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 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
+= nµ
)1(
12
1 22
−= nσ
2
σµ >
described by hypothetical Poisson distribution
µ .. 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 χ2 statistics
for random distribution:
!
)(
x
e
xP
x µ
µ −
=
Random distribution
2
σµ =
described by hypothetical negative binomial distribution
µ .. 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 





+−
−+






−=
−
µ
µµ
)!1(!
)!1(
1)(
µσ
µ
−
≈ 2
2
k
Aggregated distribution
x
s
CD
2
=
2
σµ <
• Geographic range - radius of space containing 95% of
individuals
• individual makes blind random walk
• random walk of a population undergoes diffusion in space
- radial distance moved in a random walk
is proportional to
- area occupied (radius2)
is proportional to time
Spread of muskart in Europe
time
Elton 1958
N t=0
t=1
t=2
t=3







 −
=
DtDt
N
tN
4
exp
4
),(
2
0 ρ
π
ρ
Pure dispersal
radius
Dt4=ρ
t
D
4
2
ρ
=
• Difussion model
- solved to
2dimensional
Gaussian distribution
N0- initial density
ρ .. radial distance from point of release (range)
D - diffusion coefficient (distance2/time)
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 ρ
π
ρ
• Skellam‘s model
- added exponential
population growth
r .. intrinsic rate of increase
c - expansion rate [distance/time]
Skellam 1951
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 core-satellite model
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
m ..proportion of open sites colonised per unit time
e ..proportion of sites that become unoccupied per unit time
eppmp
dt
dp
−−= )1(
Levin‘s model
p .. proportion of patches occupied
m .. colonisation rate
e .. extinction rate
Levin 1969
Time
p
0.5
0.1
equilibrium is found for dp/dt = 0
- sub-populations will persist (p* > 0) only if colonisation is
larger than extinction
- all patches can be occupied only if e = 0
- K ..is fraction of patches
- defined by balance between m and e
m
e
m
em
p −=
−
= 1*
K
In a field the abundance of spiders on leaves was studied.
The following counts per leaf were made:
1. What is the distribution of spiders per leaf and per plant?
2. If aggregated, what is the coefficient of dispersion (CD) and the
degree of aggregation (k)?
Plant Counts
1 0, 0, 1, 5, 7
2 0, 1, 1, 4, 1
3 0, 0, 2, 0, 0
4 3, 1, 8, 1, 1
5 1, 2, 6, 3, 2
spider<-c(0,0,1,5,7,0,1,1,4,1,0,0,2,3,0,0,1,6,1,1,1,2,6,3,2)
table(spider)
CD1<-var(spider)/mean(spider); CD1
k1<-mean(spider)^2/(var(spider)-mean(spider)); k1
plant<-c(rep(1,5),rep(2,5),rep(3,5),rep(4,5),rep(5,5))
a<-tapply(spider,plant,mean)
CD2<-var(a)/mean(a); CD2
A dragonfly is spreading along a river. The spreading is anisotropic faster
down the stream than up the stream. During 6 years the dragonfly
has spread as follows:
1. Estimate D in both directions.
2. Estimate expansion rate in both directions if finite growth rate λ = 1.4.
2. Model the spread using Skellam’s model.
Rok
po proudu proti proudu
0 0 0
1 3 0.2
2 7 0.5
3 13 1
4 17 1.4
5 26 1.8
6 30 2.2
Plocha [km2]
year<-0:6
po<-c(0,3,7,13,17,26,30)
rho1<-sqrt(po)
plot(year,rho1)
m1<-lm(rho1~year-1)
abline(m1)
m1
pro<-c(0,0.2,0.5,1,1.4,1.8,2.2)
rho2<-sqrt(pro)
plot(year,rho2)
m2<-lm(rho2~year-1)
abline(m2)
m2
r<-0:50
y<-10*exp(0.34*1-r^2/(4*0.25*1))/(4*pi*0.25*1)
plot(r,y,type="l")
y<-10*exp(0.34*10-r^2/(4*0.25*10))/(4*pi*0.25*10);lines(r,y)
y<-10*exp(0.34*20-r^2/(4*0.38*20))/(4*pi*0.38*20);lines(r,y)
y<-10*exp(0.34*30-r^2/(4*0.38*30))/(4*pi*0.38*30);lines(r,y)
y<-10*exp(0.34*40-r^2/(4*0.38*40))/(4*pi*0.38*40);lines(r,y)
y<-10*exp(0.34*50-r^2/(4*0.38*50))/(4*pi*0.38*50);lines(r,y)
A population of toads has been split into two sub-populations by a
new highway. One has 100 and the other 10 individuals. The first
one has exploited its resources so their finite rate of population
increase (λ1) is 0.8. The other has a lot of resources, therefore their
λ2 = 1.2. Is it necessary to built a corridor connecting populations?
If so how large it should be in terms of the rate of exchange (d)
between sub-populations.
1. Use discrete density-independent models to simulate fate of
populations for 20 years that are completely isolated (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 NddNN ,21,111,1 )1( λλ +−=+ ttt NddNN ,12,221,2 )1( λλ +−=+
N12<-data.frame(N1<-numeric(1:20),N2<-numeric(1:20))
N12[,1]<-100
N12[,2]<-10
d=0
for(t in 1:20) N12[t+1,]<-{
N1<-0.8*((1-d)*N12[t,1]+0.8*d*N12[t,2])
N2<-1.2*((1-d)*N12[t,2]+1.2*d*N12[t,1])
c(N1,N2)}
matplot(N12, type="l",lty=1:2)
legend(1,200,c("N1","N2"),lty=1:2)
d=0.2
for(t in 1:20) N12[t+1,]<-{
N1<-0.8*((1-d)*N12[t,1]+0.8*d*N12[t,2])
N2<-1.2*((1-d)*N12[t,2]+1.2*d*N12[t,1])
c(N1,N2)}
matplot(N12, type="l",lty=1:2)
legend(1,150,c("N1","N2"),lty=1:2)
d=0.4
for(t in 1:20) N12[t+1,]<-{
N1<-0.8*((1-d)*N12[t,1]+0.8*d*N12[t,2])
N2<-1.2*((1-d)*N12[t,2]+1.2*d*N12[t,1])
c(N1,N2)}
matplot(N12, type="l",lty=1:2)
legend(15,100,c("N1","N2"),lty=1:2)