1. Community-Level Patterns: Species Richness & Diversity
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Image from:
“Paradise” by Suzanne Duranceau

2. Species Diversity & Richness
S = Species richness – the number of species in a collection of
organisms
Sd = Species density – the number of species per area
D = Species diversity – a simultaneous index of both S and
the evenness with which individuals are distributed among
species (a.k.a. equitability)
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3. Diversity Indexes Shannon’s (a.k.a. Shannon-Wiener)
Based on information theory / entropy
H’ = – Σ(pi * ln pi)
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The decomposition of each equation helps understand the relative emphasis on rare vs. common species.
See Magurran (1988 – Shannon pp , Simpson pp ; 2004)
Shannon, C. E. (1948) A mathematical theory of communication. The Bell System Technical Journal, 27, 379–423 and 623–656.
Simpson, E. H. (1949) Measurement of diversity. Nature, 163, 688.

4. Diversity Indexes Simpson’s
Based on the probability of conspecific encounters in
an infinitely large collection:
D = Σ(pi2)
For a finite community, use:
D = Σ((ni (ni-1))/(N(N-1)))
The index is generally expressed as the probability that
two individuals differ:
DSimpson = 1 - D or DSimpson = 1 / D
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The decomposition of each equation helps understand the relative emphasis on rare vs. common species.
See Magurran (1988 – Shannon pp , Simpson pp ; 2004)
Shannon, C. E. (1948) A mathematical theory of communication. The Bell System Technical Journal, 27, 379–423 and 623–656.
Simpson, E. H. (1949) Measurement of diversity. Nature, 163, 688.

5. Diversity at Different Scales
R. H. Whittaker (1972) proposed the following measures
of S and species turn-over:
Sα = Alpha “diversity” – the number of species in a local area
(or habitat)
Sβ = Beta “diversity” – the turn-over rate of species from local
area to local area (e.g., from habitat to habitat)
Sγ = Gamma “diversity” – the number of species in a region
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A good example comes from Condit et al. (2002), in which they compared beta diversity between Ecuador and Panama.

6. New World Alpha Diversity Birds
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C. Jenkins:

7. New World Alpha Diversity Mammals
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C. Jenkins:

8. New World Alpha Diversity Birds
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The image of bird diversity is from Hawkins et al. (2006); I opened the pdf of the full article in Adobe Acrobat, selected the image, copied it, and pasted it into this PowerPoint slide.
See also: MacArthur’s (1972) Geographical Ecology
Hawkins et al. (2006)

9. New World Alpha Diversity Mammals
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The image came directly from the pdf of Willig et al. (2003).
Willig et al. (2003)

10. Biodiversity across the Isthmus of Panama Free-Standing Trees & Shrubs
~ 120 species / ha
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For detailed comparisons of species richness of free-standing trees and shrubs in 1-ha plots across the Isthmus of Panama, see: Richard Condit…
Image from biogeodb.stri.si.edu/bioinformatics/maps
~ 70 species / ha
Image from biogeodb.stri.si.edu…

11. Major Determinants of Global Climate
1. Shape of the Earth – causes unequal heating
(energy per area) with latitude
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Since many of the regional-to-global-scale gradients that interested Whittaker and others in the early stages of the development of Community Ecology as a discipline were climate-based, let’s discuss the major determinants of global climate, as an aside, since abiotic, physical factors are often correlated with species’ ranges & local abundances, as well as community-level diversity. Even so, correlation does not necessarily mean causation.

12. Major Determinants of Global Climate
1. Shape of the Earth – differential heating & cooling
causes air masses to rise & sink: Ferrel & Hadley cells
Polar cell
Ferrel
cell
Ferrel cell
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Hadley cell
Ferrel cell
Ferrel
cell
Image from NASA

13. Major Determinants of Global Climate
1. Shape of the Earth
2. Revolution of the Earth around the Sun on a tilted axis
– Ferrel & Hadley cells move latitudinally, tracking seasonal changes
in the position of the solar equator, with a slight time lag
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Southern Hemisphere
is tilted towards the Sun
Northern Hemisphere
is tilted towards the Sun

14. Major Determinants of Global Climate
1. Shape of the Earth
2. Revolution of the Earth around the Sun on a tilted axis
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Notice that wet, warm band at the Intertropical Convergence Zone (ITZ) corresponds to highest levels of both alpha and gamma diversity for most major groups of organisms. The correlation may indicate causation, but does not do so necessarily.

