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
∑=
= 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
Smith’s index (FT):
- qk .. proportion of available individuals in class k
- 0 < D, 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+ αN2) where α .. coefficient of competition
α = 0 .. no interspecific competition
α < 1 .. species 2 has lower effect on species 1 than species 1 on itself
α = 0.5 .. one individual of species 1 is equivalent to 0.5 individuals of
species 2)
α = 1 .. both species has equal effect on the other one
α > 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 α





 +
−=
1
2121
11
1
1
K
NN
rN
dt
dN α
examination of the model behaviour on a phase plane
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 + α12N2] / K1) = 0
r1N1 ([K1 - N1 - α12N2] / K1) = 0
trivial solution if r1, N1, K1 = 0
and if K1 - N1 - α12N2 = 0
then N1 = K1 - α12N2
if N1 = 0 then N2 = K1/α12
if N2 = 0 then N1 = K1
species 2
r2N2 (1 - [N2 + α21 N1] / K2) = 0
N2 = K2 - α21N1
trivial solution if r2, N2, K2 = 0
if N2 = 0 then N1 = K2/α21
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
α
K
12
1
α
K
1. Species 2 drives species 1 to extinction
K and α determine the model behaviour
disregarding initial densities species 2 (stronger competitor) will
outcompete species 1 (weaker competitor)
equilibrium (0, K2)
K1 = K2
α12 > α21
12
1
2
α
K
K >
21
2
1
α
K
K <
N1
N2
K2
K1
12
1
α
K
21
2
α
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
α
K
K <
21
2
1
α
K
K >
N1
N2
K2
K1
12
1
α
K
21
2
α
K
K1 = K2
α12 < α21
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
α12, α21 < 1
N1
N2
K2
K1
12
1
α
K
21
2
α
K
stable equilibrium
12
1
2
α
K
K <
21
2
1
α
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
α
K
21
2
α
K
12
1
2
α
K
K >
21
2
1
α
K
K >
N01 < N02
0
species 2
species 1
time
N
K2
α12, α21 > 1
N01 > N02
0
species 1
species 2
time
N
K1
Jacobian matrix of partial derivations
evaluation of the derivations for densities close to equilibrium
estimate eigenvalues of the matrix
- if all eigenvalues < 0 .. locally stable
Lotka-Volterra system is stable for α12α21 < 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
α





 −−
+ = 2
,121,22
2
,21,2
K
NNK
r
tt
tt
eNN
α
dynamic (multiple) regression is used to estimate parameters from a
series of abundances
ar =
1
121
,2
1
1
,11
,1
1,1
ln
K
r
N
K
r
Nr
N
N
tt
t
t α
−−=







 +
b
r
K =
r
Kc
=α
2
212
,1
2
2
,12
,2
1,2
ln
K
r
N
K
r
Nr
N
N
tt
t
t α
−−=







 +
solution of the differential model – Ricker’s model: