1. introduction n = f(n ,nt-2,nt-3

1

Year

Pop

ulat

ion

inde

x

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1810 1870 1930 1990

Time Series Analysis

Nt = f(Nt-1,Nt-2,Nt-3)1. Introduction

3. Global wader dynamics

2. Sarcoptic mange-fox dynamics

Year

Pop

ulat

ion

inde

x

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1810 1870 1930 1990

Time Series Analysis 1. Introduction

-1

1

3

5

7

1820 1860 1900 1940 1980

Time Series Analysis (TSA)

1. Introduction

2

4

6

8

1965 1970 1975 1980 1985

grow

th ind

ex

2

4

6

8

1965 1970 1975 1980 1985

grow

th ind

ex Cassiope tetragonaCassiope tetragona

40

80

120

160

20 30 40 50 60 70 80

year

Julia

n fl

ower

dat

e TussilagoTussilago

Population Time Series


1. Introduction

-1

1

3

5

7

1820 1860 1900 1940 1980

Popu

latio

n si

ze (

stdz

)

year

•  We have seen and expect changes in ecology parallel to changes in climate •  Interesting as this may be, we need to go further - to go behind the patterns to expose the processes …

… direct, indirect, multi-trophic, cascading, feedback dynamics

temporally

spat

ially

16 pops


1. Introduction

•  Time series analysis (TSA) is a pure statistical tool designed to disentangle the autocovariate patterns in time series

•  We have seen and expect changes in ecology parallel to changes in climate •  Interesting as this may be, we need to go further - to go behind the patterns to expose the processes …

… direct, indirect, multi-trophic, cascading, feedback dynamics


Analysis of the lynx 10-year cycle

•  Boreal forest the Arctic Ocean

•  Boreal forest to the Arctic Ocean •  Food: berry (summer), birch, willow (winter)

•  Food: snowshoe hare, squirrel, grouse

•  Predators: lynx, fox, coyote, owl

Lepus americanus Lynx canadensis

2


An example: the lynx 10-year cycle

Distinct 10-year cycle (harvest data)

Processer?: obscure!

Sun spots lynx

Sun spots

+

EXTRINSIC (DID): Lunar cycles (moonlight quality), Weather, Forest-fire (plant) INTRINSIC (DD): Hare population, Plant-Hare, Lynx-Hare




Sun spots lynx

Sun spots


-





Sun spots lynx

Sun spots


+

Pure correlation analyses not good!



What to do?


Nt = f(Nt-1,Nt-2,..., Nt-11)!...

Kluane indicates that hare-predator interactions are central.

lynx

hare

year

dens

ity

f(Nt-1,Nt-2) decrease Nt =

f(Nt-1,Nt-2) increase

… dynamics non-linear!

High dependence (80%) on hare density ...


Time Series Analysis (TSA) Time Series Analysis (TSA)

1. Introduction

1935 1945 1955 1965year

abun

danc

e

An example: grouse population dynamics

3


1. Introduction

1935 1945 1955 1965year

abun

danc

e te

mpe

ratu

re

rtemp,grouse = 0.90

Does temperature explain 81%?



1. Introduction

1935 1945 1955 1965year

abun

danc

e te

mpe

ratu

re

rtemp,grouse = 0.01

at,t-1 = 0.46



1. Introduction

•  TSA makes no sense without an ecological framework

(i) Scale is important: data must reflect biology (ii) Data often have an ”internal dependence” (populations and phenology)

John Maynard Smith ... ”mathematics without ecology are sterile” ...


1. Introduction

bt

tt aN

RNN

1

1

1 −

−

+=

Maynard Smith – Slatkin population model

Dynamics of Natural Populations

1. Introduction

R: fundamental net reproductive rate a: susceptibility of crowding b: degree of intra-specific competition

“a” gives the level about which fluctuations occur

bt

tt aN

RNN

1

1

1 −

−

+=


1. Introduction

bt

tt aN

RNN

1

1

1 −

−

+=

Maynard Smith – Slatkin population model

( )btett aNrXX 11 1log −− +−+=

1)1())(log( −−+−= tet XbabrXAutoregressive model (AR)

4


1. Introduction

intra-specific

predator

inter-specific

forage

inter-specific

Combine ecological theory with time series analysis

Nt = f(Nt-1, Nt-2, Nt-3)


1. Introduction

predator

forage

intra-specific

inter-specific

inter-specific

Nt = f(Nt-1, Nt-2, Nt-3)



1. Introduction

predator

forage

intra-specific

inter-specific

inter-specific

Nt = f(Nt-1, Nt-2, Nt-3)



1. Introduction

predator

forage

intra-specific

inter-specific

inter-specific

Nt = f(Nt-1, Nt-2, Nt-3)


Statistical dimension of time series indicates number of trophic interactions!!!

