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Ecological Archives M081-021-A3

Philip J. Harrison, Ilkka Hanski, and Otso Ovaskainen. 2011. Bayesian state-space modeling of metapopulation dynamics in the Glanville fritillary butterfly. Ecological Monographs 81:581–598.

Appendix C. Prior distributions and additional mark-recapture results.

The definitions of the parameters and states described in this Appendix are given in Appendix A.

Table C1 gives the priors used for the model parameters apart from those related to the movement

model. These priors were based on the literature (Kuussaari 1998, Moilanen 2002) and the expert

opinions of butterfly biologists in the Metapopulation Research Group. We took a conservative

approach whereby all the priors were made quite dispersed and were well able to generate the full

range of dynamics thought possible for the species.

Based on results of Ovaskainen et al. (2008) on the movements of the Glanville fritillary in

another patch network, we set the medians and 95% confidence intervals of the movement model

priors (in the mark-recapture analysis) to k=102 (1…104), 1D =105 (104...106), 2D =103 (102...104),

m =0.1 (0.05...0.2), , 1MRp =0.3 (0.16...0.5), , 2MRp =0.3 (0.16...0.5). The posterior distribution from

the mark-recapture analysis was approximated by a multivariate normal distribution and used as a

prior distribution in the main model, with parameter values described in Table C2. We also need

priors for the initial states: In the IBM with probability ½ ,1 0iN and with probability ½

,1 ~ 1 (prob 2 (2 ))i iN Geom K . For the SPOM we simply assume that with probability ½

,1 0iO and with probability ½ ,1 1iO .

In line with our expectation, when fitting the diffusion model to the mark-recapture data, the

higher capture effort in one of the patches led to a higher estimate of capture probability,

,2 ,1MR MRp p (see Fig. 3 in the main manuscript). The posterior densities for the log-transformed

movement parameters and logit-transformed capture probabilities closely resembled multivariate

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normal distributions. As observed previously by Ovaskainen et al. (2008), it is difficult to estimate

the parameters 1D and k separately, as they are highly positively correlated in the posterior (Table

C2).

Figure C1 shows an assessment of the fit of the mark-recapture model, where we compare

the data with posterior predictive distributions for the days from first recapture to last recapture, the

movement distances, and the number of recaptures in a given patch. The overall fit of the model is

very good suggesting that our model assumptions are reasonably well in line with the data. In

Figure C2 we compare the times spent in patches as a function of their area and the number of

patches visited by an individual coming from a patch of a certain area (derived from the median

estimates of the diffusion model parameters). These comparisons show that females are more

mobile than males, and that individuals spend substantially more time in large patches than in small

patches.

LITERATURE CITED

Kuussaari, M. 1998. Biology of the Glanville fritillary butterfly (Melitaea cinxia). PhD thesis,

Department of Ecology and Systematics. University of Helsinki, Helsinki, Finland.

Moilanen, A. 2002. Implications of empirical data quality to metapopulation model parameter

estimation and application. Oikos 96:516–530.

Ovaskainen, O., H. Rekola, E. Meyke, and E. Arjas. 2008. Bayesian methods for analyzing

movements in heterogeneous landscapes from mark-recapture data. Ecology 89:542–554.

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TABLE C1. Mean and standard deviation of the normally distributed marginal prior distributions

(assumed to be independent of each other in the joint prior) for the parameters used in the IBM and

the SPOM, apart from the diffusion model priors which are given in Table C2. The priors are based

on the literature (Kuussaari 1998, Moilanen 2002) and the expert opinions of butterfly biologists in

the Metapopulation Research Group.

Parameter Mean Standard

deviation

logit ( ) 0.944 0.400

e 1.815 0.700

log ( )e 0.000 1.000

0log ( )hn 1.609 0.500

t 1.099 0.500

log ( )t -1.000 0.800

p 0.000 1.000

logit ( )z -2.197 1.000

log ( )p 0.000 1.000

log ( ) -1.609 1.000

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log ( )kq 0.000 1.000

q 3.000 1.000

rq

0.500 0.200

log( )eq

-1.000 1.000

i 0.000 1.000

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TABLE C2. Mean and Variance-Covariance-Correlation matrix for the diffusion model parameters

estimated from the mark-recapture data.

Sex Diffusion

Parameter

Mean and Variance-Covariance-Correlation matrix (VCC)†

Female 1

2log

D

D

m

k

12.476 0.388 0.036 0.031 0.428

4.428 0.186 0.097 0.005 0.010Mean , VCC

2.004 0.264 0.086 0.035 0.054

4.334 0.925 0.044 0.389 0.552

Male 1

2log

D

D

m

k

13.691 0.390 0.012 0.006 0.418

5.449 0.084 0.052 0.003 0.041Mean , VCC

1.877 0.154 0.218 0.004 0.012

7.598 0.961 0.259 0.248 0.485

† The variance and covariance are given in the diagonal and the upper parts of the VCC matrix and

the correlation in the lower parts

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FIG. C1. Model fit for the mark-recapture data (females, first column; males, second column). Large

dots depict the real data and the small dots with error bars show the mean and 95% Bayesian

credibility interval of the posterior predictive distributions. Panels (a) and (b) show the number of

days from first to last recapture; (c) and (d) show the movement distances (in meters) and; (e) and

(f) show the number of recaptures in a given patch.

FIG. C2. A comparison of movement statistics calculated for females (black) and males (gray)

derived from the posteriors median values for the diffusion model parameters: (a) area of patch i in

m2 against iU , the expected number of patches an individual coming from patch i will visit in its

lifetime. Where i ij

j ì

U R

and ijR gives the probability that an individual initially in patch i will

visit patch j in its lifetime and; (b) patch area against iT , the time that an individual presently in

patch i is expected to spend in patch i in its lifetime.

Femalesa

Days from first to last capture

Fre

qu

en

cy

Movement distance (m)

c

# recaptures in a given patch

e

Malesb

Movement distance (m)

d

# recaptures in a given patch

f

Days from first to last capture

4 5 6 7 8 9 10

1

2

3

4

4 6 8 10

-4

-3

-2

-1

0

1

log (area)log (area)

U

log (T

)

a b