1. Equations
Defining
Metapopulations

2. The variables
Time: best measured in generations but most convenient for us
to measure time in years. Tm=1/emin
pe = probability of local Extinction
1" pe = probability of local Persistence
pc = probability of local colonization
! x = number of patches
Px = probability of regional persistence
f = fraction of sites occupied
! i = effect of increasing patch occupancy
r = intensity of rescue
N e = effective (breeding) population size
!

3. df
Levins (1969) equation was basically dt
= cf (1" f ) " ef
Note that C " E # $ is analogous to G = B " D
Bottom line, df = C ! E , and !
dt
there are four typical models
for estimating C and E:
! !
Extinction
Independent Rescue
Colonization
df df
External dt
= pc (1" f ) " pe f dt
= pc (1" f ) " e(1" f ) f
Internal df
= if (1" f ) " pe f df
= if (1" f ) " e(1" f ) f
! dt ! dt
! !

4. The method
Persistence of one patch over time
1" pe
Persistence of one patch over two time periods is:
2
(1" pe )(1" pe ) = (1" pe )
Persistence of one patch over n time periods is:
! n
(1" pe )
Persistence of two patches over time is:
!
1" pe1 pe 2
Persistence of many patches over time is:
! x
Px = 1" ( pe )
!

5. Assumptions
Patches are homogenous in size, distance from
each other, habitat quality, food, CC
All patches have same pc and pe over all time
periods
pc and pe are independent of patch occupation
! !
Instantaneous response to !
No diffusion effect and no spatial structure
!

6. f = fraction of occupied patches
(1" f ) = fraction of unoccupied patches
C = pc (1" f )
! rate of colonization in one time period thru
immigration. We use it as though it were a
probability. C is dependent on patch suitability
(area, critical habitat, food, predators,
competitors, disease, distance from other
occupied patches) & proportion of unoccupied
patches.
E = pe f
rate of extinction in one time period (we must use
as though it’s a probability).
df
dt = pc ( ! f )! pe f
1

7. One External Source (Propagule Rain)
A source that is outside the metapopulation
pc is constant
df
If stable, dt = 0 then solve equation for zero
0 = pc (1" f ) " pe f
ˆ pc
f =
( pc + pe )
!

8. Multiple Internal Sources
Each occupied internal site produces an excess of
propagules that can colonize unoccupied patches
i = effect of increasing patch occupancy
pc = if because C depends only on patch occp’ncy
! If stable, df
dt = 0 then solve equation for zero
!
0 = if (1" f ) " pe f
ˆ = 1" pe
f
i
!

9. 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
pe = r(1" f ) because E depends on breeding pop’n
df
If stable, dt = 0 then solve equation for zero
0 = pc (1" f ) " r(1" f ) f
ˆ = pc
f
r
!

10. Closed
Propagules arise only from w/in the metapop’n
& patches rescue each other
df
If stable, dt = 0 then solve equation for zero
0 = if (1" f ) " r(1" f ) f
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.

11. Making models realistic
All metapop’n models begin with these
fundamental equations and then add
procedures for modeling the variables and
factors affecting the variables.
b = per capita birth rate
d = per capita death rate
! = P of catastrophic destruction of a patch
! = P of migrant making it to a patch
! (x )= lacunarity (index of l’scape texture)
µij = enemy-victim relationship