Modelling complex migration

Michael Bode

Migration in metapopulations

Metapopulation dynamics are defined by the balance between local extinction and

recolonisation.

Overview

1. Metapopulation migration needs to be modelled as a complex and heterogeneous process.

2. We can understand metapopulation dynamics by direct analysis of the migration structure using

network theory.

Different migration models

1. Time invariant models.

2. Well-mixed migration.

3. Distance-based migration.

4. Complex migration.

Time invariant models

• Re-colonisation probability is constant

• Probability of metapopulation extinction is underestimated.

Well-mixed migration (the LPER assumption)

• All patches are equally connected.

• The resulting metapopulation is very homogeneous

Distance-based migration(The “spatially real” metapopulation)

• Migration strengths are defined by inter-patch distance.

• The result is symmetric migration,

where every patch is connected.

)),(()()( jidfijmjim

Will complex migration patterns really affect metapopulation persistence?

Pr(

Ext

inct

ion)

Amount of migration

Both metapopulation (a) and (b) have the

• same total migration

• same number of migration pathways

Only the migration pattern is different

Complex migration

1. Metapopulations can be considered networks

2. We can directly analyse the structure of the metapopulations to determine their dynamics

3. Using these methods we can rapidly analyse very large metapopulations

Network metrics

How can we characterise a migration pattern?

• Clustered/Isolated?

• Asymmetry?

Determining the importance of network metrics

Construct a complex migration pattern

Use Markov transition metrics to determine the

probability of metapopulation persistence

Calculate the network metrics

Do the metrics predict metapopulation

dynamics?

Predicting metapopulation extinction probability

• Average Path Length ( )

• Asymmetry of the metapopulation migration (Z)

(Where M is the migration matrix)

N

i

N

jijLN

L1 1

2

1L

TMMMZ 2

1

Asymmetry (Z)

Symmetric

Asymmetric

Predicting metapopulation extinction probability

Predicting incidence using patch centrality

• Ci = (shortest paths to i)

0.3

0.8

0.4

0.4

84.1

8.03.08.04.04.0

Predicting patch incidence using Centrality

Bars indicate 95% CI

Implications: patch removal

Centrality of patch removedHighLow

Single patch removed

Pro

babi

lity

of r

emai

ning

met

apop

ulat

ion

extin

ctio

n

Implications: sequential patch removalP

roba

bilit

y of

rem

aini

ng m

etap

opul

atio

n ex

tinct

ion

Number of patches removed

32 41

Average strategy

Unperturbed metapopulation

Single strategy

Removal by Centrality

Limitations and extensions

• Lack of logical framework.

• Incorporating differing patch sizes.

• Modelling abundances.

Simulating metapopulation migration patterns

Regular Lattice Complex network