Download - Species diversity
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Species diversity
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Concept of diversity
• Informally … variety• Wallace’s traveler
– A traveler in Amazonia encounters a species of tree (1 individual); If he looks for another member of that same species he will seek it for a long time, encountering many other species before he finds another. This is true for most if not all the tree species in Amazonia.
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Concept of diversity
• In contrast … cattail marsh– for one species, number of encounters
between successive encounters with the same species will be small (often 0)
• Intuitively, Amazon forest is more diverse than the cattail marsh
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Diversity
• Equivalent to – probability of interspecific encounter– average rarity
• What determines these?– 1) number of species (S ) … species richness– 2) evenness, or equitability, or relative
abundances (E )
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Kinds of diversity
• a diversity: variety of species within one community
• b diversity: extent of replacement of species with changes in environment from place to place
• g diversity: combination of a and b diversity
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Numbers, measurements, etc.
• S = number of species• N = number of individuals• ni = number of individuals of species i
– i = 1, 2, 3, … , S
• Si=1ni = N
• pi = ni / N = relative abundance of species i
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Graphical representation of ESn = number of species with abundance ni,
ni
Sn
ni
Sn
ni
Sn
ni
Sn
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Quantifying evenness
• (observed diversity / maximal diversity) for a given S
• Calculate some measure of overall diversity (D)
• Hold S constant and set relative abundances of all species to 1/S
• Calculate Dmax= maximal diversity
• E = / D Dmax
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Diversity indices
• Single number combining species number and evenness
• >60 different formulas• differ in relative weight given to evenness
or species number
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Three examples
• S - 1 = D 1
–gives 0 weight to evenness
• Shannon-Weiner: - S i=1 pi ln(pi) = D2
– intermediate weight to evenness
• Simpson’s: 1 - Si pi
2 = D3
–gives major weight to evenness• Used to compare communities
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Imaginary communities: which is more diverse?
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Different comparisons
• Indices do not measure a single quantity• Diversity indices combine 2 inherently
different quantities• Weights chosen are arbitrary• Indices are related in a complex way
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General form of diversity indices
• Average rarity = diversity– a community
composed of many rare species is diverse
– Rarity ( R (pi)) is a decreasing function of relative abundance (pi)
pi
R(pi)
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General form of diversity indices
Average rarity =
= S pi (R (pi)) S
S ni (R (pi)) S
N
= D [A diversity index]
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What is R ( pi ) ?Rarity indicated by number of encounters y with other species that occur between encounters of a given species
encounters:i … j … k … m… j … x … z … i y = 6
y is amenable to analysis via probability theory
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From y to R ( pi )
• In general, probability theory yields the general function:
R ( pi ) = (1 - pib ) / b
• where b is a constant chosen by investigator
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D = average rarity(a general diversity index)
• D = S pi R( pi ) = S pi [ (1 - pib) / b ]
• constant b that is chosen determines which of the diversity indices (S-1, Shannon, Simpson) results
• as b increases, greater weight is given to evenness
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Three diversity indices
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Diversity indices
• Shows that they are related• Differ in R( pi )
• Does not solve the problem… which weighting is correct
• Solution: don’t bother with diversity indices
• implies that diversity as a single thing doesn’t exist
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Quantifying diversity
• Report S– in a sample S depends on N– to compare samples of different sizes
use rarefaction• Report E
–many measures depend on S–choose those least dependent on S
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Rarefaction
• two samples of different N• lower N, lower S in the sample, regardless
of S in the community itself• Rarefaction … estimating expected
number of species in a sample of size n :• E (S )n
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RarefactionE (S)n =
s (N - ni)! N !
S 1 - n! (N - ni - n)! n! (N - n) !
N = number of individuals in entire samplen = number of individuals in the subsampleni = number of individuals in species i
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Example
• Communities A and B
• differ in S and N in sample
• How many species are expected in B if only 16 individuals were sampled?
