Stano Pekár“Populační ekologie živočichů“
 dN
= Nr
dt
Spatial ecology - describes changes in spatial pattern over time
 processes - colonisation / immigration and local extinction /
emigration
 local populations are subject to continuous colonisation and
extinction
 wildlife populations are fragmented
Metapopulation - a population consisting of many local
populations (sub-populations) connected by migrating
individuals with discrete breeding opportunities (not patchy
populations)
 population density changes also in space
 for migratory animals (salmon) seasonal movement is the dominant
cause of population change
 movement of individuals between patches can be density-dependent
 distribution of individuals have three basic models:
 most populations in nature are aggregated (clumped)
Regular distribution
 described by hypothetical discrete uniform distribution
n .. is number of samples
x .. is category of counts (0, 1, 2, 3, 4, ...)
 all categories have similar probability
 mean:
 variance:
 for regular distribution:
n
xP
1
)( 
)1(
2
1
 n
)1(
12
1 22
 n
2
 
 described by hypothetical Poisson distribution
 .. is expected value of individuals
x .. is category of counts (0, 1, 2, 3, 4, ...)
 probability of x individuals at a given area usually decreases with x
 observed and expected frequencies are compared using 2 statistics
 for random distribution:
!
)(
x
e
xP
x 
 

Random distribution
2
 
 described by hypothetical negative binomial distribution
 .. is expected value of individuals
x .. is category of counts (0, 1, 2, 3, 4, ...)
k .. degree of clumping, the smaller k (0) the greater degree of clumping
 approximate value of k:
 for aggregated:
Coefficient of dispersion (CD)
CD < 1 … uniform distribution
CD = 1 … random distribution
CD > 1 … aggregated distribution
xk
kkx
xk
k
xP 

















)!1(!
)!1(
1)(



 2
2
k
Aggregated distribution
x
s
CD
2

2
 
• species occupy new areas
• expansion – increase in geographic range of native species
• invasion - increase in geographic range of allien species
Elton 1958
Spread of
muskart in
Europe
~ 5 km/year
• individual makes blind random walk
• random walk of a population undergoes
diffusion in space
• diffusion (Brownian motion) model in
2dimensional space:
where D .. diffusion coefficient
(distance2/time)
- radial distance is related to time
- slope of the line is rate of expansion














2
2
2
2
y
N
x
N
D
t
N
time0
area
N t=0
t=1
t=2
t=3







 

DtDt
N
tN
4
exp
4
),(
2
0 


Pure dispersal
radius
Dt4
t
D
4
2


• Diffusion model
- solved to
2dimensional
Gaussian distribution
- assuming all individuals are dispersers
- range expanses linearly with time
- no reproduction
N0 .. initial density
ρ .. radial distance from point of release (range)
0
N
t=0
t=1
t=2
t=3
Dispersal + population growth
radius
rDc 2
0









Dt
rt
Dt
N
tN
4
exp
4
),(
2
0 


• Skellam‘s model
- includes diffusion
and exponential
population growth
r .. intrinsic rate of increase
c - expansion rate [distance/time]
- for species with R0 < 1.5 ….
Skellam 1951
)ln(2 0R
T
c


Van den Bosch et al. 1990
 types of spatially distributed populations:
- Source-sink (mainland-island), panmictic (clumped), nonequilibrium
(fragmented), metapopulation
 the degree of isolation may vary depending on the distance among
patches
 unlike growth models that focus on population size, metapopulation
models concern persistence of a population - ignore fate of a single subpopulation
and focus on fraction of sub-population sites occupied
Metapopulation assumptions:
- Suitable habitat is in discrete patches
- Population is reproductively active
- Subpopulations have high risk of extinction
- Subpopulations are not too isolated such that
recolonization is likely
- Dynamic are not synchronised across subpopulations
- All patches are alike for extinction and colonization
 Levins (1969) distinguished between dynamics of a single
population and a set of local populations which interact via
individuals moving among populations
Hanski (1997) developed the theory - suggested coresatellite
model
 assumptions
- sub-populations are identical in size, distance, resources, etc.
- extinction and colonisation are independent of p
- many patches are available
- natality and mortality is ignored
eppmp
t
p
 )1(
d
d
Levin‘s model
p .. proportion of patches occupied
m .. colonisation (immigration) rate - proportion of open sites colonised
per unit time
e .. extinction (emigration) rate - proportion of sites that become
unoccupied per unit time
Levin (1969)
Time
p
0.5
0.1
 equilibrium is found for
- sub-populations will persist (p* > 0) only if colonisation is larger
than extinction (m > e)
- Equilibrium state will always include empty patches if if e > 0
- K ..is fraction of patches
- defined by balance between m and e
 metapopulations are found in
insects, Daphnia, frogs, birds
rescue-effect model,
incidence model
m
e
m
em
p 

 1*
K
0
d
d

t
p