15. Major Determinants of Global Climate
1. Shape of the Earth
2. Revolution of the Earth around the Sun on a tilted axis
Please do not use the images in these PowerPoint slides without permission.
Notice that wet, warm band at the Intertropical Convergence Zone (ITZ) corresponds to highest levels of both alpha and gamma diversity for most major groups of organisms. The correlation may indicate causation, but does not do so necessarily.

16. Major Determinants of Global Climate
1. Shape of the Earth
2. Revolution of the Earth around the Sun on a tilted axis
3. Rotation of the Earth on Earth’s axis creates Coriolis forces (actually conservation of momentum)
Currents in air & water are deflected; right in N. Hemisphere, left in S. Hemisphere
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17. Major Determinants of Global Climate
Polar cell
Ferrel
cell
Ferrel cell
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Notice that it is warm & wet at the equator, dry in the Sonoran Desert (home of Saguaro), and that moisture-laden northeasterly trade winds cross the Isthmus of Panama from the Caribbean toward the Pacific to the southwest.
Hadley cell
Ferrel cell
Ferrel
cell
Image from NASA

18. Species Diversity – Accumulation & Rarefaction Curves
E.g., estimating tree diversity within a large study plot:
Individual-based assessment: Choose trees at random from
the plot; sum the number of species as each new tree is added
Sample-based assessment: Establish a set of quadrats in the plot; sum the total number of species as each new quadrat is added
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19. Species Diversity – Accumulation & Rarefaction Curves
(or higher taxa)
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Rarefaction curves provide the number of species expected for a given sample size. These are especially useful when you have sufficient data to plot them.
Rarefaction is a method to generate smooth species-accumulation curves, since for each value on the x-axis (either number of individuals or number of samples), you take the average of multiple random samples from your data set. 
See Bravo et al. (2008) for an example.
Note that Gotelli & Colwell (2001) provide a useful criticism of Hubbell et al. (1999).
The use of species-per-individual is discouraged, since the curves for 2 communities could cross (e.g., once, twice), giving rise to different rank species richness depending on sample size.
Kyle considers all 4 of their curves to be "species-accumulation" curves, since in all 4 cases we are looking at how species accumulate with numbers (either numbers of individuals or numbers of samples).  Gotelli & Graves "Individuals: Rarefaction" = Kyle's "Rarefied, individual-based species-accumulation" Gotelli & Graves "Individuals: Accumulation" = Kyle's “Random- or encounter-order, individual-based species-accumulation" Gotelli & Graves "Samples: Rarefaction" = Kyle's "Rarefied, sample-based species-accumulation" Gotelli & Graves "Samples: Accumulation" = Kyle's “Random- or encounter-order, sample-based species-accumulation"
Sample-based species richness accumulates more slowly than individual-based species richness. Why?
Population-level spatial autocorrelation!
Figure from Gotelli & Colwell (2001)

20. Species-Area Relationships log (S) = log (c) + z * log (A)
Emerging from a sample-based approach, the relationship between species number & area is asymptotically increasing
Botanist Olaf Arrhenius (1921) first formalized the species-area curve
The Arrhenius equation is a power function: S = cAz
log (S) = log (c) + z * log (A)
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The species-area curve appears with almost law-like consistency in empirical data sets.
See Lawton and Hutchings for the “law-like” character of the species-area relationship.
Log (Number of species)
Two constants:
Intercept = log (c)
Slope = z
Number of species
Area
Log (Area)

21. Species-Area Relationships
The power function S = cAz typically works well for islands
E.g., Darlington (1957) proposed that a ten-fold increase in island area results in a two-fold increase in S
Land birds in the West Indies:
log (S) = * log (A)
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Map of Caribbean islands from Wikimedia Commons

