Patterns in time

-3-2

-10123

1964 1968 1972 1976 1980

Oulu-3-2-1

0123

1964 1968 1972 1976 1980

Vaasa

-3-2-1

0123

1964 1968 1972 1976 1980

Häme-3-2-1

0123

1964 1968 1972 1976 1980

Turku-Pori

-3-2-1

0123

1964 1968 1972 1976 1980

Central Finland

-3-2-1

0123

1964 1968 1972 1976 1980

Uusimaa

-3-2-10

123

1964 1968 1972 1976 1980

Lapland-3-2-10

123

1964 1968 1972 1976 1980

Kuopio-3-2-10

123

1964 1968 1972 1976 1980

North Karelia

-3

-2-10123

1964 1968 1972 1976 1980

Mikkeli-3-2

-101

23

1964 1968 1972 1976 1980

Kymi

Population dynamics (1964 to 1983) of the red squirrel in 11 provinces of Finland (Ranta et al. 1997)

0

10000

20000

30000

40000

50000

60000

Numb

er o

f lyn

x fu

rs tr

aded

1830 1840 1850 1860 1870 1880 1890 1900 1910 1820 1930

0

10

20

30

40

50

60

70

80

90

100

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

Study time (years)

Ab

un

da

nc

e i

nd

ex

Lynx fur in Canada

Voles in Norway

Mean abundance

Upper limit (carrying capacity)

Lower limit (extinction treshold)Elton and Nicholson (1942 )

GenerationSpecies1.00 0.71 1.25 1.27 6.36 12.50 14.52 50.84 49.072.00 0.93 1.10 3.43 0.55 14.30 15.06 31.47 27.853.00 0.09 0.83 2.08 2.45 12.34 15.45 26.01 61.754.00 0.97 0.06 0.35 3.51 13.08 27.56 5.57 44.995.00 0.14 0.60 0.56 4.33 13.47 9.86 28.64 30.706.00 0.33 1.44 3.22 4.89 15.39 27.84 34.93 39.657.00 0.63 0.41 0.94 0.87 14.89 2.87 49.46 61.228.00 0.96 0.06 2.31 6.81 12.76 28.50 48.08 60.169.00 0.20 1.48 0.90 3.43 8.08 25.14 60.29 51.97

10.00 0.30 1.49 2.80 4.60 15.75 1.52 46.60 24.2611.00 0.96 0.36 2.49 1.40 7.67 25.94 51.83 48.5712.00 0.38 0.19 3.52 5.56 1.64 28.19 24.89 4.7813.00 0.09 1.52 2.39 7.02 9.82 5.54 18.72 12.0614.00 0.88 1.10 0.43 1.23 6.92 24.29 55.69 34.6615.00 0.16 1.30 3.69 1.43 4.06 1.99 1.03 10.1516.00 0.45 1.76 2.24 4.43 8.46 17.37 7.02 36.2517.00 0.20 1.27 0.11 6.41 9.32 6.64 12.77 1.6318.00 0.83 1.60 0.12 0.04 9.55 17.38 19.19 22.0819.00 0.97 0.99 0.59 4.49 14.33 6.77 46.32 47.9120.00 0.36 0.59 1.29 0.32 0.40 4.29 46.40 7.69

Mean 0.53 0.97 1.74 3.51 10.24 15.34 33.29 33.87Variance 0.12 0.30 1.48 5.40 20.10 98.09 334.99 380.96

Taylor’s power law

Assume an assemblage of species, which have different mean abundances and fluctuate at random but proportional to their abundance.

The relationship between variance and mean follows a power function of the form

2 2a

Going Excel

Taylor’s power law; proportional rescaling

0.00

1.00

2.00

3.00

4.00

0.00 5.00 10.00 15.00 20.00

Generation

Ab

un

da

nce

z

y = 0.34x2.0

R2 = 0.99

0

500

1000

1500

0 20 40 60 80

MeanV

aria

nce

z

Taylor’s power law

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.2 0.6 1 1.4 1.8 2.2 2.6 3 3.4

Variance category

Per

cent

age

Taylor’s power law in aphids (red), moths (green) and birds (blue). In all three groups the exponent z of the relation s2 = a mz peakes around 2.

