Demography - study of organisms with special attention to stage
or age structure
processes associated with age, stage or size
x .. age/stage/size category
px .. age/stage/size specific survival
mx .. reproductive rate (expected average number of offspring per
female)
x
x
x
S
S
p 1+
=
main focus on births and deaths
immigration & emigration is
ignored
no adult survive
one (not overlapping)
generation per year
egg pods over-winter
despite high fecundity they just
replace themselves
Chorthippus
Richards & Waloff (1954)
Annual speciesAnnual species
breed at discrete
periods
no overlapping
generations
BBiennaliennal speciesspecies
breed at discrete periods
adult generation may overlap
adults
adults
0
birth
t0
t1
adults
pre-adults
0
birth
t0
t1
adults t2
p
pre-adults
birth
t0
t1adults
t2
0
pre-adults
p
breed at discrete periods
breeding adults consist of
individuals of various ages (1-5 years)
adults of different generations are
equivalent
overlapping generations
PerennialPerennial speciesspecies
Parus major
Perins (1965)
age/stage
classification is based
on developmental time
size may be more
appropriate than age
(fish, sedentery
animals)
Hughes (1984) used
combination of
age/stage and size for
the description of coral
growth
Age-size-stage life-tableAge-size-stage life-table
Agaricia agaricites
show organisms‘ mortality and reproduction as a function of age
examination of a population during
one segment (time interval)
- segment = group of individuals of
different cohorts
- designed for long-lived organisms
ASSUMPTIONS:
- birth-rate and survival-rate are constant
over time
- population does not grow
DRAWBACKS: confuses age-specific changes in e.g.
mortality with temporal variation
Static (vertical) life-tables
Cervus elaphus
Sx- number of survivors
Dx- number of dead individuals
lx- standardised number of
survivors
qx- age specific mortality
Lowe (1969)
0S
S
l x
x =
x
x
x
S
D
q =
x Sx Dx lx px qx mx
1 129 15 1.000 0.884 0.116 0.000
2 114 1 0.884 0.991 0.009 0.000
3 113 32 0.876 0.717 0.283 0.310
4 81 3 0.628 0.963 0.037 0.280
5 78 19 0.605 0.756 0.244 0.300
6 59 -6 0.457 1.102 -0.102 0.400
7 65 10 0.504 0.846 0.154 0.480
8 55 30 0.426 0.455 0.545 0.360
9 25 16 0.194 0.360 0.640 0.450
10 9 1 0.070 0.889 0.111 0.290
11 8 1 0.062 0.875 0.125 0.280
12 7 5 0.054 0.286 0.714 0.290
13 2 1 0.016 0.500 0.500 0.280
14 1 -3 0.008 4.000 -3.000 0.280
15 4 2 0.031 0.500 0.500 0.290
16 2 2 0.016 0.000 1.000 0.280
1+−= xxx SSD
examination of a population in a cohort = a group of individuals born
at the same period
followed from birth to death
provide reliable information
designed for short-lived organisms
only females are included
Cohort (horizontal) life-table
Vulpes vulpes
x Sx Dx lx px qx mx
0 250 50 1.000 0.800 0.200 0.000
1 200 120 0.800 0.400 0.600 0.000
2 80 50 0.320 0.375 0.625 2.000
3 30 15 0.120 0.500 0.500 2.100
4 15 9 0.060 0.400 0.600 2.300
5 6 6 0.024 0.000 1.000 2.400
6 0 0 0.000
survival and reproduction depend on stage / size rather than age
age-distribution is of no interest
used for invertebrates (insects, invertebrates)
time spent in a stage / size can differ
Lymantria dispar
Campbell (1981)
x Sx Dx lx px qx mx
Egg 450 68 1.000 0.849 0.151 0
Larva I 382 67 0.849 0.825 0.175 0
Larva II 315 158 0.700 0.498 0.502 0
Larva III 157 118 0.349 0.248 0.752 0
Larva IV 39 7 0.087 0.821 0.179 0
Larva V 32 9 0.071 0.719 0.281 0
Larva VI 23 1 0.051 0.957 0.043 0
Pre-pupa 22 4 0.049 0.818 0.182 0
Pupa 18 2 0.040 0.889 0.111 0
Adult 16 16 0.036 0.000 1.000 185
display change in survival by plotting log(lx) against age (x)
logarithmic transformation allows to compare survival based
on different population size
sheep mortality increases with age
survivorship of lapwing (Vanellus) is independent of age
Pearls (1928) classified hypothetical age-specific mortality:
Type I .. mortality is concentrated at the end of life span (humans)
Type II .. mortality is constant over age (seeds, birds)
Type III .. mortality is highest in the beginning of life (invertebrates,
fish, reptiles)
ln(Survivorship)
0
Type I
Type II
Type III
1
Time
fecundity - potential number of offspring
fertility - real number of offspring
semelparous .. reproducing once a life
iteroparous .. reproducing several times during life
birth pulse .. discrete reproduction
(seasonal reproduction)
birth flow .. continuous
reproduction
Numberofoffsprings
0
Time
reproductivepre-reproductive post-reproductive
0
0.1
0.2
0.3
0.4
0.5
0.6
0 20 40 60 80 100 120 140
Time [Days]
Fecundity
Triaeris stenapsis
Geospiza scandens
number.ofbirths/individual
0
0.4
age 16
Cervus elaphus
Odocoileus
numberofbirths/individual
0 6age
0.8
k-value - killing power - another measure of mortality
k-values are additive unlike q
Key-factor analysis - a method to identify the most important
factors that regulates population dynamics
k-values are estimated for a number of years
important factors are identified by regressing kx on log(N)
)log(pk −=
x
kK ∑=
over-wintering adults emerge in June → eggs are laid in
clusters on the lower side of leafs → larvae pass through 4 instars
→ form pupal cells in the soil → summer adults emerge in August
→ begin to hibernate in September
mortality factors overlap
Leptinotarsa decemlineata
Harcourt (1971)
highest k-value indicates the role of a factor in each generation
profile of a factor parallel with the K profile reveals the key
factor
emigration is the key-factor
Summary over 10 years
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 births and deaths
used for modelling of density-independent processes (exponential
growth)
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
px .. age/stage specific survival
class 0 is omitted
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