22. Species-Distance Relationships
Diamond (1972) compared species richness on islands with that expected for an island “near” (< 500 km) a “mainland” source
“Mainland” = New Guinea
Islands = Bismark Archipelago
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Map of Oceania from Wikimedia Commons

23. Theory of Island Biogeography
Joint consideration of area and distance led to the
Equilibrium Theory of Island Biogeography
(Munroe 1948; MacArthur & Wilson 1963, 1967;
for a good description see Gotelli 2001, chapter 7)
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Although not a law in Community Ecology, the Theory of Island Biogeography was developed as a process-based potential explanation for the repeated patterns of species richness being related to area (species-area curves) and isolation.
Photos of MacArthur & Wilson from Wikimedia Commons

24. Theory of Island Biogeography
s = (-I / P) * S + I
Immigration rate (s)
(e.g., new species
per yr) I
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For a given island, notice that the immigration rate must be zero at Smax (or P, the total number of species in the mainland species pool).
Equation notation from Gotelli (2001) P
Number of species (S)

25. Theory of Island Biogeography
µS = (E / P) * S E
Extinction rate (µS) (e.g., number of species per yr)
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Equation notation from Gotelli (2001) P
Number of species (S)

26. Theory of Island Biogeography
dS/dt = (immigration rate) – (extinction rate) = (-I/ P)S + I - (E/P)S
Equilibrium S when dS/dt = 0  S* = IP/(I+E)
Immigration rate (s)
(e.g., new species
per yr) I E
Equil. turn-over rate (T*)
Extinction rate (µS) (e.g., number of species per yr)
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Equation notation from Gotelli (2001) S* P
Number of species (S)

27. Theory of Island Biogeography
Turnover is a key feature of this model because there is no fixed stable composition of species, even though S is constant
Therefore, the model is simultaneously both equilibrial (species number) and non-equilibrial (species composition)
Notice that the model doesn’t (so far) “explain” the
species-area relationship
What do we need?
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28. Theory of Island Biogeography
Why does the probability of extinction for each
species vary with island size?
Immigration rate (s)
(e.g., new species
per yr)
Small island
Large island
TSmall
TLarge
Extinction rate (µS) (e.g., number of species per yr)
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Equation notation from Gotelli (2001)
SSmall
SLarge
Number of species (S)

29. Theory of Island Biogeography
Why does the probability of immigration for each
species vary with island isolation?
Immigration rate (s)
(e.g., new species
per yr)
Near island
Far island
TNear
TFar
Extinction rate (µS) (e.g., number of species per yr)
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Equation notation from Gotelli (2001)
SFar
SNear
Number of species (S)

30. Theory of Island Biogeography
Some of the simplifying assumptions:
- Fixed source pool of species from which colonists are drawn
- Source pool species have the same colonization & extinction probabilities
- Population sizes scale with island size
- Immigration rate is inversely proportional to distance
- The probability of extinction is inversely proportional to population size
- The probability of immigration & extinction is independent of species
composition on the island (i.e., no effects of species interactions)
- Habitat heterogeneity is constant relative to island size
There is no species-specific biology in this theory!
The radical idea is that species are identical!
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Some of these assumptions do not significantly
alter the model’s predictions…

31. Theory of Island Biogeography
Predictions are fairly robust to non-linear
extinction and immigration functions
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Equation notation from Gotelli (2001)
Figure modified from Gotelli (2001)

32. Theory of Island Biogeography
Predictions are fairly robust to non-linear
extinction and immigration functions
Small
ELarge
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Equation notation from Gotelli (2001)
Figure modified from Gotelli (2001)

33. Theory of Island Biogeography
Predictions are fairly robust to non-linear
extinction and immigration functions
Near
IFar
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Equation notation from Gotelli (2001)
Figure modified from Gotelli (2001)

34. Theory of Island Biogeography
Target effect: Larger islands present larger immigration targets
(Gilpin & Diamond 1976)
Recall that without
target effect
I Small = I Large
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Remember that in the original formulation, immigration did not vary as a function of island size, so either immigration curve on the figure could be considered the immigration curve for both island types in the absence of a target effect.
Equation notation from Gotelli (2001)
Notice the influence
on T*
Figure modified from Gotelli (2001)