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Example• N = 62, n = 16, ni = {10, 20, 10, 20, 2}
• E (S )n = 0.962 + 0.999 + 0.962 + 0.999 + 0.453
• = 4.376• i.e., 4 or 5 species from community B• Even with rarefaction, community B has
greater species richness
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When to use rarefaction
• If NA and NB are close, rarefaction makes little difference
• Rule of thumb: if NA / NB > 10 or < 0.1 then use rarefaction
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Evenness
• Quantification should be independent of S
• Smith & Wilson 1996 tested 14 indices• Most, including common ones, fail
– note: Morin p. 18 J = H’ / Hmax
– = [ -S pi ln pi ]/[ln S ]
– one of the worst for independence of S
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Evenness
• example: Modified Hill’s Ratio• claimed to be less dependent on S than
most• E = {(1 / S pi
2) - 1} / {exp(-S pi ln pi ) - 1}
• dominance by 1 species … E = 0• maximal evennesss … E = 1• Smith & Wilson show it is independent
of S only for S > 10
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Evenness• among those that are independent of S:
E1/D = 1 / (S S pi2)
&
Evar = 1 – (2/ )p {arctan[VAR(ln(ni))]}
= 1 – (2/ )p {arctan S[ln(ni) – Sln(ni)/S]2 / S}
• seem to be good choices
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Evenness
• E1/D … simple
• Evar … derived from variance, hence derived from the conceptual basis of evenness
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Dominance-diversity plot
0.0001
0.001
0.01
0.1
1
0 5 10 15 20
Ab
un
dan
ce
Rank Abundance
Dominance diversity plot
Comm #1
Comm. #2
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Data for dominance-diversity plot
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b diversity
• Extent of replacement of species from place to place
• May be equated to dissimilarity between locations– If all locations have identical species list, b
diversity is 0
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b diversity
• Whittaker defined b diversity as;– = /b g a
• Where g is regional diveristy• And a is local diversity • However
– Debate about additive vs. multiplicative relationship of a b g
• =g ab vs. = +g a b
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Quantifying b diversity
• Inversely related to similarity– Two samples
• a=species unique to community A• b=species unique to community B• c=species shared
• Jaccard’s similarity J = c/(a+b+c)• Sørensen’s similarity Ø = 2c/(a+c+b+c)• 1-similarity = distance or turnover
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Quantifying b diversity
• Whittaker’s index– b = (a+b+c)/{[a+c+b+c]/2} – 1– = (Stotal /S) -1
• Works with >2 samples.• NOTE: none of these weight species by
abundances – (i.e., none incorporate evenness)
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b diversity
• Thorough mathematical treatments of b diversity– Tuomista 2010 a, b– Jost 2007
• But the more interesting question is why do we care about b diversity?– component of biodiversity that is at least in
part independent of a diversity– Relationships of biotic variables to a b g are
not consistent
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Diversity
Productivity
Stability
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How is S related to primary productivity?
• Small scale– fertilize plots - plant diversity declines– Tilman 1996 (Fig. 2c)
• Lakes– Eutrophication - diversity declines
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Unimodal diversity-productivity gradients
productivity
spec
ies
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Monotonic diversity-productivity gradients
productivity
spec
ies
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Unimodal can look like monotonic
productivity
spec
ies
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Why should diversity decline with productivity?
• High productivity reduced spatial heterogeneity in resources
• Spatial heterogeneity fosters diversity– reduces competitive exclusion– variance in resource ratio hypothesis (VRR)– each species does best on a particular ratio of
resources– e.g., plants and soil nutrients
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Spatial heterogeneity of resources
• Tilman’s fertilizer experiment– add N to soil– changes resource ratio– as N goes up greatly, ratio of N to other
nutrients gets larger and more constant• Varying resource ratios do foster
coexistence among plants– other taxa?
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VRR Hypothesis
• Assumes –species occur in patches–competition is local– resource competition
• Generality?–Rodents? –Benthic invertebrates?–Tropical mammals?
