Stano Pekár“Populační ekologie živočichů“
 dN
= Nr
dt
Acarus Cheyletus
continuous model of Lotka & Volterra (1925-1928) used to explain
decrease in prey fish and increase in predatory fish after World War I
assumptions
- continuous predation (high population density)
- populations are well mixed
- closed populations (no immigration or emigration)
- no stochastic events
- predators are specialised on one prey species
- populations are unstructured
- reproduction immediately follows feeding
H .. density of prey P .. density of predators
r .. intrinsic rate of prey population m .. predator mortality rate
a .. predation rate b .. reproduction rate of predators
 in the absence of predator, prey grows exponentially 
 in the absence of prey, predator dies exponentially 
 predation rate is linear function
of the number of prey .. aHP
 each prey contributes identically
to the growth of predator .. bHP
rH
t
H

d
d
mP
t
P

d
d
aHPrH
t
H

d
d
mPbHP
t
P

d
d
do not converge, has no asymptotic
stability (trajectories are closed lines)
 neutral stability
 unstable system, amplitude of the cycles
is determined by initial numbers
Zero isoclines:
 for prey population:
 for predator population:
H
P
prey isocline
predator isocline
0
0
d
d

t
P
aHPrH 0
a
r
P 
mPbHP 0
b
m
H 
0
d
d

t
H
time
density
prey
predator
0
Analysis of the model
a
r
b
m
 in the absence of the predator prey population reaches
carrying capacity K
Addition of density-dependence
 for given parameter values: r = 3, m = 2, a = 0.1, b = 0.3, K = 10
HP
H
H
t
H
1.0
10
13
d
d






 PHP
t
P
23.0
d
d

aHP
K
H
rH
t
H






 1
d
d
mPbHP
t
P

d
d
Zero isoclines:
 for prey population:
if H = 0 (trivial solution) or if
 for predator population: 0.3HP - 2P = 0
if P = 0 (trivial solution)
or if 0.3H - 2 = 0
 gradient of prey isocline is negative
0
d
d

t
H
HP
H
H 1.0
10
130 






0
d
d

t
P
P = 30 - 3H
P
H
1.0
10
130 






H = 6.667
H
P
30
6.670 10
time
density
prey
predator
0
K
H
P
30
6.70 10
 has single positive asymptotically stable equilibrium
defined by crossing of isoclines
converges to the stable equilibrium
functional response Type II:
 rate of consumption by all predators:
Addition of functional response of Type II
 for parameters: rH = 3, a = 0.1, Th = 2, K = 10
prey isocline: predator isocline:
h
a
aHT
aHT
H


1
h
a
aHT
aHP
T
PH


1
h
H
aHT
aHP
K
H
Hr
t
H








1
1
d
d
0
d
d

t
H
21.01
1.0
10
130
H
HPH
H








2
6.0630 HHP 
mPbHP
t
dP

d
H = constant
b
m
H 
.. damped oscillations
 predator exploits prey
close to K
- isocline: H = 9
time
density
time
density
time
density  predator exploits
prey close to K/2
- isocline: H = 5
 predator exploits
prey at low density
- isocline: H = 2
Rosenzweig & MacArthur (1963)
H
P
H
P
H
P
K
prey
predator
0 0 0
0 0 0K/2
K
Damped oscillations Sustained oscillations Extinction
K K
 logistic model with carrying capacity proportional to H
 k .. parameter of carrying capacity of the predator
 rP = bH - m
Addition of predator’s carrying capacity
 for parameters: rP = 2, k = 0.2
predator isocline:
prey isocline:
mPbHP
t
P

d
d







kH
P
Pr
t
P
P 1
d
d
0
d
d

t
P







H
P
P
2.0
120
H = 5P
2
6.0630 HHP 
h
H
aHT
aHP
K
H
Hr
t
H








1
1
d
d
H
P
K0
time
density
prey
predator
0
K
H
P
K0
quick approach to stable equilibrium
Zatypota Theridion
 discrete model of Nicholson & Bailey (1935)
- discrete generations
- attack happens at reproduction
- 1, .., several, or less than 1 host
- random host search and functional response Type III
- lay eggs in aggregation
Ht = number of hosts in time t
Ha = number of attacked hosts
 = finite rate of increase of the host
Pt = number of parasitoids
c = conversion rate, no. of parasitoids for 1 host
)(1 att HHH  
aat HcHP 1
 parasitoid searches randomly
 encounters (x) are random (Poisson distribution)
p0 = proportion of not encountered,  .. mean number of encounters
Et = total number of encounters
a = searching efficiency
Et = a Ht Pt
 proportion of encounters (1 or more times): p = (1– p0)
Incorporation of random search
x = 0, 1, 2, ...
!x
e
p
x
x

 


 ep0
 taP
ta eHH 
 1
 t
t
t
aP
H
E taP
ep 
0
)1( taP
ep 

 highly unstable model for all parameter values:
- equilibrium is possible but the slightest disturbance leads to divergent
oscillations (extinction of parasitoid)
taP
tt eHH 
  1
 taP
tt eHP 
  11
time
density
H
P
0
0
)(1 att HHH  
at HP 1
 exponential growth of hosts is replaced by logistic equation
H*.. new host carrying capacity
 depends on parasitoids’ efficiency
- when a is low then q  1
- when a is high then q  0
 density-dependence have
stabilising effect for moderate r and q
Stability boundaries
Addition of density-dependence
Beddington et al. (1975)
t
t
aP
K
H
tt eHH







 
1
1 
 taP
tt eHP 
  11
K
H
q
*

Addition of the refuge
 if hosts are distributed non-randomly in the space
Fixed number in refuge: H0 hosts are always protected
 have strong stabilising effect
even for large r
Hassell & May (1973)
taP
tt eHHHH 
  )( 001 
 taP
tt eHHP 
  1)( 01
 distribution of encounters is not random but aggregated (negative
binomial distribution)
- proportion of hosts not encountered (p0):
where k = degree of aggregation
 very stable model system if k  1
Stability boundaries:
a) k=, b) k=2, c) k=1, d) k=0
Addition of aggregated distribution
Hassell (1978)
k
tt
k
aP
K
H
tt eHH














 
11
1 

















k
t
tt
k
aP
HP 111
k
t
k
aP
p







 10