Stano Pekár"Populační ekologie živočichů"
dN
= Nr
dt
+ + .. mutualism (plants and pollinators)
0 + .. commensalism (saprophytism, parasitism, phoresis)
- + .. predation (herbivory, parasitism), Batesian 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
based on the logistic model of Lotka (1925) and Volterra (1926)
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
dt
dN
1
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
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 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 growth
rate 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
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
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)
K1 = K2
12 > 2112
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)
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
K1 = K2
12, 21 < 1
N1
N2
K2
K1
12
1

K
21
2

K
stable equilibrium
0
species 1
species 2
time
N
K
12
1
2

K
K <
21
2
1

K
K <
r1 < r2
N01 = N02
one species will drive other to extinction
depending on the initial conditions
coexistence for a short time
both species are strong competitors
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
when Tribolium and Oryzaephilus were reared separately both species
increased to 420-450 individuals (= K)
when reared together Tribolium 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
Tribolium versus Oryzaephilus
N1 Tribolium
N2Oryzaephilus
K1
K2
0
1
2 3
1: N1 < N2
2: N1 = N2
3: N1 > N2
Crombie (1947)
equilibrium





 -
+
= 1
,212,1
1 1
,11,1
K
NN
r
tt
tt
eNN






 -
+
= 2
,121,2
2 1
,21,2
K
NN
r
tt
tt
eNN

dynamic regression analysis is used to estimate parameters from
abundances
tt
t
t
cNbNa
N
N
,2,1
,1
1,1
ln ++=







 +
ar =
1
121
,2
1
1
1,1
,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
1,2
,2
1,2
ln
K
r
N
K
r
Nr
N
N
tt
t
t 
--=







 +
tt
t
t
cNbNa
N
N
,1,2
,2
1,2
ln ++=







 +
solution of the differential model:
Two species of Tribolium beetles were kept together in a jar with
flour. Their densities were recorded once a week. The following
abundances were observed:
A: 10, 6, 5, 4, 3, 4, 6, 8, 10, 12, 15, 16
B: 20, 18, 16, 11, 6, 6, 5, 3, 2, 2, 1, 1
1. Estimate r1, r2, K1, K2, 12, 21.
2. Simulate the dynamics using estimated parameters and initial densities
of 20 individuals for each species within POPULUS.
3. Find such combinations of r1, r2 and 12, 21 so that the two species
would coexist.
a<-c(10, 6, 5, 4, 3, 4, 6, 8, 10, 12, 15, 16)
b<-c(20, 18, 16, 11, 6, 6, 5, 3, 2, 2, 1, 1)
a1<-a[-1]/a[-12]
b1<-b[-1]/b[-12]
summary(lm(log(a1)~a[-12]+b[-12]))
0.60443/0.02992
20.20154*0.04106/0.60443
summary(lm(log(b1)~b[-12]+a[-12]))
0.399980/0.005052
79.1726*0.011438/0.399980
Two species of spiders, Pardosa and Ero, occur together and were
found to feed on the following prey:
1. Estimate and plot niche breadth for each species.
2. Estimate niche overlap (a12, a21) for each species.
3. Simulate the population dynamic of the two species using
estimated 12, 21 and given r1=0.7, r2=0.8, K1=300, K2=200, N01=20,
N02=20.
=
= n
k
kp
1
2
1
breadth

= 2
1
21
12
k
kk
p
pp
a

= 2
2
21
21
k
kk
p
pp
a
Araneae Collembola Isopoda Hemiptera Ensifera
Pardosa 0.61 0.15 0.12 0.07 0.05
Ero 0.93 0.05 0.01 0 0.01
Par<-c(0.61,0.15,0.12,0.07,0.05)
Ero<-c(0.93,0.05,0.01,0,0.01)
both<-rbind(Par,Ero)
barplot(both,beside=T,legend.text=c("Par","Ero"))
1/sum(Par^2)
1/sum(Ero^2)
a12<-sum(Par*Ero)/sum(Par^2); a12
a21<-sum(Par*Ero)/sum(Ero^2); a21
comp<-function(t,y,param){
N1<-y[1]
N2<-y[2]
with(as.list(param),{
dN1.dt<-r1*N1*(1-(N1+a12*N2)/K1)
dN2.dt<-r2*N2*(1-(N2+a21*N1)/K2)
return(list(c(dN1.dt,dN2.dt)))})}
N1<-20;N2<-20
param<-c(r1=0.7,r2=0.8,a12=1.4,a21=0.7,K1=300,K2=200)
time<-seq(0,50,0.1)
library(deSolve)
out<-data.frame(ode(c(N1,N2),time,comp,param))
matplot(time,out[,-1],type="l",lty=1:2,col=1)
legend("right",c("N1","N2"),lty=1:2)