Year

Pop

ulat

ion

inde

x

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1810 1870 1930 1990

Time Series Analysis 2. Sarcoptic mange – fox dynamics


2. Sarcoptic mange – fox dynamics

•  The disease sarcoptic mange is caused by the skin-dwelling mite (Sarcoptes scabiei var. vulpes) and has been reported in red fox populations in Europe, North America and Russia.

•  Approximately one month after exposure, infected foxes commonly develop skin lesions characteristic of hyperkeratosis. Severe loss of hair and progressive deterioration of body condition then follows and, in the majority of observed cases, infected foxes eventually die from starvation.

5





12110 −− ++= ttt XXX βββDensity

dependence Effect of mange



12110 −− ++= ttt XXX βββ

<1 0

1 <0

AR(1)

AR(2)

Density dependence

Effect of mange



β1

β2


2. Prey dynamics with climate

Forchhammer et al. 1

Figure 1

fox: Yt = ln(Pt)

prey: Xt = ln(Nt)

Climate: Ut

a1(t-1)

b1(t-1)

b3(t)

b2(t) a2(t-2)

a3(t) Pt = Pt−1 exp(a0 + a1Yt−1 + a2X t−2 + a3Ut )

Nt = Nt−1 exp(b0 + b1Xt−1 + b2Yt + b3Ut )

Xt = (b0a0 − a1b0 )+ (2 + a1 + b1)Xt−1 + (b2a2 − b1 − a1 − a1b1 −1)Xt−2

+(b3 + b2a3)Ut + (−b3a1 − b3)Ut−1

Xt = β0 + β1Xt−1 + β2Xt−2 +ω1Ut +ω 2Ut−1ARMA (2,2):




Figure 1

fox: Yt = ln(Pt)

prey: Xt = ln(Nt)

Climate: Ut

a1(t-1)

b1(t-1)

b3(t)

b2(t) a2(t-2)

a3(t)

14

Table 1: The statistical (i.e., ARMA in Equation 3) coefficients expressed as a

function of the ecological interaction coefficients in Figure 1 and their ecological

interpretation.

Statisticalcoefficients

Ecologicalcoefficients

Ecologicalinterpretation

β1 2+a1+b1 direct density dependence: function of intra-trophic interactions {a1,b1} only.

β2 b2a2-b1-a1-a1b1-1 delayed density dependence: function of inter-trophic interactions {a2,b2} and a complex function of intra-trophic interactions {a1,b1}.

ω1 b3+b2a3 additive effect of direct {b3} and indirect climatic influence through predator dynamics {a3,b2}.

ω2 -b3a1-b3 direct climatic influence {b3} relative to its interaction with fox density dependence {a1}.

Forchhammer et al.

6




Figure 1

fox: Yt = ln(Pt)

prey: Xt = ln(Nt)

Climate: Ut

a1(t-1)

b1(t-1)

b3(t)

b2(t) a2(t-2)

a3(t)


Figure 2

-0.04

0

0.04

0.08

0.12

roe deer hare partridge

0.6

0.8

1

1.2

roe deer hare partridge-0.4-0.3-0.2-0.100.10.2(a)

(b)

Ɣ E1’ ż E2’

Ɣ Z1’ ż Z2’

Z1’

and

Z2’

E1’ E2’ *

** *

*

*

*




Figure 1

fox: Yt = ln(Pt)

prey: Xt = ln(Nt)

Climate: Ut

a1(t-1)

b1(t-1)

b3(t)

b2(t) a2(t-2)

a3(t)

•  Hare and partridge dynamics displayed delayed DD and climate mediated through fox.

•  Roe deer dynamics displayed direct DD with both direct and indirect climatic effects.