Data from Taylor et al. (1980).

Major results from this database are that

The variance – mean relationship of most populations follows Taylors power law

z = 2 is equivalent to a random walk

Z =<< 2 is required for population regulation

2 zN

The majority of species has 1.5 < z < 2.5

Long term studies of population variability

Most populations, in particular invertebrate populations are not regulated!

They are not in equilibrium

0.00

1.00

2.00

3.00

4.00

0.00 5.00 10.00 15.00 20.00

Generation

Ab

un

da

nce

z

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 5.00 10.00 15.00 20.00

Generation

Ab

un

da

nce

z

0.001.002.003.004.005.006.007.00

0.00 5.00 10.00 15.00 20.00

Generation

Ab

un

da

nce

z

2 1a

2 2a

2 0a

Ecological implications

2 za

Temporal variability is a random walk in time

Abundances are not regulated

Extinctions are frequent

Temporal species turnover is high

Temporal variability is intermediate

Abundances are or are not regulated

Extinctions are less frequent

Temporal species turnover is low

Temporal variability is low

Abundances are often regulated

Extinctions are rare

Temporal species turnover is very low

2ln( ) ln( )(ln( )) ln( )

2ER

N NT N K

Var

Mean time to extinction Extinction probability

Under the assumption of Taylor’s power law (a simple random walk in time without density dependent population regulation and lower extinction boundary)

we can calculate the frequency of local extinction

1.42)2

)20000ln(100000(ln

42.5)20000ln(2 ET

)(ln1 NT

t

tEeP

02.01 1.42

1

1 eP

)(ln2

1

1

rsVar

N

Nr

rNN

R

t

t

tt

Abundances (ind.m-2)

YearEustochus atripennis

Exallonyx ater

1981 1.78 1.81982 4.15 2.51983 5 2.71984 36 1.31985 0.8 0.81986 8.8 0.81987 2 0.5

     K 10 3

Abundances (ind.ha-2)

Eustochus atripennis

Exallonyx ater

17800 1800041500 2500050000 27000

360000 130008000 8000

88000 800020000 5000

   100000 30000

  Reproduction rate

  ln r ln r

0.85 0.330.19 0.081.97 -0.73-3.81 -0.492.40 0.00

  -1.48 -0.47

Variance 5.42 0.17

TE 42.101 1056.342

K 0.023 0.001

0

0.1

0.2

0.3

0.4

0.5

1 2 4 8 16 32 64 128256512

Number if individuals

Nor

mal

ized

num

ber

of

extin

ctio

ns

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Mean number of nesting pairs

Ext

inct

ion

prob

abili

ty

y = 0.06 / x y = 0.96x + 0.55

R2 = 0.46

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5

ln (number of nesting pairs)

ln (

ext

inct

ion

tim

e)

How many individuals do populations need to survive (lower extinction boundary)?

Orb web spiders on the Bahama islands (Schoener 1983)

Birds on small islands off the British coast (Pimm 1991)

4 6 8 10 12 15 20 25 300.1

0.40.80

0.2

0.4

0.6

0.8

1

Ext

inct

ion

prob

abili

ty

Number of patchesEmigrationrate

Parasitic Hymenoptera (Hassell et al. 1991)

The species – time relationship

Local species area and species time relationships in a temperate Hymenoptera community studied over a period of eight years.

