env. Noise

Neutrality is kept onTEMPORAL AVERAGE

EXPLICIT TIME-DEPENDENCY?PROBLEM: DETAILS MATTER!!!

Environment affects mortality: Environment affects fitness:

Linear fitness case For constant environments(mean-field)

EXTINCTION TIMES

In agreement with:

(Environment modulatesGROWTH RATES

linearly)

Relative fitness case For constant environments(mean-field)

EXTINCTION TIMES

In agreement with:

(Environment modulatesspecies fitness,

but effective fitness is re-scaled by the total one)

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

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)

THANKSFOR YOUR ATTENTION!