Biodiversity: periodic boundary conditions and spatiotemporal stochasticity

Uno WennergrenIFM Theory and Modelling,

Division of Theoretical BiologyLinköping University

Outline • Biodiversity-

– Is the ’amount’ of species in an area and over a specific time– Depends on the amount of niches in the area and over the timeperiod

• We need to know/handle-– Niches in space – how to distribute resources– Niches in time – how to distribute resources– The population/individuals behaviour to disperse to utilize the resources in the

area/space– The population/individuals way to grow to utilize the resources over time– The interactions between populations, competition of resources

• We know that the mathematical models, systems of ODE’s, cannot not be both large and have stable equilibriums

• Developed methods to analyse data and to generate systems to test the dynamics

Outline• Conceptual framework of methods

• Example by biodiversity question:– How can there be such high biodiversity?

• Not included– Spatial kernels and Bayesian MCMC to asses

dispersal kernels from data om movements between habitats of different quality.

Spatio temporal stochasticity of resources

• A resource may vary – over time– over space

• A single population may track this variation over time and space more or less.

• There may become resource left overs for other species to exist on – a new niche!

• What promotes left overs for other species?• What combinations of species characteristics are

complementary in respect to spatiotemporal stochasticity of resources

Firstly

• WE have to consider a way to model spatio temporal stochasticity.

• 2-3 dim Fourier transform

Conceptual framework

• What characteristics of in signal relates to specific characteristics of out signal (increase risk of explosion or extinction)?

• What impact do the characteristics of the population have on this relation on in and out signal?

In signal (time):TemperatureHumidityOther population densitiesetc

Population filter:ReproductionSurvivalGrowthDispersal

Out signal (time):Population density

Conceptual frameworkadding complexity

In signalTemperatureHumidityOther population densitiesetc

Population filter:ReproductionSurvivalGrowthDispersal

Out signal:Population density

Spatial domain:Populations exist in a 2 dimensional heterogeneous landscape (or even 3D). Hence the signals are in 2D.

Characteristics of 2D signals?

Predation and competition between populations:Sets of interacting populations is the filter:

Characteristics of sets of out signals?The effect of the characteristics of interactions, feedbacks?

Conceptual frameworkmethodological questions, part I

In signalTemperatureHumidityOther population densitiesetc

Population filter:ReproductionSurvivalGrowthDispersal

Out signal:Population density

Spatial domain and sets of population

What defines the characteristics of the signals? What characteristics are important (extinction/explosion)? variance

mean autocorrelation/aggregation

synchronization

Conceptual frameworkmethodological questions, part II

In signal Population filter: Out signal:

Spatial domain and sets of populations

What defines the characteristics of the signals?What characteristics are important (extinction/explosion)? variance mean autocorrelation-1/f noise-flicker noise , in time and space synchronization between subpopulations

How to generate and analyze:variance meanautocorrelation

synchronization

In 1 dim, 2 dim and….. FFT

FFT vs Science in Theoretical Biology

• Analyzing time series to estimate 1/f noise of densities• Testing different in signals and measuring impact on

probability of extinction • Few studies on the relation between insignal and outsignal

measured by change of frequency spectrum • Few studies (one or two) on resonance

– within system populations– between system and insignal

• Few studies on how to generate or analyse time series and landscapes by FFT with desired properties

• No studies made on landscape of resources (in signal) and landscapes of densities (out signal) by FFT – single populations– Sets of populations

Generating Coordinates

Generate by starting with random (white noise) tilt the line in the frequency planeBy inverse Fourier Transform go back to landscape

Example on generating

• Different slopes in the frequency plane

• Continous or ’binary’ landscapes

• Different amount of primary habitat

Gamma=0 Gamma=1 Gamma=2 Continuous landscapes

viewed from the

side

Continuous landscapes

viewed from above

Digitalized landscapes with 10% preferred habitat

Digitalized landscapes with 40% preferred habitat

Environmental noise in time and space

• Landscape of old oaks.• A system of patches

that:– vary over time, and– are synchronized in their

variation. • Extinction risk,

in general, in this kind of system?

Environmental noise; the method

• 1/f noise i 2D:– Time, noise color– Space, synchrony

• Fourier transform, compare with generating landscape.

Extinction risk → resources

• Resource utilization as a measure of extinction risk?

• Resources left – other species?

0 20 40 60 80 100 120 140 160 180 2000

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Ricker noise in K (over comp.): Regional

Resources leftDensity

0 20 40 60 80 100 120 140 160 180 2000

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Ricker noise in K (over comp.): Local

Local1: resources leftLocal2Local3Local4Local5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

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Environmental noise colour

Resource utilization in a spatially subdivided population

sync=0.1sync=0.2sync=0.4sync=0.6sync=0.8sync=0.9

Conclusions

• Need to handle both time and space (synchrony) without mixing up with the variance

• Yes, there is a great potential for higher diversity when including spatial joint with temporal niche separation

Next concept:periodic boundaries in population interactions

• Periodic boundaries: handling infinity.

• Population exists and interact in an infinite space.

• Any model of interactions that impose boundaries may impose an error.

• Periodic boundaries: will it promote higher biodiversity???

A foodweb, set of populations with interactions, with stable oscillations

The system can be more, or less stable, when introducing space-time-periodic boundaries

More webs, only introducing spatio temporal stochasticity, no periodic boundaries

γ - noise colour

Periodic boundaries

• Set of periodic have same properties as single webs: when no stochasticity

• Adding stochasticity may change the picture

• Stochasticity – temporal and not synchronized- impose that at any time the web units are not the same, hence a diversity of species.

An example of temporal stochasticity on foodwebs linked as periodic units with periodic boundaries

Final conclusion

• High Biodiversity– Can be explained by

spatio-temporal niche separationinfinite foodwebs

• Studying populations/ecology ought to include– Spatiotemporal aspects of resources and

populations– Infinite boundaries of population interactions (-

foodwebs)