Stano Pekár“Populační ekologie živočichů“
 dN
= Nr
dt
 model of Leslie (1945) uses parameters (survival and fecundity) from
life-tables
 where populations are composed of individuals of different age, stage
or size with specific natality and mortality
 generations are not overlapping
 reproduction is asexual
 fertility and mortality is constant in time
Nx,t .. no. of organisms in age x and time t
Gx .. probability of persistence in the same size/stage
Fx .. age/stage specific fertility (average no. of offspring per female),
px .. age/stage specific survival
 class 0 is omitted
 individuals cannot persist in an age class
 number of individuals in the first age class
 number of individuals in the remaining age class

 
n
x
ttxtxt FNFNFNN
1
2,21,1,1,1 ...
xtxtx pNN ,1,1 
N1 N2 N3 N4
Age-structured
p12 p23 p34
F4
F3
F2
F1
















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
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









1,4
1,3
1,2
1,1
,4
,3
,2
,1
34
23
12
4321
000
000
000
t
t
t
t
t
t
t
t
N
N
N
N
N
N
N
N
p
p
p
FFFF
 each column in A specifies fate of an organism in a specific age:
3rd column: organism in age 2 produces F2 offspring and goes to age 3
with probability p23
 A is always a square matrix
 Nt is always one column matrix = a vector
transition matrix A age distribution vectors Nt
1 tt NAN
calculation of fertilities/fecundities (F) and survivals (p)
depend on census and reproduction type
- discrete pulses post-reproductive census: census of offspring
shortly after birth (class 0)
x
x
x
l
l
p 1
 1 xxx mpF
includes p of
reproductive stages
age
class
0 1 2
1 2
- for birth-flow
- for birth-pulse
1 xx mF
- discrete pulses pre-reproductive census: census of offspring
born last year (class 1), class 0 is omitted
- continuous reproduction: each class is composed of early and
older age class













xx
xx
x
ll
ll
p
1
1  
2
11 

xxx
x
mpml
F
x
x
x
l
l
p 1
 10  xx mpF includes p of the
youngest stage
Egg Larva Pupa Imago
Stage-structured
p2 p3p1
F4












000
000
000
000
3
2
1
4
p
p
p
F
 in species where parameters are function of developmental stage
 when inter-moult intervals vary in duration (but residence time
per stage is considered identical for all)
 may contain persistence
 only imagoes reproduce thus F1,2,3 = 0
 no imago survives to another reproduction period: p4 = 0
Size-structured
Tiny Small Medium Large
p1
p2 p3
F4
F3
G11
G22 G33 G44












443
332
221
43211
00
00
00
Gp
Gp
Gp
FFFG
 model of Lefkovitch (1965) uses 3 parameters (mortality, fecundity
and persistence)
 parameters are a function of size
 F1 = 0
 above diagonal elements can include p of shrinkage
F2
 multiplication by a vector
by a scalar
Matrix operations
 determinant
 eigenvalue (λ)
λ1 = 2.41
λ2 = -0.41a
acbb
2
42
2,1


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2115
96
3
75
32
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


55
23
5745
5342
5
4
75
32
23472
74
32






012)425.0()0()2(
025.0
42 2











0)det(  IA uAu 






025.0
42
0
t
t NAN 
12 ANN 
23 ANN 
ttt NAAANN 2
2 
 parameters are constant over
time and independent of population
density
 follows constant exponential
growth after reaching stable age
distribution (following initial
damped oscillations)
Density-independent model
Net reproductive rate (R0)
 average total number of offspring produced by a female in her
lifetime
 equals to finite growth rate
Average generation time (T)
average age of females when they give birth
not valid for populations with generation overlap
Expectation of life
age specific expectation of life – average age that is expected
for particular age class
 o .. oldest age


n
x
xxmlR
0
0
0
0
R
mxl
T
n
x
xx

2
1
 xx
x
ll
L
o
x
xx LT
x
x
x
l
T
e  where
Growth rates
 Discrete time/generations
- estimate of  (finite growth rate) from the life table:
where is vector at stable age distribution
 is dominant positive eigenvalue of A
- or
 Continuous time
- r can be estimated from 
- by approximation
or by Euler-Lotka method
- valid only for population with SCD
tt NNA
~~

T
R
r
)ln( 0

T
R0

)ln(r
tN
~
0)det(  IA 
 


x
rx
xx eml1
- 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
w1 .. right eigenvector (vector of the dominant
eigenvalue)
- provides stable age distribution
- scale w1 by sum of individuals
Stable Class distribution (SCD)

 S
i iw
SCD
1 1
1w
111 wAw 
Reproductive value (vx)
measures relative reproductive potential and identifies age class
that contributes most to the population growth (Fisher 1930)
such class is under highest selection force
sum of all expected offspring produced in age x and further
when population increases then early offspring contribute more
to vx than older ones
is a function of fertility and survival
v1 .. left eigenvector (vector of the dominant
eigenvalue of transposed A)
- v1 is proportional to the reproductive values
and scaled to the first category
(class 1 = 1)
vx
age
1
0
111 vAv 
11
1
v
v
v x
x  1x
Sensitivity (s)
identifies which process (p, F, G) has largest effect on the
population increase (λ1)
measures absolute change
- examines change in λ1 given small change in processes (aij)
- sensitivity is larger for survival of early, and for fertility of older
classes
- not used for postreproductive census with class 0
Elasticity (e)
 weighted measure of sensitivity
- measures relative contribution to the population increase
- impossible transitions = 0
wv,
ijij
ij
wv
s


ij
ij
ij s
a
e
1

 sum of pairwise products
to adopt means for population promotion (threatened) or
control (pests) or sustainable yield
in populations with short generation time and higher natality
population decline stabilisation will take some delay
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 - on parameters with high
elasticities
4. Development and application of management plan
5. Prediction of future