23

Figure 5


Figure 5

hare:partial R2 = 0.52

partridge:partial R2 = 0.49

-0.05

0

0.05

0.1

0.15

0.2

-1 -0.5 0 0.5

partual R2 = 0.53

-0.05

0

0.05

0.1

0.15

01020304050

(a)

(b)

habitat quality(% sand in soil)

goodpoor

Z 2

Z 1

b 2

Forchhammer et al.

dire

ct D

D

Year

Pop

ulat

ion

inde

x

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1810 1870 1930 1990

Time Series Analysis 3. Global wader dynamics



H

L Cold & dry

warm & wet

H

L Cold & dry warm

& wet

High NAO winter

Low NAO winter

-6

-3

0

3

6

1850 1900 1950 2000



-4

-2

0

2

4

1970 1980 1990 2000

Stdz

UK

win

ter

popu

latio

ns

Eurasian (Bar-tailed Godwit, Curlew, Dunlin, Grey Plover, Oystercatcher)

Nearctic (Black-tailed Godwit, Knot, Turnstone)

Mixed (Redshank, Ringed Plover, Sanderling)



wintering breeding

migration

NAOt-1 NAOt

A model:

dd dd

other species

Nearctic Eurasian

7



Partial Autocorrelation Plot

0 10 20 30Lag

-1.0

-0.5

0.0

0.5

1.0

Cor

rela

tion

Lag (years)

Corr

elat

ion,

ρ(N

t, N

t-n)

Black-tailed Godwit Autocovariate structure: lagged influence of N on N

truncation

AR(1) process!: Nt = f(Nt-1, NAOt, NAOt-1)



Autocovariate structure: Nt = f(Nt-1, NAOt, NAOt-1)

β 1

(1st

ord

er A

R c

oefic

ient

)

stable (-2<β1≤1)

unstable (β1>1)

-0.4

0

0.4

0.8

1.2

Bar-tai

led G

odwit

Curlew

Dunlin

Grey Plov

er

Oysterc

atche

r

Black-t

ailed

God

witKno

t

Turns

tone

Redsh

ank

Ringed

Plover

Sande

rling

Eurasian Nearctic Mixed



*

Covariate structure: Nt = f(Nt-1, NAOt, NAOt-1)

ω1 ω

2 (N

AO c

oeff

icie

nts)

-8-6-4-202468

Bar-tai

led G

odwit

Curlew

Dunlin

Grey Plov

er

Oysterc

atche

r

Black-t

ailed

God

witKno

t

Turns

tone

Redsh

ank

Ringed

Plover

Sande

rling

* *

* * * *

*

Eurasian Mixed Nearctic

Assumes a linear relationship between NAO and N!



Non-linear effects of the NAO: a TAR model

-6

-4

-2

0

2

4

6

1970 1980 1990 2000

NAO

win

ter

inde

x

early (low NAO) phase

late (high NAO) phase

-0.5

0

0.5

1

early late

early late

early late

-12

-8

-4

0

4

8

β 1

ω1 ω

2

AR(1) NAOt NAOt-1

*

*

1989

Tree-regression analysis (reduce deviance by splitting)

Bar-tailed Godwit



-8-6-4-202468

Bar-tai

led G

odwit

Curlew

Dunlin

Grey Plov

er

Oysterc

atche

r

Black-t

ailed

God

witKno

t

Turns

tone

Redsh

ank

Ringed

Plover

Sande

rling

*

* *

* * * *

*

Covariate structure: Nt = f(Nt-1, NAOt, NAOt-1)

Eurasian Mixed

ω1 ω

2 (N

AO c

oeff

icie

nts)

Nearctic



Non-linear effects of the NAO: a TAR model

-6

-4

-2

0

2

4

6

1970 1980 1990 2000

NAO

win

ter

inde

x

-0.5

0

0.5

1

early late

early late

early late

-12

-8

-4

0

4

8

1989

early (low NAO) phase

late (high NAO) phase

β 1

ω1 ω

2

AR(1) NAOt NAOt-1

*

*

-1.5

-1

-0.5

0

0.5

1

early late

early late

early late

-4

-2

0

2

4AR(1) NAOt NAOt-1

*

β 1

ω1 ω

2

Bar-tailed Godwit

Redshank

8



Direct negative temporal dependence was found in most populations (fluctuating stability)

ln(Nt-1)

ln(N

t) (i)

NAOt

ln(N

t)

High NAO current year winters decreased the number of wintering waders in 3 species (windier and wetter)

(ii)

NAOt-1

ln(N

t) High NAO in previous year’s winter

increased the number of waders (Eurasian) the following winter (improved conditions when arriving)

(iii)

Non-linear response to changes in the NAO



-4

-2

0

2

4

1970 1980 1990 2000

Stdz

UK

win

ter

popu

latio

ns

Eurasian (Bar-tailed Godwit, Curlew, Dunlin, Grey Plover, Oystercatcher)

Nearctic (Black-tailed Godwit, Knot, Turnstone)

Mixed (Redshank, Ringed Plover, Sanderling)

Mads C. Forchhammer Aarhus University

Department of Bioscience Arctic Research Centre (ARC)

CIRCE

email: [email protected]

1. introduction n = f(n ,nt-2,nt-3

Documents