Stano Pekár“Populační ekologie živočichů“
 dN
= Nr
dt
 aim: to simulate (predict) what can happen
 model is tested by comparison with observed data
 realistic models - complex (many parameters), realistic, used to
simulate real situations
 strategic models - simple (few parameters), unrealistic, used for
understanding the model behaviour
 a model should be:
1. a satisfactory description of diverse systems
2. an aid to enlighten aspects of population dynamics
3. a system that can be incorporated into more complex models
 deterministic models - everything is predictable
 stochastic models - including random events
 discrete models:
- time is composed of discrete intervals or measured in generations
- used for populations with synchronised reproduction (annual species)
- modelled by difference equations
 continuous models:
- time is continual (very short intervals) thus change is instantaneous
- used for populations with asynchronous and continuous overlapping
reproduction
- modelled by differential equations
STABILITY
 how population changes in time
 stable equilibrium is a state (population
density) to which a population will
move after a perturbation
stable equilibrium
unstable equilibrium
focus on rates of population processes
 number of cockroaches in a living room increases:
- influx of cockroaches from adjoining rooms  immigration [I]
- cockroaches were born  birth [B]
 number of cockroaches declines:
- dispersal of cockroaches  emigration [E]
- cockroaches died  death [D]
 population increases if I + B > E + D
 rate of increase is a summary of all events (I + B - E - D)
 growth models are based on B and D
 spatial models are based on I and E
Blatta orientalisEDBINN tt 1
Population processes are independent of its density
Assumptions:
 immigration and emigration are none or ignored
 all individuals are identical
 natality and mortality are constant
 all individuals are genetically similar
 reproduction is asexual
 population structure is ignored
 resources are infinite
 population change is instant, no lags
Used only for
relative short time periods
closed and homogeneous environments (experimental chambers)
 for population with discrete generations (annual reproduction), no
generation overlap
time (t) is discrete, equivalent to generation
exponential (geometric) growth
Malthus (1834) realised that any species can potentially increase in
numbers according to a geometric series
N0 .. initial density
b .. birth rate (per capita)
Discrete (difference) model
11   tt dNbNN
11 )(   ttt NdbNN
1)1(  tt NdbN
N
B
b 
N
D
d 
d .. death rate (per capita)
 db1 Rdb 
R1
R .. demographic growth rate
- shows proportional change (in percentage)
λ .. finite growth rate, per capita rate of growth
λ = 1.23 then R = 0.23
.. 23% increase
number of individuals is multiplied each time - the larger the
population the larger the increase
time0
Nt
t
t
tt
i
i
1
21
1
1
)...(  





 
λ < 1
λ > 1
λ = 1
Average of finite growth rates
- estimated as geometric mean
1 tt NN
t
t NN 0
 012 NNN 
if λ is constant, population number in generations t is equal to
Comparison of discrete and continuous generations
populations that are continuously reproducing, with overlapping
generations
when change in population number is permanent
derived from the discrete model
Nt
time
Continuous (differential) model
t
t NN 0
)ln()ln()ln( 0 tNNt 
)ln()ln()ln( 0 tNNt 
)ln(
1
d
d

Nt
N
)ln(
d
d
N
t
N

Solution of the differential equation:
- analytical or numerical
at each point it is possible to determine the rate of change by
differentiation (slope of the tangent)
when t is large it is approximated by the exponential function
time
N
r .. intrinsic rate of natural increase,
instantaneous per capita growth rate
r < 0
r > 0
r = 0
Nr
t
N

d
d
)ln(rif
Nr
t
N

d
d
r
Nt
N

1
d
d
 
TT
trN
N 00
dd
1
)0()ln()ln( 0  TrNNT
rT
N
NT








0
ln
rt
t eNN 0
rTT
e
N
N

0
t
t NN 0 rt
t eNN 0
rtt
e
)ln(r
r versus λ
 r is symmetric around 0, λ is not
r = 0.5 ... λ = 1.65
r = -0.5 ... λ = 0.61
 doubling time: time required for a
population to double
r
t
)2ln(

 Demography - study of organisms with special attention to stage
or age structure
 processes are associated to 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 species
 breed at discrete
periods
 no overlapping
generations
Biennal species
 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
Perennial species
Parus major
Perins (1965)
 age/stage
classification is based
on developmental time
 size may be more
appropriate than age
(fish, sedentary
animals)
 Hughes (1984) used
combination of
age/stage and size for
the description of coral
growth
Age-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 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
px .. age-specific survival
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
x
x
x
l
l
p 1

 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)
 sheep mortality increases with age
 survivorship of lapwing (Vanellus) is independent of age
but survival of sheep is age-dependent
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 scandensnumber.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 natality and mortality
 generations are not overlapping
 reproduction is asexual
 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 (average no. of offspring per female)
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 census and
reproduction
- populations with discrete pulses post-reproductive census
- populations with discrete pulses pre-reproductive census
- for pre-reproductive census 0 age is omitted
- for populations with continuous reproduction
x
x
x
l
l
p 1














xx
xx
x
ll
ll
p
1
1  
2
11 

xxx
x
mpml
F
1 xxx mpF
x
x
x
l
l
p 1
 10  xx mpF
includes p of
reproductive stages
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
 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
 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

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
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012)425.0()0()2(
025.0
42 2
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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 initial damped
oscillations