=














×












+
+
+
+
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
fertilities/fecundities (F) and survivals (p) depend on whether
population has discrete or continuous reproduction
- for populations with discrete pulses post-reproductive survivals and
fertilities are
- for populations with continuous reproduction post-reproductive
survivals and fertilities are
x
x
x
S
S
p 1+
=








+
+
≈
−
+
xx
xx
x
SS
SS
p
1
1 ( )( )
4
1 101 ++
= xx
x
mpmS
F
xx mpF 0=
Egg Larva Pupa Imago
Stage-structured
p2 p3p1
F4












000
000
000
000
3
2
1
4
p
p
p
F
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)
F1 = 0
F2
addition / subtraction
multiplication by a vector
by a scalar
Matrix operations
determinant
eigenvalue (λ)
λ1 = 2.41
λ2 = -0.41a
acbb
2
42
2,1
−±−
=λ






=





+





1510
73
85
41
75
32






=×





2115
96
3
75
32






=





×+×
×+×
=





×





55
23
5745
5342
5
4
75
32
23472
74
32
=×−×=





12)425.0()0()2(
025.0
42
025.0
42 2
−−=×−−×−=





−
−
=





λλλλ
λ
λ
t
0t ANN =
ANN 12 =
ANN 23 =
2
2 ANAANN ttt ==+
parameters are constant over
time and independent of population
density
follows constant exponential
growth after initial damped
oscillations
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∑=
=
2
1++
= xx
x
ll
L ∑=
o
x
xx LT
x
x
x
l
T
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 the only dominant positive eigenvalue of the
transition matrix A (λ1.. finite growth rate)
∑ =−
x
rx
xx eml 1
T
R
r
)ln( 0
≈
T
R0
≈λ
)ln( 1λ=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 Class distribution (SCD)
∑ −
−
=
x
rx
x
rx
x
x
el
el
c
∑ =
= S
i
SCD
1 1
1
w
w
111 wAw λ=
MODULARIZACE VÝUKY EVOLUČNÍ A EKOLOGICKÉ BIOLOGIE
CZ.1.07/2.2.00/15.0204
S. Pekár
podzim 2011
Cvičení z Populační
ekologie
Stage Number Factor
Eggs 562 Overwintering
Larvae 240 Parasites
Pupae 112 Predation
Imagoes 64
Factors causing decline in the population of a moth:
1. Estimate k-value for each factor.
2. Simulate change in population density given the following
estimated linear models of ki on log(N):
overwintering: k1 = 0.48 - 0.04 log(NE)
parasites: k2 = 0.55 - 0.09 log(NL)
predation: k3 = 0.30 - 0.03 log(NP)
The sex ratio is 1:1. Female has average fecundity 17 eggs.
Perform demographic study of the common fox using life table
menu in POPULUS. The fox breeds in pulses and the data were
collected using post-breeding census.
Plot standardised survival (lx) with age.
Which survival curve type it corresponds to?
Plot fecundity (mx) and reproductive value (RV) with age.
Construct Leslie transition matrix and project
it over a period of another 20 years using initial vector (25, 18, 9,
5, 4). When does the population reach stable age distribution?
x lx mx
0 1 0
1 0.8 0
2 0.3 2
3 0.1 3
4 0.07 0