Stano Pekár“Populační ekologie živočichů“
dN
= Nr
dt
if then
logistic growth due to density dependent
changes in fecundity and survival
K .. carrying capacity,
upper limit of population growth, where λ = 1
change in λ depends on N
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
−
+=
+
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
rtat
e
K
N −
+
=
1







 −
=
0
0
ln
N
NK
a
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
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 −=
• plot ln(λ) against Nt
• estimate λ and K using
0
10
20
30
40
50
0 5 10 15 20 25
Time
N
Hassell (1975) proposed general model for DD
Effect of θ on population density
0
200
400
600
800
1000
1200
1400
0 10 20 30 40 50 60
Time
Density
- where θ.. the strength of competition
θ >> 1 .. scramble competition (over-compensation)
θ = 1 .. contest competition (exact compensation)
θ < 1 .. under-compensation
( )θ
λ
t
t
t
aN
N
N
+
=+
1
1
N K
r
0
θ = 1
θ = 3
θ = 0.2
θ = 0.5
θ = 1
θ = 2
species response to resource change is not immediate but delayed due to
maternal effect, seasonal effect
appropriate for species with long generation time where reproductive rate
is dependent on density of a previous generation
time lag (d, 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 4ττττ
dt
t
t
aN
N
N
−
+
+
=
1
1
λ
K
NK
rN
dt
dN t
t
τ−−
=
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
to attain maximum sustainable yield (MSY)
local maximum of the model for N
01 =





−=
K
N
Nr
dt
dN
2
* K
N =
time
Nt
0
K
K/2
N*





 −
=
2
KK
aMSY
λ
where a = 0.6 for L < 5
a = 0.4 for L = (5,10)
a = 0.2 for L > 10
K2 .. extinction threshold, unstable equilibrium
population increase is slow at low density but fast at high density






−





−= 11
21 K
N
K
N
Nr
dt
dN
population sizeK2 K1
r
birth
death
0
Simulate population dynamics using density-dependent model for
discrete population growth for a period of 40 generations with
N0=10.
1. With deterministic λ (=1.2) and K (=500).
2. With stochastic λ (=1.2 ±0.2) but deterministic K (=500).
3. With stochastic K (=500 ±50) but deterministic λ (=1.2).
4. With stochastic λ (=1.2 ±0.2) and K (=500 ±50).
N<-41
for(t in 1:40) N[t+1]<-{
N[t]*1.2/(1+N[t]*(1.2-1)/500)}
plot(0:40,N,type="b")
for(t in 1:40) N[t+1]<-{
N[t]* runif(1,1,1.4)/(1+N[t]*(runif(1,1,1.4)-1)/500)}
plot(0:40,N,type="b")
for(t in 1:40) N[t+1]<-{
N[t]*1.2/(1+N[t]*(1.2-1)/runif(1,450,550))}
plot(0:40,N,type="b")
for(t in 1:40) N[t+1]<-{
N[t]* runif(1,1,1.4)/(1+N[t]*(runif(1,1,1.4)-
1)/runif(1,450,550))}
plot(0:40,N,type="b")
You have observed the following population dynamic of yearly
censuses of aphids:
180, 531, 277, 296, 828, 329, 397, 772, 625, 318, 567, 881, 386
1. Plot the population dynamic. Is there evidence for density-
dependence?
3. Estimate λmax and K.
aphid<-c(180, 531, 277, 296, 828, 329, 397, 772, 625, 318, 567,
881, 386)
plot(aphid,type="b")
lambda1<-aphid[-1]/aphid[-13]
plot(aphid[-13], log(lambda1))
m2<-lm(log(lambda1)~aphid[-13])
coef(m2)
abline(m2)
exp(1.3057703)
-1.30577030/-0.00248398
On an African market wild game animals are sold. You know
carrying capacities (K), finite growth rates (λ), and longevities (L)
for each species:
1. Compute MSY for each species:
2. Is the observed harvest sustainable in each species?
K lambda Longevity Harvest
monkey 49000 1.17 31 781
pangolin 22000 2.01 13 192
porcupine 110000 1.82 23 1580
duiker 45000 1.63 7 732
0.2*(1.17*49000-49000)/2
0.2*(2.01*22000-22000)/2
0.2*(1.82*110000-110000)/2
0.4*(1.63*45000-45000)/2