Patch based Approaches

A classification of multi-site scenarios

Correlations

in population

Movement rates

Vital rates Essentially none Low to medium

High

Significantly

negative

Separate pops, multiple strongly beneficial

Highly effective

Metapop

Multiple sites, very different habitat

Non different from zero

Separate pops multiple strongly beneficial

Somewhat effective Matapop

Multiple sites

Somewhat different habitat

Significantly

Positive

Separate pops

Multiple not very effective

Ineffective

Metapop

One population

Metapopulation models

• Most theoretical metapopulation models assume that all populations have identical extinction rates, and that they are all equi-distant from one another (e.g. Harrison and Quinn 1989)

• But these models are too far removed from realities of specific multi-site situations to be of practical use for particular species.

Incidence function models

• Developed by Hanski (1991, 1994)

• Sjögren-Gulve and Ray 1996, Moilanen 1999, 2000 and Kindvall 2000)

Ilkka Hanski

Atte Moilanen

The Incidence function model

Use patterns of patch occupancy over time and space (“incidence’) to estimate:

Ei, the probability of extinction for habitat patch i when it is occupied, and

Ci, the probability that patch i becomes colonized when it is occupied

Taking in account the patch’s habitat area, habitat quality, distance to other populations, and other potentially important characteristics

The Incidence function model

• We start assuming specific function forms for the effects of the causal factors (patch area and quality, proximity of other populations, etc) on Ei and Ci.

Extinction

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120

Area

P(e

xtin

ctio

n )

in p

atch

i e=1

x=0.5

xi

i A

eE

A= area

Colonization

Arriving migrants

P(c

olon

izat

ion)

in p

atch

i

22

2

yM

MC

i

ii

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

M= number of migrants

y=1

The number of migrants

• depends on:

• The probability of other patches having extant populations, their population sizes (which is assumed to be proportional to the area of each patch), the rate at which dispersers leave the patch, and the distances from each other patch to focal patch i.

The number of migrants

jij

N

ij

bji pDAM )exp(

Mi= number of migrants

β = per-unit-area migrant production rate

A= patch i area

Scaled by parameter b to allow for nonlinearity

pj= 0 = empty and pj=1 if occupied

α= scaling factor

Number of migrantsA= area

distance

Num

ber

of m

igra

nts

(M)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6

A=10

A=20

The incidence function model

)1( iii

ii CEC

CJ

The probability that site i is occupied in any one site is predicted by

in cases where rescue effects are thought to occur

The incidence function model

• A snap-shot of the pattern of patch occupancy can be used to estimate the parameters governing Ci and Ei

However, it means assuming that the occupancy patterns seen in the field are very near their equilibrium values

The incidence function model

)'

1(

1

2 xii

i

ASe

J

Where e’=ey’ and y’=y/β

jij

N

ij

bji pDAS )exp(

Count based Approaches

Information needed

• Mean population growth rate

• Variance in population growth rate

• Covariance in population growth rate

• Probabilities of movement between populations

• Estimates of density dependence

The California clapper rail

Harding et al 2001

The California clapper rail

Population Numbers μ σ2 P(ex)

Mowry 70 0.043 0.051 0.06

Faber 29 -0.002 0.041 0.79

Laumeister 33 0 0.051 0.72

Harding et al. 2001

0.06*0.79*0.72=0.034

Pearson correlation coefficients

Population Mowry Faber

Mowry 1

Faber 0.995 1

Laumiester 0.896 0.938

The transition matrix

• Determining counts at different sites

nM(t+1)

nF(t+1)

nL(t+1)

=

nM(t)

nF(t)

nL(t)

λMt 0 0

0 λFt 0

0 0 λLt

The transition matrix

• Determining counts at different sites

nM(t+1)

nF(t+1)

nL(t+1)

=

nM(t)

nF(t)

nL(t)

(1-d)λMt da da

da (1-d)λFt da

da da (1-d)λLt

d=constant probability of an individual dispersing

a=constant probability of an individual arriving

Effect of correlations

0

0.1

0.2

0.3

0.4

0.5

0.6

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

Years in the future

Cu

m p

rob

of

(Exti

n)

.

With correlations

Without correlations

Effect of levels of dispersal

0

0.1

0.2

0.3

0.4

0.5

0.6

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99

Years in the future

Cu

m p

rob

of

(Exti

n)

.

20 % move

10 % move

5 % move

none move

Demographic Approaches

Coryphantha robbinsorum

The basic model

0 0 0 f4s4

s1 s2(1-g2) 0 0

0 0 s3g3 s4

0 s2g2 s3(1-g3) 0A=

Seeds

Small juveniles

Large juveniles

Adults

Vital rates

Site s1 s2 s3 s4 g2 g3 f4

a 0.0173 0.8545 0.9875 0.9692 0.2145 0.1411 33

0.0175 0.0524 0.025 0.0386 0.1304 0.0528 0

b 0.0028 0.6197 0.9645 0.9852 0.1834 0.2466 33

0.0017 0.1924 0.0416 0.0295 0.2135 0.1711 0

c 0.0073 0.767 0.9135 0.9563 0.4376 0.3348 33

0.0057 0.0905 0.0512 0.0361 0.3198 0.071 0

The multi-site model0 0 0 f4as4a

s1a s2a(1-g2a) 0 0

0 0 s3ag3a s4a

0 s2ag2a s3a(1-g3a) 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 f4bs4b

s1b s2b(1-g2b) 0 0

0 0 s3bg3b s4b

0 s2bg2b s3b(1-g3b) 0

0 0 0 f4c4c

s1c s2c(1-g2c) 0 0

0 0 s3cg3c s4c

0 s2cg2c s3c(1-g3c) 0

G=

The multi-site model0 0 0 f4as4a(1-m-m)s1a s2a(1-g2a) 0 0

0 0 s3ag3a s4a

0 s2ag2a s3a(1-g3a) 00 0 0 0

0 0 0 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 f4as4a,mBA

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 00 0 0 0

0 0 0 0

0 0 0 0

0 0 0 f4bs4b(1-m-m)

s1b s2b(1-g2b) 0 0

0 0 s3bg3b s4b

0 s2bg2b s3b(1-g3b) 0

0 0 0 f4c4c (1-m-m)s1c s2c(1-g2c) 0 0

0 0 s3cg3c s4c

0 s2cg2c s3c(1-g3c) 0

G=

0 0 0 f4as4a,mcA

0 0 0 f4as4a,mAB

0 0 0 f4as4a,mCB

0 0 0 f4as4a,mAC

0 0 0 f4as4a,mBC