biodiversity: periodic boundary conditions and spatiotemporal stochasticity
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Biodiversity: periodic boundary conditions and spatiotemporal stochasticity. Uno Wennergren IFM Theory and Modelling, Division of Theoretical Biology Linköping University. Outline. Biodiversity- Is the ’amount’ of species in an area and over a specific time - PowerPoint PPT PresentationTRANSCRIPT
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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1000
1200
N
Ricker noise in K (over comp.): Regional
Resources leftDensity
0 20 40 60 80 100 120 140 160 180 2000
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Time
N
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
0.1
0.2
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0.8
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Res
ourc
e ut
ilizat
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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)