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Metapopulations. All finite populations are vulnerable to extinction. Demographic and environmental stochasticity. Frequency. r. Even populations with an have some probability of going extinct!. The probability of extinction, p e , depends on…. The current population size

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slide2

All finite populations are vulnerable to extinction

  • Demographic and environmental stochasticity

Frequency

r

  • Even populations with an have some probability of going extinct!
slide3

The probability of extinction, pe, depends on…

  • The current population size
  • The probability with which individuals die (d) and give birth (b)
slide4

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!
slide5

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!
slide6

Adding multiple populations

N=30

N=30

N=30

N=30

N=30

N=30

N=30

N=30

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

slide7

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!
slide8

Metapopulations

N=30

N=30

N=30

N=30

  • Metapopulation – A population of populations (Levins, 1970)
  • How does movement between populations shape the dynamics of extinction and recolonization?
slide9

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?

slide10

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
slide11

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
slide12

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
slide13

An example of the mainland-island scenario

(The Bay Checkerspot Butterfly)

Euphydryas editha

Plantago erecta

Lives primarily on serpentine soils

slide14

An example of the mainland-island scenario

(The Bay Checkerspot Butterfly)

Euphydryas editha

slide15

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?
slide16

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!

slide17

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.
slide18

An example of internal colonization

(The Glanville Fritillary on the Aland Islands)

Glanville fritillary

(Melitaea cinxia)

Plantago lanceolata

Veronica spicata

slide19

An example of internal colonization

(The Glanville Fritillary on the Aland Islands)

F

N

S

Glanville fritillary

(Melitaea cinxia)

slide20

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?
slide21

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
slide22

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

0

1

f

  • We can consider this case by setting pe = e(1-f)
slide23

The rescue effect in the mainland-island scenario

  • Adding the rescue effect to the mainland-island model:
  • What is the equilibrium of this model?
slide24

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 > 0
slide25

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?
slide26

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?

slide27

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

2.5

5

1

Patch size (km2)

slide28

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
slide29

Exam 3 Results

Average: 149 points (75%)

# Students

Score (in points)