1. Models pretend that nature is simple, predictable and unchanging:

2. The real world is messy. Saether et al. 2002

3. Does this “messiness” matter to populations and communities? Causes of variability Patterns of variability How does variability matter to populations? (“messiness” = heterogeneity, variability)

4. Causes of variability weather related: temperature, precipitation, light,… site related: soil properties, slope, vegetation cover,… biological: the number of competitors, predators,… Environmental Demographic stochasticity Random variability in the growth rates of small populations arising chance events in individual survival and reproduction. Can lead to chance extinction in small populations. Chaotic dynamics Random-like variability from strong and non-linear interactions between variables.

5. Patterns in space or time spatial variation: when sites differ temporal variation: when conditions change over time Deterministic or random variability deterministic: predictable variation that follows precise rules random: has only statistical properties: mean, variance, etc. Types of deterministic variability periodic (in time or space) progressive change (spatial gradient, temporal trend) Environmental variability

6. Patterns in time: Month of the year 123456789101112 Temperature (C) 0 10 20 30 40 Average monthly temperature in Tucson Precipitation in Moab, Utah Year 19301940195019601970198019902000 Precipitation (mm) 0 50 100 150 200 250 300 Winter Summer

7. Patterns in time:

9. Patterns in space:

12. Demographic stochasticity: High per-capita variability in growth rate due to random fluctuation in individual survivorship, and fecundity. Small populations can go extinct, even though average growth rates are positive.

13. Excel Worksheets: Demographic Stochasticity

14. Exponential model with b = 0.01 per year d = 0.002 per year Demographic stochasticity N N 0 = 5 N N 0 = 20

15. A documented extirpation Three-spined stickleback (Gasterosteus aculeatus) The population disappeared from a pond in Wales in 2000 (Wootton 2010)

16. Chaotic fluctuations: Some entirely deterministic processes can produce fluctuations indistinguishable from random fluctuations. This was first discovered by the meteorologist Edward Lorenz who worked the prediction of weather patterns. Edward N. Lorenz (1917-2008)

17. The Butterfly Effect Whether or not a hurricane forms may depend on whether or not a butterfly flapped its wings several weeks before Chaotic dynamics have extreme sensitivity to initial conditions.

18. The Butterfly Effect

19. (Slingo and Palmer 2011)

20. The Discrete Logistic Model: Equilibrium defined as N t+1 = N t N*,1 = 0 N*,2 = K

21. Excel Worksheets: Discrete logistic equation

22. The existence of the various forms and changing patterns of variability drives evolution and creates and maintains biodiversity.

23. A prairie

24. Six species of Solidago (golden rod) are found in old fields and prairies. nemoralis missouriensis speciosa canadensis gigantea graminifolia

25. Species usually occupy only a subset of an environments range of conditions. species Frequency of occurrence

26. Species are active only during a fraction of the total available time. Hour of day Oklahoma Prairie

27. Summary: There are various sources and patterns of variability in nature producing spatial and temporal heterogeneity. Forms of patterns: random, cyclic, or persistent trends (both in space and time). A special case of variability is called: demographic stochasticity: the random fluctuations of small populations (which can have large and important consequences). Chaos is a special case of variability that is seemingly random but actually caused by deterministic processes. Much of ecology is concerned with understanding species’ responses to variability in its various forms in evolutionary and ecological time.