Biodiversity: periodic boundary conditions and spatiotemporal stochasticity

1. Biodiversity: periodic boundary conditions and spatiotemporal stochasticity Uno Wennergren IFM Theory and Modelling, Division of Theoretical Biology Linköping University

2. 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

3. 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.

4. Spatio temporal stochasticity of resources • A resource may vary • over time • over space • A single population maytrack this variation over time and spacemore or less. • Theremaybecomeresourceleftovers for other species to exist on – a new niche! • Whatpromotesleftovers for other species? • What combinations of species characteristics are complementary in respect to spatiotemporalstochasticity of resources

5. Firstly • WE have to consider a way to modelspatio temporal stochasticity. • 2-3 dimFourier transform

6. Conceptual framework In signal (time): Temperature Humidity Other population densities etc Population filter: Reproduction Survival Growth Dispersal Out signal (time): Population density • 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?

7. Conceptual frameworkadding complexity In signal Temperature Humidity Other population densities etc Population filter: Reproduction Survival Growth Dispersal 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?

8. Conceptual frameworkmethodological questions, part I In signal Temperature Humidity Other population densities etc Population filter: Reproduction Survival Growth Dispersal 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

9. 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 mean autocorrelation synchronization In 1 dim, 2 dim and….. FFT

10. 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

11. Generating Coordinates Generate by starting with random (white noise) tilt the line in the frequency plane By inverse Fourier Transform go back to landscape

12. Example on generating • Different slopes in the frequency plane • Continous or ’binary’ landscapes • Different amount of primary habitat

13. Environmentalnoise in time and space • Landscape of oldoaks. • A system of patchesthat: • vary over time, and • are synchronized in theirvariation. • Extinction risk, in general, in this kind of system?

14. Environmentalnoise; the method • 1/f noise i 2D: • Time, noisecolor • Space, synchrony • Fourier transform, compare with generating landscape.

15. Extinction risk → resources • Resourceutilization as a measure of extinction risk? • Resources left – other species?

16. Conclusions • Need to handleboth time and space (synchrony) withoutmixing up with the variance • Yes, there is a great potential for higherdiversitywhenincluding spatial joint with temporal niche separation

17. Nextconcept:periodicboundaries in population interactions • Periodicboundaries: handling infinity. • Population exists and interact in an infinite space. • Anymodel of interactions that imposeboundariesmayimpose an error. • Periodicboundaries: will it promotehigherbiodiversity???

18. A foodweb, set of populations with interactions, with stable oscillations The system can be more, or less stable, whenintroducingspace-time-periodicboundaries

19. Morewebs, onlyintroducingspatio temporal stochasticity, no periodicboundaries γ - noisecolour

21. Periodicboundaries • Set of periodichave same properties as singlewebs: when no stochasticity • Addingstochasticitymaychange the picture • Stochasticity – temporal and not synchronized-impose that at any time the webunits are not the same, hence a diversity of species.

22. An example of temporal stochasticity on foodwebs linked as periodicunits with periodicboundaries

23. Final conclusion • High Biodiversity • Can be explained by spatio-temporalniche separationinfinite foodwebs • Studyingpopulations/ecologyought to include • Spatiotemporalaspects of resources and populations • Infinite boundaries of population interactions (-foodwebs)