35. Theory of Island Biogeography
Rescue effect: Higher rates of continued immigration to near vs. far islands results in larger N (or more patches of populations) & potentially greater genetic diversity (Brown & Kodric-Brown 1977)
Recall that without
rescue effect
E Near = E Far
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Remember that in the original formulation, extinction did not vary as a function of island isolation, so either extinction curve on the figure could be considered the extinction curve for both island types in the absence of a rescue effect.
Equation notation from Gotelli (2001)
Notice the influence
on T*
Figure modified from Gotelli (2001)

36. Island Biogeography in the Real World Floral & Faunal Relaxation in Habitat Fragments
Departures from predictions of the “null model” may be the most important contribution of this theory to modern ecology and management
How do newly created islands respond to fragmentation & isolation?
What are the best strategies within the SLOSS debate?
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37. A subset of tree species is favored on small islands
Island Biogeography in the Real World Floral & Faunal Relaxation in Habitat Fragments
Leigh et al. (1993) sampled species composition on small islands (< 2 ha) in Lake Gatun ~ 80 yr after construction of the Panama Canal
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Tree diversity on small islands < equivalent sized areas of mainland or large islands like Barro Colorado Island
A subset of tree species is favored on small islands

38. Island Biogeography in the Real World Floral & Faunal Relaxation in Habitat Fragments
Terborgh et al. (2001) studied forest-savanna ecosystems on islands within Lago Guri, a 4300 km2 hydroelectric lake in Venezuela, formed by damming the Caroní Rio in 1986
Species loss has been rapid, but the loss of species through local extinction has not been random
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39. Island Biogeography in the Real World Floral & Faunal Relaxation in Habitat Fragments
The Brazilian government mandated in the 1970s that a fraction of each Amazonian cattle ranch had to be retained as forest
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40. Island Biogeography in the Real World Floral & Faunal Relaxation in Habitat Fragments
The Biological Dynamics of Forest Fragments Project isolated 1, 10, 100 &1000 ha fragments and continues to compare them to forested control plots on ranches north of
Manaus, Brazil
Tom Lovejoy, Bill Laurance, Robb Bierregaard, Phil Stouffer,
Bruce Williamson, etc. have demonstrated dramatic changes,
especially in the smallest fragments, as a function of size,
degree of isolation, and type of intervening matrix
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Vellend (2010, pg. 199) outlines multiple hypotheses for species-area relationships: (1) prob. extinction may decrease with island area; (2) larger islands may provide larger target; (3) environ. heterogeneity may scale positively with island area; (4) prob. speciation may increase with island area.
“A paradigm like IBT that considers only changes in fragment size
and isolation while ignoring other anthropogenic effects…
is dangerously inadequate for conservation purposes”
(Laurance 2008, pg. 1739)

41. Spatial & Temporal Scale in Ecology
“It is argued that the problem of pattern and scale is the central problem in ecology, unifying population biology and ecosystems science, and marrying basic and applied ecology”
S. Levin (1992)
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Photo from Princeton U.

42. Spatial & Temporal Scale in Ecology
Spatial & temporal patterns change with the scale of measurement
For example, the slope of the species-area curve changes across scales
Focus
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From Willig et al. (2003), pg. 275: “The focus of a research design is defined by the inference space to which each datum applies, whereas the extent of a research design relates to the inference space to which
the entire collection of data applies in an analysis.”
Extent
See Willig et al. (2003, pg. 275)
Figure from Hubbell (2001, pg. 158)

43. Spatial & Temporal Scale in Ecology
Processes that impact organisms, populations & communities
act on a variety of spatial & temporal scales
Processes occurring at any given scale differentially determine patterns at increasing – or decreasing – scales
Spatial & temporal patterns change
with the scale of measurement
Spatial & temporal variability change
with the scale of measurement
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“How can we meaningfully extrapolate ecological information across spatial scales? This is one of the central issues in… ecology”
P. Turchin (1996)

44. Scale in Ecology
We seek mechanistic links among patterns and processes across scales
E.g., how can we extrapolate from one scale to another
(e.g., leaf-level gas exchange and photosynthesis 
forest productivity  global climate change)?
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Photos from Wikimedia Commons