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Scale
• Local plots Continental areas– relationships of diversity to productivity
differ– Chase & Leibold 2002; Gross et al 2000– Unimodal patterns at local scale– Monotonic* patterns at regional scales– Implication: b diversity increases with
productivity
* Increasing
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Scale
• Same mechanism at all scales?– VRR works best at local plot scale– Applicability at larger scales unknown– Does natural variation in productivity affect
species the same way as experimental manipulation?
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Alternative interpretations (Abrams 1995)
• Challenges interpretation of monotonic patterns as artefacts of inadequate sampling
• Questions the assumption that unimodal patterns must be due to competition– need for experimental data on competition– experimental test of VRR
• Alternative hypotheses predict monotonic relationships
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How is S related to stability?
• What is stability?• Mathematically, a stable equilibrium…
– for variables i = 1 to n: dXi / dt = 0
– if the system is perturbed away from equilibrium (X1, X2, … Xn)* it returns• GLOBAL STABILITY: returns from any
perturbation• LOCAL STABILITY: returns from a limited set of
perturbations
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Ecological stability has multiple facets
• Constancy: lack of change in a variable• Resiliency: continued functioning despite
change• Recovery: return to original state
– elasticity greater if return in more rapid• Inertia: resistance to change via
perturbation• Persistence: survival of the system despite
changes (no extinctions)
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Ecological stability
• Populations may be stable in number of individuals
• Communities may be stable in:– species number– total biomass– Gross primary productivity– species abundance patterns
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Why stability matters
• Mathematically tractable, interesting• Value judgement … lack of change = good• Practical utility … we have an interest in
exploiting populations, ecosystem services– HOWEVER: some desirable properties
depend on inconstancy and periodic disturbance
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Diversity Stability
• Elton: Diverse or complex communities are more stable– Theory: simple models oscillations– Outbreaks of pests in simple agricultural
systems– Population cycles in “simple” arctic– Lack of cycles in diverse tropics
• Actual data few
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Diversity Stability • MacArthur: models of simple communities• Feeding relationships (food webs)
COMPLEX
SIMPLE
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May (1973)• Used models of communities to test
stability-complexity• Randomly assembled food webs• Simple: low connectance (trophic links /
species)• Complex: high connectance• Complex webs less likely to be stable
– Extinctions more common
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Empirical study• Lawler (1993):
Experimental microbial communities– bacteria
• (4 edible species)– protist bacteriovores
• (1-4 spp)– protist predators
• (1-4 spp)– simple 3-taxa chains vs. 5-
or 9-taxa communities
PCB
P1 P2
C1 C2
B
P1 P2 P3 P4
C1 C2 C3 C4
B
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Lawler 1993
• Extinctions– 9-taxa > 5-taxa > 3-taxa communities– consistent with May’s model results– diversity (or complexity) leads to instability– but population variation was not greater with
greater diversity (for most of the species)
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How did ecologists conclude complexity stability?
• Elton– particularly concerned about human impact– simplification systems– need to preserve natural systems
• Complexity and stability both valued• Assumed observed natural systems were
stable (return to equilibrium)
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Any resolution?
• Complexity inconstancy– numbers of species change readily
• Complexity resiliency– function despite change
• Stability & complexity (or diversity) may be related, but not simply– abiotic stability or predictability may foster
evolution of complexity
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Tilman 1996• Experimental manipulation of productivity
– Soil N– Produces a range of diversities (S ) in plots
• Diversity and productivity are related– How do plots respond to natural perturbations?
• annual variation, particularly drought (1988)
• Three measures of biomass– change in biomass pre-drought - peak drought– CV biomass over 10 years– CV biomass in non-drought years
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Whole community response:Total biomass
• All measures indicate greater stability with greater S– change in biomass lower with greater S– CV’s less with greater S
• McNaughton obtained similar results with respect to grazing as a disturbance
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Tilman, Fig. 5
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Individual species responses:Biomass
• Averaged across species (39)• Lower stability of populations with
greater S– CV’s greater with greater S– both drought and non-drought years
• Population stability (average) decreases with diversity
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Tilman, Fig. 9A