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 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

5. Firstly WE have to consider a way to model spatio temporal stochasticity. 2-3 dim Fourier transform

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

7. Conceptual framework adding 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 framework methodological 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 framework methodological questions, part II In signalPopulation 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. 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?

14. Environmental noise; the method 1/f noise i 2D: – Time, noise color – Space, synchrony Fourier transform, compare with generating landscape.

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

16. 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

17. 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???

18. A foodweb, set of populations with interactions, with stable oscillations The system can be more, or less stable, when introducing space-time-periodic boundaries

19. More webs, only introducing spatio temporal stochasticity, no periodic boundaries γ - noise colour

21. 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.

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

23. Final conclusion High Biodiversity – Can be explained by spatio-temporal niche separation infinite foodwebs Studying populations/ecology ought to include – Spatiotemporal aspects of resources and populations – Infinite boundaries of population interactions (- foodwebs)