0

100

200

300

400

500

600

700

0 50 100 150Area

Nu

mb

er

of

spe

cie

s

A

0

100

200

300

400

500

600

0 50 100 150Area

Nu

mb

er

of

spe

cie

sB

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 5 10t

Tu

rno

ver

C

S = S0Az S = S0tt

S = S0Aztt

The accumulation of species richness in space and time follws a power function model

S = (73.0 ± 1.7)A(0.41 ± 0.01) t(0.094 ± 0.01)

The mean extinction probability per year is about 9%

Photo E. G. Vallery

Coeloides pissodis (Braconidae)

-3-2-10123

1964 1968 1972 1976 1980

Oulu-3-2-10123

1964 1968 1972 1976 1980

Vaasa

-3-2-10123

1964 1968 1972 1976 1980

Häme-3-2-10123

1964 1968 1972 1976 1980

Turku-Pori

-3-2-10123

1964 1968 1972 1976 1980

Central Finland

-3-2-10123

1964 1968 1972 1976 1980

Uusimaa

-3-2-10123

1964 1968 1972 1976 1980

Lapland-3-2-10123

1964 1968 1972 1976 1980

Kuopio-3-2-10123

1964 1968 1972 1976 1980

North Karelia

-3-2-10

123

1964 1968 1972 1976 1980

Mikkeli-3-2-10

123

1964 1968 1972 1976 1980

Kymi

Population dynamics (1964 to 1983) of the red squirrel in 11 provinces of Finland

The Moran effect

Regional sychronization of local abundances due to correlated environmental effects

Patrick A.P. Moran 1917-1988

t t 1 t 2 A

t t 1 t 2 B

N (A) f (N (A), N (A))

N (B) f (N (B), N (B))

A B

Moran assumed:1. Linear density dependence2. Density dynamics are identical3. Stochastic effects are correlated

0

100000

200000

300000

400000

500000

600000

700000

Acr

es D

efol

iate

d Maine

0

500000

1000000

1500000

2000000

2500000

3000000

Acr

es D

efol

iate

d

02000040000

6000080000

100000120000140000

Acr

es D

efol

iate

dVermont

0

500000

1000000

1500000

2000000

2500000

Acr

es D

efol

iate

d

New Hampshire

Massachusetts

Year20 30 40 50 60 70 80 90

Defoliation by gypsy moths in New England states

Lymantria dispar

Data from Williams and Liebhold (1995)

Species turnover rates differ between groups of animals and plants

Larger animal species have lower turnover rates

Despite high turnover rates total species numbers of

habitats remain largely constant.

This constancy holds for ecological, historical and

evolutionary times

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E-04 1.E-03 1.E-02 1.E-01 1.E+001.E+011.E+021.E+03

Generation time

Tur

nove

r ra

te (

%/y

r)Protozoa

Sessile marine organisms

Arthropoda

Birds

Lizards

Vascular plants

Body weight

0

5

10

15

20

25

1977 1982 1987 1992 1997

YearN

umbe

r of

spe

cies

0

5

10

15

20

25

02000400060008000

Year

Num

ber

of s

peci

es

0

20

40

60

80

100

120

140

1940 1950 1960 1970 1980 1990 2000

Year

Num

ber

of s

peci

es

0

10

20

30

40

50

60

030006000900012000

Year

Num

ber

of s

peci

es

Desert rodents

Birds

Plants

Plants

Species turnover rates (Brown et al. 2001)

Speciation rates, latitudinal gradients, and macroecology

What causes the latitudinal gradient in species diversity?TemperatureHow does temperature influences species richness?SpeciationExtinction

Metabolic theory predicts that generation time t should scale to body weight and temperature to

1/ 4 E / kTW e

The theory predicts further that mutation rate a should scale to

body weight and temperature to

How does mean generation time decreases if we increase mean

environmental temperature from 5º to 30 º?

Mutation rates are predicted to increase by the same factor

Evolutionary speed can be seen as the product of mutation rates and generation turnover (1/t).

Still unclear is how temperature influences extinction rates.

E1/ 4 kTt w e 11.0

38.91

)5()30(

)2735/(7541

)27330/(7541

ee

tt

Today’s reading

Minimum viable population size: http://en.wikipedia.org/wiki/Minimum_viable_population

Long term ecological research: http://www.lternet.edu/

Kinetic effects of temperature on speciation: http://www.pnas.org/content/103/24/9130.full.pdf

Paleobiology: http://findarticles.com/p/articles/mi_m2120/is_n5_v77/ai_18601045