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
Two approaches:
landscape ecology - focus on communities at large
geographic scale
metapopulation ecology - focus on metapopulations
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
regular distribution is described by hypothetical uniform distribution
n .. is number of samples
x .. is category of counts (0, 1, 2, 3, 4, ...)
all samples have similar probability
mean:
variance:
for regular distribution:
n
xP
1
)( =
)1(
2
1
+= n
)1(
12
1 22
-= n
2
 >
random distribution is 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
 =
aggregated distribution is 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:
exact value is estimated iteratively
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
 <
first model developed by MacArthur and Wilson (1967)
the theory:
- does not predict stable populations
- there is ongoing colonisation and stochastic extinction
- species composition is continually changing but the total number of
species is constant
- colonisation-extinction equilibrium
- mainland serves as continuous source of migrants
mainland
larger islands/areas support more species
Darlington's rule = tenfold increase in area result in
double increase of species
S .. number of species
A .. island area
c, z .. constants
larger area contain
more habitat types
larger habitats support
larger populations
(lower probability of extinction) 1 ln(area[ha])
ln(speciesnumber)
10
10000
Specie-area relationship for breeding
land-birds of the West Indies
Gotelli & Abele (1982)
y = 0.94 + 0.11x
Species-area relationship
)ln()ln()ln( AzcS +=
z
cAS =
slope of the regression line for islands ranges between 0.24-0.34 for
ants, birds, plants, etc.
species diversity is a function of sampled size and habitat diversity
slope of linear regression ranges for inland between 0.12-0.17
there is exchange of organisms with surrounding areas, not isolated
as islands
Number of plant species found in sampled areas
of England
0
200
400
600
800
1000
1200
1400
1 10 100 1000 10000 100000
sample area
no.ofspecies
Number of amphibians & reptiles on West
Indies islands
MacArthur & Wilson (1967)
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
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
assumptions
- sub-populations are identical in size, distance, resources, etc.
- extinction and colonisation are independent of p
- many patches are available
equilibrium is found for dp/dt = 0
- sub-populations will persist only if
colonisation is larger than extinction
- all patches can not be occupied
eppmp
dt
dp
--= )1(
m
e
m
em
p -=
=
1*
Levins model
p .. proportion of patches occupied
m .. colonisation rate
e .. extinction rate
Time
p
0.5
0.1
)1()1( peppcp
dt
dp
---=
)1()(0 ppec --=
Hanski model - ,,core-satellite"
p .. proportion of patches occupied
c .. external colonisation rate
e .. extinction rate
e decreases with increasing p .. ep(1-p)
- rescue effect - re-colonisation of sites on verge of extinction
equilibrium is found for dp/dt = 0
- if c > e .. all sites will be occupied - stable
equilibrium
- if c < e .. extinction - stable equilibrium
- if c = e .. stable equilibrium
Time
p
1
0
In a field the abundance of spiders was studied by means
of 20 pitfall traps. The following counts were recorded:
0, 0, 1, 5, 7, 0, 1, 1, 4, 1, 0, 0, 2, 0, 0, 3, 1, 8, 1, 1
1. Is the distribution of spiders in the field random or aggregated?
2. If aggregated, what is the coefficient of dispersion (CD) and the
degree of aggregation (k)?
=
-
=
I
i
i
x
xx
1
2
2 )(
ab<-c(0, 0, 1, 5, 7, 0, 1, 1, 4, 1, 0, 0, 2, 0, 0, 3, 1, 8, 1, 1)
table(ab)
hist(ab)
mean(ab)
chi<-sum((ab-1.8)^2/1.8);chi
1-pchisq(60.67,19)
chisq.test(ab)
var(ab)
CD<-var(ab)/mean(ab); CD
k<-mean(ab)^2/(var(ab)-mean(ab)); k
library(MASS)
fitdistr(ab,"negative binomial")
A new protected area, 48 700 ha large, is going to be established. You
need to know expected number of mammal species that occur in this
new area before species inventory will be performed.
1. Use information from surrounding areas to estimate the expected
species richness.
2. The minimum number of species that a protected area must have is
90. How large you expect the new protected area must be?
Area [ha] Species
224 24
740 39
1720 45
4304 56
10806 71
34931 80
area<-c(224, 740, 1720, 4304, 10806, 34931)
spec<-c(24, 39, 45, 56, 71, 80)
plot(area,spec)
plot(log(area),log(spec))
lm(log(spec)~log(area))
exp(2.0211+0.2352*log(48700))
There are two sub-populations of deer. One has 100 and the other 5
individuals. The first one has exploited its resources so their finite
rate of population increase (1) is 0.8. The other has a lot of
resources, therefore their 2 = 1.2. There is a possible rate of
exchange (d) between sub-populations.
1. Use discrete density-independent models to simulate fate of
populations that are not connected, i.e. d = 0.
2. Simulate the dynamics of the two sub-populations for 20 years
with various levels of exchange, d = 0.1 to 1.
ttt dNdNN ,2,111,1 )1( +-=+ 
ttt dNdNN ,1,221,2 )1( +-=+ 
N12<-data.frame(N1<-numeric(1:20),N2<-numeric(1:20))
N12[,1]<-100
N12[,2]<-5
d=0
for(t in 1:20) N12[t+1,]<-{
N1<-0.8*((1-d)*N12[t,1]+d*N12[t,2])
N2<-1.2*((1-d)*N12[t,2]+d*N12[t,1])
c(N1,N2)}
matplot(N12, type="l",lty=1:2)
d=0.2
for(t in 1:20) N12[t+1,]<-{
N1<-0.8*((1-d)*N12[t,1]+d*N12[t,2])
N2<-1.2*((1-d)*N12[t,2]+d*N12[t,1])
c(N1,N2)}
matplot(N12, type="l",lty=1:2)
d=0.8
for(t in 1:20) N12[t+1,]<-{
N1<-0.8*((1-d)*N12[t,1]+d*N12[t,2])
N2<-1.2*((1-d)*N12[t,2]+d*N12[t,1])
c(N1,N2)}
matplot(N12, type="l",lty=1:2)