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Equations
Defining
Metapopulations
The variablesTime: best measured in generations but most convenient for us
to measure time in years. Tm=1/emin
= probability of local Extinction
= probability of local Persistence
= probability of local colonization
= number of patches
= probability of regional persistence
= fraction of sites occupied
= effect of increasing patch occupancy
= intensity of rescue
= effective (breeding) population size
!
pe
!
x
!
pc
!
1" pe
!
f
!
i
!
Px
!
r
eN
!
df
dt= pc 1" f( ) " pe f
!
df
dt= pc 1" f( ) " e 1" f( ) f
!
df
dt= if 1" f( ) " pe f
!
df
dt= if 1" f( ) " e 1" f( ) f
Extinction
Independent Rescue
External
InternalColo
niz
ation
Note that is analogous to
Levins (1969) equation was basically
!
df
dt= cf 1" f( ) " ef
!
C " E # $
!
G = B "D
Bottom line, , and there are four typical models
for estimating C and E:EC
dt
df!=
The method
Persistence of one patch over time
Persistence of one patch over two time periods is:
Persistence of one patch over n time periods is:
Persistence of two patches over time is:
Persistence of many patches over time is:
!
1" pe
!
1" pe( ) 1" pe( ) = 1" pe( )2
!
1" pe( )n
!
1" pe1pe2
!
Px =1" pe( )x
Assumptions
Patches are homogenous in size, distance from
each other, habitat quality, food, CC
All patches have same and over all time
periods
and are independent of patch occupation
Instantaneous response to !
No diffusion effect and no spatial structure!
pe
!
pc
!
pe
!
pc
= fraction of occupied patches
= fraction of unoccupied patches
rate of colonization in one time period thruimmigration. We use it as though it were aprobability. C is dependent on patch suitability(area, critical habitat, food, predators,competitors, disease, distance from otheroccupied patches) & proportion of unoccupiedpatches.
rate of extinction in one time period (we must useas though it’s a probability).
!
f
!
1" f( )
!
C = pc 1" f( )
!
E = pe f
( ) fpfp ecdt
df!!= 1
One External Source (Propagule Rain)
A source that is outside the metapopulation
is constant
If stable, then solve equation for zero0=dt
df
!
0 = pc 1" f( ) " pe f
!
ˆ f =pc
pc + pe( )
!
pc
Multiple Internal Sources
Each occupied internal site produces an excess of
propagules that can colonize unoccupied patches
= effect of increasing patch occupancy
because depends only on patch occp’ncy
If stable, then solve equation for zero
!
0 = if 1" f( ) " pe f
!
ˆ f =1"pe
i
!
i
!
pc = if
!
C
0=dt
df
Rescue
If propagules land in occupied sites, they can "Ne
which # pe. If more sites are occupied then more
propagules will be available for rescue
r = combination of Ne and migration rate
because E depends on breeding pop’n
If stable, then solve equation for zero
!
0 = pc 1" f( ) " r 1" f( ) f!
pe = r 1" f( )
!
ˆ f =pc
r
0=dt
df
Closed
Propagules arise only from w/in the metapop’n
& patches rescue each other
If stable, then solve equation for zero
Oops, can’t solve for f so we must weigh possible
results based on likely values of i and r. Barbour &
Pugliese ’05 show that there are thresholds, below
which all solutions indicate total extinction of the
metapop’n. Thus, in the end, most closed
metapopulations will expire without a stabilizing
influence from outside.
!
0 = if 1" f( ) " r 1" f( ) f
0=dt
df
Making models realistic
All metapop’n models begin with thesefundamental equations and then addprocedures for modeling the variables andfactors affecting the variables.
= per capita birth rate
= per capita death rate
= P of catastrophic destruction of a patch
= P of migrant making it to a patch
= lacunarity (index of l’scape texture)
= enemy-victim relationship
b
d
!
!
( )x!
ijµ
