Stano Pekár“Populační ekologie živočichů“
 dN
= Nr
dt
+ + .. mutualism (plants and pollinators)
0 + .. commensalism (saprophytism, parasitism, phoresis)
- + .. predation (herbivory, parasitism), mimicry
- 0 .. amensalism (allelopathy)
- - .. competition
Increase Neutral Decrease
Increase + +
Neutral 0 + 0 0
Decrease + - - 0 - Effect
of species 1 on fitness of species 2
Effectofspecies2on
fitnessofspecies1
DIRECT
INDIRECT Apparent competition
Facilitation
Exploitation competition

 n
k
kp
D
1
2
1
 

kk
kk
pp
pp
a
21
21
Niche breadth
Levin’s index (D):
- pk .. proportion of individuals in class k
- does not include resource availability
- 1 < D < ∞
Smith’s index (FT):
- qk .. proportion of available individuals in class k
- 0 < FT < 1
Niche overlap
Pianka’s index (a):
- does not account for resource availability
- 0 < a < 1
Lloyd’s index (L):
- 0 < L < ∞


n
k
kk qpFT
1

k
kk
q
pp
L 21
 based on the logistic differential model
species 1: N1, K1, r1
species 2: N2, K2, r2





 

1
21
11
1
1
K
NN
rN
dt
dN





 

2
21
22
2
1
K
NN
rN
dt
dN







K
N
Nr
t
N
1
d
d
assumptions:
- all parameters are constant
- individuals of the same species are identical
- environment is homogenous, differentiation of niches is not possible
- only exact compensation is present
 model of Lotka (1925) and Volterra (1926)
 total competitive effect (intra + inter-specific)
(N1+ N2) where  .. coefficient of competition
 = 0 .. no interspecific competition
 < 1 .. species 2 has lower effect on species 1 than species 1 on itself
 = 0.5 .. one individual of species 1 is equivalent to 0.5 individuals of
species 2)
 = 1 .. both species has equal effect on the other one
 > 1 .. species 2 has greater effect on species 1 than species 1 on itself
species 1:
species 2:
 if competing species use the same resource then interspecific
competition is equal to intraspecific





 

2
2121
22
2
1
K
NN
rN
dt
dN 





 

1
2121
11
1
1
K
NN
rN
dt
dN 
 examination of the model behaviour using null isoclines
 used to describe change in any two variables in coupled differential
equations by projecting orthogonal vectors
 identification of isoclines: a set of abundances for which the change
in populations is 0:
N1
N2
K1
0
dt
dN
N1
N2
K2
species 1 species 2
0
dt
dN
0
dt
dN
0
dt
dN
0
dt
dN
 species 1
r1N1 (1 - [N1 + 12N2] / K1) = 0
r1N1 ([K1 - N1 - 12N2] / K1) = 0
trivial solution if r1, N1, K1 = 0
and if K1 - N1 - 12N2 = 0
then N1 = K1 - 12N2
if N1 = 0 then N2 = K1/12
if N2 = 0 then N1 = K1
 species 2
r2N2 (1 - [N2 + 21 N1] / K2) = 0
N2 = K2 - 21N1
trivial solution if r2, N2, K2 = 0
if N2 = 0 then N1 = K2/21
if N1 = 0 then N2 = K2
 above isocline i1 and below i2 competition is weak
 in-between i1 and i2 competition is strong
N1
N2
K2
K1
21
2

K
12
1

K
1. Species 2 drives species 1 to extinction
 K and  determine the model behaviour
disregarding initial densities species 2 (stronger competitor) will
outcompete species 1 (weaker competitor)
 equilibrium (0, K2)
K1 = K2
12 > 21
12
1
2

K
K 
21
2
1

K
K 
N1
N2
K2
K1
12
1

K
21
2

K
time
0
species 2
species 1
N
K
r1 = r2
N01 = N02
2. Species 1 drives species 2 to extinction
species 1 (stronger competitor) will outcompete species 2 (weaker
competitor)
equilibrium (K1, 0)
12
1
2

K
K 
21
2
1

K
K 
N1
N2
K2
K1
12
1

K
21
2

K
K1 = K2
12 < 21
r1 = r2
N01 = N02
time
0
species 1
species 2
N
K
3. Stable coexistence of species
 disregarding initial densities both species will coexist at stable
equilibrium (where isoclines cross)
 at at equilibrium population density of both species is reduced
 both species are weak competitors
 equilibrium (K1*, K2*)
K1 = K2
12, 21 < 1
N1
N2
K2
K1
12
1

K
21
2

K
stable equilibrium
12
1
2

K
K 
21
2
1

K
K 
r1 < r2
N01 = N02
0
species 1
species 2
time
N
K
K*
one species will drive other to extinction
depending on the initial conditions
 coexistence only for a short time
 both species are strong competitors
equilibrium (K1, 0) or (0, K2)
4. Competitive exclusion
r1 = r2
K1 = K2
N1
N2
K2
K1
12
1

K
21
2

K
12
1
2

K
K 
21
2
1

K
K 
N01 < N02
0
species 2
species 1
time
N
K2
12, 21 > 1
N01 > N02
0
species 1
species 2
time
N
K1
Jacobian matrix of partial derivations for 2dimensional system
 evaluation of the derivations for densities close to equilibrium
negative
 estimate eigenvalues of the matrix (negative values indicate
approach to equilibrium):
- real parts of all eigenvalues < 0 .. globally stable
- real part of some eigenvalues < 0 .. saddle stability
- real part of all eigenvalues > 0 .. globally unstable
- imaginary parts present .. oscillations
- imaginary parts absent .. no oscillations
 Lotka-Volterra system is stable for 1221 < 1





















2
2
1
2
2
1
1
1
dddd
dddd
N
tN
N
tN
N
tN
N
tN
J
 when Rhizopertha and Oryzaephilus were reared separately both
species increased to 420-450 individuals (= K)
 when reared together Rhizopertha reached K1 = 360, while
Oryzaephilus K2 = 150 individuals
 combination resulted in more efficient conversion of grain (K12 = 510
individuals)
 three combinations of
densities converged to the
same stable equilibrium
 prediction of
Lotka-Volterra model is correct
N1 Rhizopertha
N2Oryzaephilus
K1
K2
0
1
2
3
1: N1 < N2
2: N1 = N2
3: N1 > N2
Crombie (1947)
equilibrium





 
  1
,212,11
1
,11,1
K
NNK
r
tt
tt
eNN






 
  2
,121,22
2
,21,2
K
NNK
r
tt
tt
eNN

dynamic (multiple) regression is used to estimate parameters from a
series of abundances
.. a, b, c – regression
parameters
ar 
1
121
,2
1
1
,11
,1
1,1
ln
K
r
N
K
r
Nr
N
N
tt
t
t 








 
b
r
K 
r
Kc

2
212
,1
2
2
,12
,2
1,2
ln
K
r
N
K
r
Nr
N
N
tt
t
t 








 
 solution of the differential model – Ricker’s model:
 Vandermeer & Boucher (1978)
  .. coefficient of mutalism
 Facultative (able to exist
independently) x obligatory mutualists
 Outcome depends on the type of
mutualism





 

2
1212
22
2
1
K
NN
rN
dt
dN 





 

1
2121
11
1
1
K
NN
rN
dt
dN 
N1
N2
K2
K1
Facultative mutualists