Population density of the mite Acarus siro was recorded every 3
days during 28 days. The following densities were found:
165, 145, 139, 125, 105, 101, 88, 81, 73, 69
What is the intrinsic rate of increase (r) and what was the initial
density ?
How long it takes for a population to decrease to half size?
Project population growth for another 5 weeks using estimated r
and N0 = 69.
What would be the estimated rate if you know the initial and
final density?
mite<-c(165, 145, 139, 125, 105, 101, 88, 81, 73, 69)
ti<-c(1,4,7,10,13,16,19,22,25,28)
plot(ti,mite,type="b")
lmi<-log(mite)
plot(ti,lmi)
summary(lm(lmi~ti))
exp(5.132735)
log(0.5)/-0.033217
time<-1:35
Nt<-69*exp(-0.033*time)
plot(time,Nt,type="b")
(log(69)-log(165))/27
Net reproductive rate (R0)
average number of offspring produced by a female in her lifetime
Average generation time (T)
average age of females when they give birth
Expectation of life
age specific expectation of life
o .. oldest age
=
=
n
x
xxmlR
0
0
0
0
R
mxl
T
n
x
xx=
=
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= xx
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ll
L =
o
x
xx LT
x
x
x
l
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e =
Intrinsic growth rate (Intrinsic growth rate (rr ))
when Leslie model show exponential growth the potential rate of
increase can be determined from
Euler (1760) found how to estimate r from the life table
r can be estimated from is the only dominant positive eigenvalue of
the transition matrix (1.. finite growth rate)
 =-
x
rx
xx eml 1
T
R
r
)ln( 0

T
R0

)ln( 1=r
- relative abundance of different life history age/stage/size categories
population approaches stable age distribution:
N0 : N1 : N2 : N3 :...:Ns is stable
- once population reached SCD it grows exponentially
proportion of individuals (c) in age x
w1 .. right eigenvector of the dominant eigenvalue
- provides stable age distribution
- scale w1 by sum of individuals
Stable age distribution (SCD)
 -
-
=
x
rx
x
rx
x
x
el
el
c
 =
= S
i
SCD
1 1
1
w
w
111 wAw =
Reproductive value (RV)
identifies age class that contributes most to the population
growth
measures relative reproductive potential of an individual of a
given age
when population increases then early offspring contribute more
to vx than older ones
v1 .. left eigenvector of the dominant eigenvalue
- provides reproductive values
- v1 is proportional to the reproductive values
scaled to the first category
rx
x
o
x
rx
xx
x
el
eml
v -
-

=
vx
age
1
0
111 vAv =
 =
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i
RV
1 1
1
v
v
Sensitivity (s)
identifies which process (survival or fertility) has largest effect
on the population increase (1)
- examines change in 1 given small change in processes (aij)
- sensitivity is larger for survival of early, and for fertility of older
classes
Elasticity (E)
weighted measure of sensitivity
- measures relative contribution to the population increase
- impossible transitions = 0
v.w
ijij
ij
ij
wv
a
s ==

1
ij
ij
ij
a
a
E



1
1
=
 sum of pairwise products
Perform demographic study of the common fox using life table
menu in POPULUS. The fox breeds in pulses and the data were
collected using pre-breeding census.
Plot standardised survival (lx) with age.
Which survival curve type it corresponds to?
Plot fecundity (mx) and reproductive value (vx) with age.
Construct Leslie transition matrix and project
it over a period of another 20 years using initial vector (10, 12, 6,
5, 4, 2). When does the population reach stable age distribution?
What is R0, T, and r ?
x lx mx
0 1 0.000
1 0.8 0.000
2 0.32 2.000
3 0.12 2.100
4 0.06 2.300
5 0.024 2.400
to adopt means for population promotion or control
Conservation/control procedure
1. Construction of a life table
2. Estimation of the intrinsic rates
3. Sensitivity analysis - helps to decide where
conservation/control efforts should be focused
4. Development and application of management plan
5. Prediction of future
There is a butterfly species that appears to be rare. You perform a
life-history study and gain data on survival and reproduction. You
also observe which factors determine stage-specific survival.
Estimate  using POPULUS and find whether the population
increases or decreases?
Perform sensitivity analysis in POPULUS by replacing each
stage-specific survival by 1 and identify which factor has most
dramatic effect on population increase.
Suggest a conservation plan.
stage px mx mortality
egg 0.7 0 frost in winter
larva 1 0.6 0 parasitoids
larva 2 0.4 0 parasitoids
larva 3 0.5 0 bird predation
larva 4 0.3 0 parasitoids
pupa 0.2 0 habitat destruction
adult 0 80
You observe a population decrease in a duck species. You
perform a life-history study with post-breeding census and find
that duck has birth-pulse breeding. You obtain the following data:
Make simple population projections in POPULUS.
Create transition matrix in R and find stable class distribution
and reproductive values.
Perform sensitivity analysis to identify important processes.
Suggest a conservation plan.
x lx mx mortality
0 1 0 racoons
1 0.2 2 foxes
2 0.1 3 paras ite
3 0.03 5 virus
4 0.002 1 old age
5 0