Species diversity

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.

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

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 )

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

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

Graphical representation of ESn = number of species with abundance ni,

ni

Sn

ni

Sn

ni

Sn

ni

Sn

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

Diversity indices

• Single number combining species number and evenness

• >60 different formulas• differ in relative weight given to evenness

or species number

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

Imaginary communities: which is more diverse?

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

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)

General form of diversity indices

Average rarity =

= S pi (R (pi)) S

S ni (R (pi)) S

N

= D [A diversity index]

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

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

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

Three diversity indices

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

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

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

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

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?

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

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

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

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

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

Evenness

• E1/D … simple

• Evar … derived from variance, hence derived from the conceptual basis of evenness

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

Data for dominance-diversity plot

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

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

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

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)

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

Diversity

Productivity

Stability

How is S related to primary productivity?

• Small scale– fertilize plots - plant diversity declines– Tilman 1996 (Fig. 2c)

• Lakes– Eutrophication - diversity declines

Unimodal diversity-productivity gradients

productivity

spec

ies

Monotonic diversity-productivity gradients

productivity

spec

ies

Unimodal can look like monotonic

productivity

spec

ies

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

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?

VRR Hypothesis

• Assumes –species occur in patches–competition is local– resource competition

• Generality?–Rodents? –Benthic invertebrates?–Tropical mammals?

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

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?

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

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

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)

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

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

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

Diversity Stability • MacArthur: models of simple communities• Feeding relationships (food webs)

COMPLEX

SIMPLE

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

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

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)

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)

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

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

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

Tilman, Fig. 5

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

Tilman, Fig. 9A