Wolf populations in North America

Black bear distribution:

Scouler’s willow distribution:

A landscapes mosaic:

Only some pieces of the mosaic are suitable for a given species:

The populations in the landscape mosaic are not totally isolated,But only weakly connected by migrating individuals:

A population: a group of individuals of the same species that live and breed in the same space.

A metapopulation: a group of several local populations connected by the occasional movement of individuals between populations (immigration and emigration).

A deme: a population that is part of a metapopulation.

MetapopulationDynamics: The dynamics of patch occupancy.

Local extinction: a deme goes extinct.

Colonization: an empty but suitable habitat is repopulated by emigrants.

Definitions

How long can a deme persist without immigration?

Example: probability of local extinction is 1 in 6:

pe = 1/6 = 0.1667

Probability of persisting the first year is 5 in 6:

P(t=1) = 1-1/6 = 0.8333

Probability of persisting two consecutive years:

P(t=2) = (1-1/6)*(1-1/6) = 0.6944

Probability of persisting n consecutive years:

P(t=n) = (1-1/6)n

How long can a deme persist without immigration?

Without immigration, all demes eventually go extinct, and sooner, the higher the annual extinction probability.

0

0.2

0.4

0.6

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1.2

0 5 10 15 20 25 30

0.9

0.8

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Probability of deme persistence

time (years)

pe

0.2

0.5

0.8

0.9

How long can a deme persist without immigration?

The more demes the smaller the chance of regional extinction:

(1/6)4 = 0.00077

Probability of simultaneous extinction in 4 patches:

P4 = 1-(1/6)4

Probability of persistence over 4 patches:

Pm = 1-(pe)m

Probability of persistence over m patches:

Time (years)

Without immigration, metapopulations can go also extinct, but it takes a lot longer. Many demes lower the risk of regional extinction.

Regional extinction is a lot less likely than local extinction.

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0 0.2 0.4 0.6 0.8 1

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Probability of regional persistence

no ofpatches

pe (probability of local extinction)

An example of a metapopulation:

The endangered bay checkerspot butterfly:

It’s host plant: Plantago erecta(caterpillar food)

Serpentine grasslands in Northern California

The distribution of serpentine grassland in Santa Clara County

(Harrison et al. 1988)

A severe drought in 1975-1977 caused several local extinctions.

Some empty patches were recolonized in 1986.

The Island-Mainland model:

The probability of immigration is constant.

mainlandislands

Island-mainland model: a constant “propagule rain” originating on the mainland.

How do we characterize the dynamics of metapopulations?

1) We only ask whether patches are occupied or not:

16 patches 4 occupied 12 empty

2) The state variable we follow is the fraction of occupied patches f.

f = 4/16 = 0.25

The relevant rates in metapopulation dynamics:

Probability of local extinction pe

the probability that in a given amount of time a local population will go extinct.

Probability of local colonization pi:

the probability that in a given amount of time, a site will be colonized.

pe

pi

fpe *

)1(* fpi

The rate of occupancy loss:

)1(* fpi The rate of re-colonization:

fpe *

The rate of change in f: fpfpdt

dfei *)1(

0

0.05

0.1

0.15

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0.25

0 0.2 0.4 0.6 0.8 1

I

E

Fraction of occupied sites f

Ext

inct

ion

or

Imm

igra

tion

ra

te

pe

ei

i

pp

pf

ˆ

Island-Mainland Model:

fpfpdt

dfei )1(

If pi > 0 there is always a positive equilibrium: metapopulations always persist.

The Internal Colonization model:

The probability of immigration depends on the patches occupied:

islands

ifpi The immigration probability is proportional to the occupied patches:

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6 0.8 1

i = 0.4

i = 0.2

E

ip

f e1ˆ

Internal colonization model:

fpfifdtdf

e )1(

Metapopulations may not persist. Positive equilibrium only if pe < i.

Excel Worksheets:

• Metapopulation dynamics

Summary:

Metapopulations are collection of populations (demes) linked by immigration.

Typically, not all patches of a metapopulation are occupied. Average occupancy depends on extinction risks (pe) and immigration probabilities (pi).

The mainland-island model assumes that pi is constant. Metapopulations cannot go regionally extinct and there always is an equilibrium with non-zero patch occupancy.

The internal colonization model assumes that pi =if. Metapopulations can go regionally extinct if pe > i.

Both models assume:

• All sites are exactly the same. • Extinction and colonization probabilities do not change over time.• Local extinctions and colonizations are independent events. • The spatial arrangement of the sites does not matter.• Many patches (ignoring chance fluctuations in patch occupancy).