Demographic PVAs Simulating Demographic Stochasticity and Density Dependence.
Biodiversity: periodic boundary conditions and spatiotemporal stochasticity Uno Wennergren IFM...
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![Page 1: Biodiversity: periodic boundary conditions and spatiotemporal stochasticity Uno Wennergren IFM Theory and Modelling, Division of Theoretical Biology Linköping.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d2d5503460f94a0462f/html5/thumbnails/1.jpg)
Biodiversity: periodic boundary conditions and spatiotemporal stochasticity
Uno WennergrenIFM Theory and Modelling,
Division of Theoretical BiologyLinköping University
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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
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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.
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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
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Firstly
• WE have to consider a way to model spatio temporal stochasticity.
• 2-3 dim Fourier transform
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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
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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?
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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
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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
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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
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Generating Coordinates
Generate by starting with random (white noise) tilt the line in the frequency planeBy inverse Fourier Transform go back to landscape
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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
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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?
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Environmental noise; the method
• 1/f noise i 2D:– Time, noise color– Space, synchrony
• Fourier transform, compare with generating landscape.
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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
200
400
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800
1000
1200
N
Ricker noise in K (over comp.): Regional
Resources left
Density
0 20 40 60 80 100 120 140 160 180 2000
20
40
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100
120
Time
N
Ricker noise in K (over comp.): Local
Local1: resources left
Local2
Local3Local4
Local5
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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0.4
0.5
0.6
0.7
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0.9
1
Re
sou
rce
util
iza
tion
Environmental noise colour
Resource utilization in a spatially subdivided population
sync=0.1sync=0.2sync=0.4sync=0.6sync=0.8sync=0.9
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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
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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???
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A foodweb, set of populations with interactions, with stable oscillations
The system can be more, or less stable, when introducing space-time-periodic boundaries
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More webs, only introducing spatio temporal stochasticity, no periodic boundaries
γ - noise colour
![Page 20: Biodiversity: periodic boundary conditions and spatiotemporal stochasticity Uno Wennergren IFM Theory and Modelling, Division of Theoretical Biology Linköping.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d2d5503460f94a0462f/html5/thumbnails/20.jpg)
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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.
![Page 22: Biodiversity: periodic boundary conditions and spatiotemporal stochasticity Uno Wennergren IFM Theory and Modelling, Division of Theoretical Biology Linköping.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649d2d5503460f94a0462f/html5/thumbnails/22.jpg)
An example of temporal stochasticity on foodwebs linked as periodic units with periodic boundaries
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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)