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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
0.8
1
1.2
0 5 10 15 20 25 30
0.9
0.8
0.5
0.2
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.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
1
5
10
15
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)
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
0.2
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.
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).