Stano Pekár“Populační ekologie živočichů“
 dN
= Nr
dt
includes all mechanisms of population growth that
change with density
 population structure is ignored
 extrinsic effects are negligible
 response of  and r to N is immediate
 and r decrease with population density either because
natality decreases or mortality increases or both
- negative feedback of the 1st order
 K .. carrying capacity
- upper limit of population growth
where  = 1 or r = 0
- is a constant
if then
0
1
KNt
Nt/Nt+1
1/
y = a + x b
Discrete (difference) model
K
N
N
N
t
t
t
)1(
1
1





t
t
t
aN
N
N


1
1

K
a
1


tt NN 1

1
1

t
t
N
N
 
K
N
N
N
t
t
t 

111
1



- there is linear dependence of  on N
time0
K
Nt
when Nt  0 then
• no competition
• exponential growth
when Nt  K then
• density-dependent control
• S-shaped (sigmoid) growth
when Nt > K then
• population returns to K
1
1

 taN

1
1

 taN




 taN1
- when N  K then r  0
N K
dN/dt*1/N
r
0
logistic growth
first used by Verhulst (1838) to describe growth of human population

Solution of the differential equation
Continuous (differential) model







K
N
Nr
dt
dN
1
Nr
dt
dN
 r
Ndt
dN

1
K
r
Nr
Ndt
dN

1
00
0
)( NeNK
KN
N rtt

 
- there is linear dependence of r on N
Monotonous increase (r = 0.5) Damping oscillations (r = 1.9)
Limit cycle (r = 2.3)
Chaos (r = 3.0)
1. N = 0 .. unstable equilibrium
2. N = K .. stable equilibrium .. if 0 < r < 2
 “Monotonous increase” and “Damping oscillations” has a stable
equilibrium
 “Limit cycle” and “Chaos”
has no equilibrium
r < 2 .. stable equilibrium
r = 2 .. 2-point limit cycle
r = 2.5 .. 4-point limit cycle
r = 2.692 .. chaos
 chaos can be produced by
deterministic process
 density-dependence is
stabilising only when
r is rather low
Model equilibria
N
r
a) yeast (logistic curve)
b) sheep (logistic curve
with oscillations)
c) Callosobruchus
(damping oscillations)
d) Parus (chaos)
e) Daphnia
 of 28 insect species
in one species chaos
was identified, one
other showed limit
cycles, all other were in
stable equilibrium
 in case of density-independence  is constant – independent of N
 in case of DD  is changing with N:
 plot ln() against Nt
 estimate  and K
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 10 20 30 40 50 60
Nt
ln(lambda)
  tbNa ln
a
emax b
a
K 
0
10
20
30
40
50
0 5 10 15 20 25
Time
N
t
t NN 0
t
t
t
aN
N
N


1
1

Hassell (1975) proposed general model for DD
- r is not linearly dependent on N
where θ.. the strength of competition
 θ < 1 .. scramble competition
(over-compensation), strong DD,
leads to fluctuations around K
θ = 1 .. contest competition
(exact compensation), stable density
 θ >> 1 .. under-compensation
- weak DD, population will return to K
 

t
t
t
aN
N
N


1
1
N K
r
0
θ = 1
θ = 3
θ = 0.2
















K
N
rN
t
N
1
d
d
 species response to resource change is not immediate (as in case of
hunger) but delayed due to maternal effect, seasonal effect, predator
pressure
 appropriate for species with long generation time where reproductive rate
is dependent on the past (previous generations)
 time lag (d or τ) .. negative feedback of the 2nd order
discrete model continuous model
 many populations of mammals cycle with 3-4 year periods
 time-lag provokes fluctuations of certain amplitude at certain periods
 period of the cycle in continuous model is always 4
dt
t
t
aN
N
N




1
1







 
K
N
rN
dt
dN t
t

1
r  < 1  monotonous increase
r  < 3  damping fluctuations
r  < 4  limit cycle fluctuations
r  > 5  extinction
Solution of the continuous model:
0
500
1000
1500
2000
2500
0 10 20 30 40 50
Time
Density
tau=2
tau=6
tau=8
tau=11
K = 500
r = 0.5









 K
N
r
tt
t
eNN

1
1
Maximum Sustainable Harvest (MSH)
 to harvest as much as possible with the least negative effect on N
 ignore population structure
 ignore stochasticity 01
d
d







K
N
Nr
t
N
4
MSH
rK

N0 K/2
dN/dt
2
*
K
N local maximum:
Amount of MSH (Vmax):
at K/2:





 

2
MSY
KK
a

 Surplus production (catch-effort) models
- when r,  and K are not known
- effort and catch over several
years is known
- Schaefer quadratic model
- local maximum of the function
identifies optimal effort (OE)
Robinson & Redford (1991)
- Maximum Sustainable Yield (MSY)
where a = 0.6 for longevity < 5
a = 0.4 for longevity = (5,10)
a = 0.2 for longevity > 10
OE
2
EEcatch  
 individuals in a population may cooperate in hunting, breeding –
positive effect on population increase
Allee (1931) – discovered inverse DD
 genetic inbreeding – decrease in fertility
 demographic stochasticity – biased sex ratio
 small groups – cooperation in foraging, defence, mating,
thermoregulation
K2 .. extinction threshold,
- unstable equilibrium
population increase is slow
at low density but fast at higher density












 11
21 K
N
K
N
Nr
dt
dN
N
K2 K1
r
increase
decrease
0