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 λ and r to N is immediate
λ 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 λ = 1 or r = 0
- is a constant
if then
0
1
KNt
Nt/Nt+1
1/λ
y = a + x b
Discrete (difference) model
K
N
N
N
t
t
t
)1(
1
1
−
+
=+
λ
λ
t
t
t
aN
N
N
+
=+
1
1
λ
K
a
1−
=
λ
λtt NN =+1
λ
1
1
=
+t
t
N
N
( )
K
N
N
N
t
t
t λ
λ
111
1
−
+=
+
- there is linear dependence of λ 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
λ
1
1
≈
+ taN
λ
λ
λ
≈
+ 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 λ is constant – independent of N
in case of DD λ is changing with N:
plot ln(λ) against Nt
estimate λ 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 −=λln
a
e=maxλ b
a
K −=
0
10
20
30
40
50
0 5 10 15 20 25
Time
N
t
t NN λ0=
t
t
t
aN
N
N
+
=+
1
1
λ
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
( )θ
λ
t
t
t
aN
N
N
+
=+
1
1
N K
r
0
θ = 1
θ = 3
θ = 0.2














−=
θ
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 4τ
dt
t
t
aN
N
N
−
+
+
=
1
1
λ






−= −
K
N
rN
dt
dN t
t
τ
1
r τ < 1 → monotonous increase
r τ < 3 → damping fluctuations
r τ < 4 → limit cycle fluctuations
r τ > 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
τ
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
λ
Surplus production (catch-effort) models
- when r, λ 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 γβα ++=
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