Metapopulations

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Metapopulations
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All finite populations are vulnerable to
extinction

• Demographic and environmental stochasticity

Frequency
r

• Even populations with an have some
  probability of going extinct!

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The probability of extinction, pe, depends on

• The current population size
• The probability with which individuals die (d)
  and give birth (b)

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Small populations are particularly vulnerable

• For instance, imagine an annual population where
  the probability of giving birth, b, is .1

b .1 d 1
pe()
N

• If the average of offspring produced per
  successful birth is, say 11, the population
  growth rate would be strongly positive
• Yet because of stochasticity, the population
  would still have a very high probability of
  extinction when small!

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Can we extend this result over time?

• Imagine an endangered population of annual plant
  with
• N 30
• b .1
• d 1

• This population has a probability of extinction,
  pe (1-.1)(1)30 .042, each generation
• As a consequence, if the population size does
  not change (e.g., clutch size is 10), the
  probability that this population survives for t
  years is (1-pe)t (.958)t

Probability of surviving to time, t
Time, t

• This single isolated population is doomed to a
  fairly rapid extinction!

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Adding multiple populations
N30
N30
N30
N30
N30
N30
N30
N30
How does adding multiple populations alter the
probability of species extinction?
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Adding multiple populations

• Imagine the plant we considered before now has n
  populations

• The probability that all n populations go
  extinct in a single generation is, pen
• This probability is always lower than the
  probability of a single population going extinct

Probability of species extinction
Number of populations, n

• Thus having multiple populations buffers a
  species from extinction!

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Metapopulations
N30
N30
N30
N30

• Metapopulation A population of populations
  (Levins, 1970)
• How does movement between populations shape the
  dynamics of extinction and recolonization?

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A general metapopulation model

• Follow the fraction of occupied sites, f
• Assume that unoccupied sites are colonized at
  rate, I
• Assume that occupied sites go extinct at rate, E
• Then the fraction of occupied sites changes at
  rate

But what are I and E?
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A general metapopulation model

• Assume that I pi(1-f)
• - Empty patches are colonized at a rate
  proportional to the product of the probability
  of local colonization, pi, and the proportion of
  unoccupied patches, 1-f

• Assume that E pe(f)
• - Occupied patches go extinct at a rate
  proportional to the product of the probability
  of local extinction, pe, and the proportion of
  occupied patches, f

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A general metapopulation model

• With these assumptions, we can predict the
  change in patch occupancy, f

• This general model can be used to consider
  several specific scenarios

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The mainland-island scenario

• Immigrants arrive at a constant rate, pi , from
  a mainland source population
• Island populations go extinct at a constant
  rate, pe

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An example of the mainland-island scenario (The
Bay Checkerspot Butterfly)
Euphydryas editha
Plantago erecta Lives primarily on serpentine
soils
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An example of the mainland-island scenario (The
Bay Checkerspot Butterfly)
Euphydryas editha
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The mainland-island scenario

• Using these assumptions, we can predict the
  change in patch occupancy, f

• What is the equilibrium proportion of occupied
  patches for this model?

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The mainland-island scenario
pi .5
Fraction occupied, f
pi .1
pi .02
Probability of extinction, pe
A small rate of immigration can result in a high
proportion of occupied patches. As long as there
is some immigration, persistence of the
metapopulation is assured!
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The internal colonization scenario

• Immigrants come only from those sites which are
  currently occupied
• As a consequence, pi i f .
• Here i measures how sensitive immigration is to
  the proportion of occupied patches, f.

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An example of internal colonization (The
Glanville Fritillary on the Aland Islands)
Glanville fritillary (Melitaea cinxia)
Plantago lanceolata
Veronica spicata
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An example of internal colonization (The
Glanville Fritillary on the Aland Islands)
F
N
S
Glanville fritillary (Melitaea cinxia)
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The internal colonization scenario

• Using these assumptions, we can predict the
  change in patch occupancy, f

• What is the equilibrium proportion of occupied
  patches in this model?

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The internal colonization scenario
Remember pi i f ? The frequency of immigration
increases with increasing i
Fraction occupied, f
i .4
i .2
i .1
Probability of extinction, pe

• Internal colonization is less effective at
  maintaining the metapopulation than is external
  colonization. Persistence is not assured.
• Nevertheless, internal colonization greatly
  facilitates metapopulation persistence

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The rescue effect

• To this point we have assumed that the
  probability of extinction is independent of the
  fraction of occupied patches
• If, however, immigration increases the
  population size of existing populations, the
  probability of extinction should decrease as a
  function of the proportion of occupied patches
  (e.g., due to decreased demographic
  stochasticity)

1
e is the maximum extinction probability
e
pe
0
1
0
f

• We can consider this case by setting pe e(1-f)

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The rescue effect in the mainland-island scenario

• Adding the rescue effect to the mainland-island
  model

• What is the equilibrium of this model?

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The rescue effect in the mainland-island scenario
pi .4
pi .2
Fraction occupied, f
pi .1
All approach pi
Maximum probability of extinction, e

• The rescue effect facilitates metapopulation
  persistence
• Persistence is assured as long as pi gt 0

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Why do metapopulations matter? (a hypothetical
conservation scenario and practice problem)

• A 20km2 patch of native prairie exists
• A native species of bird occurs exclusively in
  this native prairie habitat
• This species can readily disperse
• Only 5km2 can be excluded from development
• How should this 5km2 be partitioned?

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Possibilities for partitioning (a practice
problem)
Five 1km2 preserves
Two 2.5km2 preserves
How could we make a scientifically informed
decision about which is preferable?
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Available data for the bird species (a practice
problem)

• Previous work has shown that the internal
  colonization rate of this species is pi if
  .3f
• What area do we expect the species to occupy at
  equilibrium with 2 reserves of size 2.5km2?
• What area do we expect the species to occupy at
  equilibrium with 5 reserves of size 1km2?

Probability of extinction (per year)
.3
.2
0
1
5
2.5
Patch size (km2)
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Conclusions from metapopulations

• Multiple populations spread the risk of
  extinction
• In some cases, multiple small populations have a
  larger probability of survival
• than few large populations
• Dispersal can promote the persistence of
  metapopulations