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 & EQUILIBRUM
 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

)ln(
)ln(

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(
