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
A<-matrix(c(0.4,1.5,1.5,0.07,0,
0.2,0,0,0,0,
0,0.5,0,0,0,
0,0,0.3,0,0,
0,0,0,0.07,0),nrow=5,byrow=T);A
L<-eigen(A)
L1<-max(Re(L$values))
w<-Re(L$vectors[,1]);w
scd<-w/sum(w);scd
M<-eigen(t(A));M
v<-Re(M$vectors[,1]);v
RV<-v/v[1];RV
s<-v%*%t(w)
ss<-s/as.numeric(v%*%w);ss
e<-ss*(A/L1);e
A mouse species has spread dramatically. You perform a lifehistory
study and find that it breeds continuously. So you
distinguish age classes based upon 3-months intervals. You obtain
the following data:
Estimate R0 and T.
Predict how the population size would change in another 10
years using initial population structure (2, 30, 10, 5) using
transition matrix from POPULUS.
How will r change if you subsequently halve each age-specific
survival?
Suggest a management plan to control mouse population.
x lx mx
0 1 0
1 0.8 5
2 0.5 12
3 0.3 4
x<-c(0,1,2,3)
lx<-c(1,0.8,0.5,0.3)
mx<-c(0,5,12,4)
R0<-sum(lx*mx);R0
T<-sum(x*lx*mx)/R0;T
A<-matrix(c(6.15,6.51,1.8,0,
0.72,0,0,0,
0,0.62,0,0,
0,0,0.37,0),nrow=4,byrow=T);A
L<-eigen(A); r<-log(max(Re(L$values))); r
N0<-c(2,30,10,5)
N1<-A%*%N0;N1
years<-10
Nt<-matrix(0,nrow=nrow(A),ncol=years+1)
Nt[,1]<-N0
for(i in 1:years) Nt[,i+1]<-A%*%Nt[,i]
matplot(0:years,t(Nt),type="l")
legend(2,4e+9,c(1:4),lty=1:4,col=1:4)
A1<-replace(A,c(1,2),c(3.07,0.36))
L1<-eigen(A1); log(max(Re(L1$values)))
A2<-replace(A,c(5,7),c(3.25,0.31))
L2<-eigen(A2); log(max(Re(L2$values)))
A3<-replace(A,c(9,12),c(0.9,0.18))
L3<-eigen(A3); log(max(Re(L3$values)))
Stano Pekár,,Populační ekologie živočichů"
dN
= Nr
dt
model is based on the assumption that development rate is a linear
function of temperature
valid for the region of moderate temperatures (15-25°)
at low temperatures organisms die due to coldness, and at high
temperatures organisms die due to overheating
T .. development time (days)
v .. rate of development = 1/ T
tmin .. lower temperature limit
.. temperature at which
development rate = 0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40
Temperature
Developmentrate
ET.. effective temperature .. developmental temperature = t - tmin
S .. degree-days .. number of days required to complete development
.. do not depend on temperature = T*ET
tmin and S can be estimated from the regression line of v = a + b
tmin :
S :
b
a
t -=min






+=-=
b
a
tTttTS )( min
btav
T
+
==
11
bta
b
at
S
+
+
=
b
S
1
=
0min =+ bta
accumulated degree-days (S) are equal to area under
temperature curve restricted to the interval between current
temperature and tmin
for temperatures between tmin and tmax (upper threshold)
several different non-linear models (Briere, Lactin, etc.)
allow to estimate topt (optimum temperature)
easy to interpret for experiments with constant temperature
instead of using average temperature, use actual temperature
because below and above ET model is non-linear
Briere et al. (1999)
v .. rate of development (=1/T)
t .. experimental temperature
tmin .. low temperature threshold
tmax .. upper temperature threshold
Optimum temperature:
parameters are estimated using non-linear regression
tttttav -×-××= maxmin )(
10
1691634 maxmin
2
min
2
maxminmax tttttt
topt
-+++
=
Lactin et al. (1995)
v .. rate of development
t .. experimental temperature
tm, , ,  .. constants
tmax and tmin can be estimated from the formula:
topt can be estimated from the first derivative:



+-= 
-
)(
tt
t
t
m
m
eev



+-= 
-
)(
0
tt
t
t
m
m
ee
)(1
0 
-
-
×
×






+-=
TT
T
T
m
m
ee



In the laboratory the development of Diprion pini was studied. Seven
temperatures were used. For each temperature the development time (T)
of the complete development were recorded:
Estimate the minimum development temperature (tmin) and the degreedays
(S) using the linear model.
t (°C) T
5 -
10 200
15 100
20 60
25 40
30 30
35 35
t<-c(5,10,15,20,25,30,35)
T<-c(0,200,100,60,40,30,35)
v<-1/T
m<-lm(v~t,subset=2:6)
m
abline(m)
0.010667/0.001433
1/0.001433
A study on aphids, Myzus persicae, revealed that tmin = 10 and S = 100.
The aphids just laid eggs. Average day temperatures during the
following two weeks were as follows:
15, 18, 25, 23, 24, 18, 17, 15, 18, 15, 22, 25, 26, 21.
Estimate on which day the development of aphids was complete?
t<-c(15, 18, 25, 23, 24, 18, 17, 15, 18, 15, 22, 25, 26, 21)
ET<-t-10
sum(ET)
sum(ET[1:10])
sum(ET[1:11])
Effect of temperature on the development of Nephus includens was
studied in the laboratory using a range of temperatures.
Use Briere`s and Lactin`s model to find models of development
against temperature and to estimate optimum temperature.
Plot the estimated models to the data.
t T
18 23.5
20 18.5
22 13
25 7.3
28 5.5
30 5
32 10.9
t<-c(18,20,22,25,28,30,32)
T<-c(23.5,18.5,13,7.3,5.5,5,10.9)
v<-1/T
m1<-nls(v~exp(rho*t)-exp(rho*Tm-(Tm-t)/delta)+lambda,
start=c(rho=0,Tm=30,delta=1,lambda=0))
summary(m1)
library(rootSolve)
topt1<-uniroot(function(x) 0.01*exp(0.01*x)(0.01+1/0.7)*exp(0.01*33.7-(33.7-x)/0.7),lower=0,upper=40);
topt1
x<-seq(15,40,0.1)
plot(t,v,xlim=c(10,35),ylim=c(0,0.25))
lines(x,predict(m1,list(t=x)))
m2<-nls(v~a*t*(t-tmin)*sqrt(tmax-t), start=c(a=0.1,tmin=10,tmax=35))
summary(m2)
topt2<-(4*32.6+3*17+sqrt(16*32.6^2+9*17^2-16*32.6*17))/10; topt2
lines(x,predict(m2,list(t=x)),lty=2)