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
Increas e Neutral Decreas e
Increas e + +
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+ 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 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 + α12N2] / K1) = 0
r1N1 ([K1 - N1 - α12N2] / K1) = 0
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
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)
K1 = K2
α12 > α2112
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)
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
K1 = K2
α12, α21 < 1
N1
N2
K2
K1
12
1
α
K
21
2
α
K
stable equilibrium
0
species 1
species 2
time
N
K
12
1
2
α
K
K <
21
2
1
α
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
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
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
multiple 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
,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
−−=







 +
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. The species breed in discrete periods. 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, α12, α21.
2. Simulate the dynamics using difference model system for a period of
20 years. Use estimated values of parameters and initial densities of 20
individuals.
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]
coef(lm(log(a1)~a[-12]+b[-12]))
0.60443/0.02992
20.20154*0.04106/0.60443
coef(lm(log(b1)~b[-12]+a[-12]))
0.399980/0.005052
79.1726*0.011438/0.399980
N12<-data.frame(N1<-numeric(1:20),N2<-numeric(1:20))
N12[,1]<-20
N12[,2]<-20
for(t in 1:20) N12[t+1,]<-{
N1<-N12[t,1]*exp(0.6*(20.2-N12[t,1]-1.4*N12[t,2])/20.2)
N2<-N12[t,2]*exp(-0.4*(79.2-N12[t,2]-2.3*N12[t,1])/79.2)
c(N1,N2)}
matplot(N12, type="l",lty=1:2)
legend(1,80,c("N1","N2"),lty=1:2)
Two species of spiders, Pardosa and Pachygnatha, occur together
and were found to feed in the field on the following prey:
1. Estimate and plot niche breadth (D) for each species.
2. Estimate niche overlap (a12, a21) for each species.
∑=
= n
k
kp
D
1
2
1
∑
∑= 2
1
21
12
k
kk
p
pp
a
∑
∑= 2
2
21
21
k
kk
p
pp
a
Druh Colle mbola He mipte ra Ensife ra Dipte ra Isopoda
Pardosa 0.61 0.15 0.12 0.07 0.05
Pachygnatha 0.93 0.05 0.01 0 0.01
Par<-c(0.61,0.15,0.12,0.07,0.05)
Pach<-c(0.93,0.05,0.01,0,0.01)
both<-rbind(Par,Pach)
barplot(both,beside=T,legend.text=c("Par","Pach"))
1/sum(Par^2)
1/sum(Pach^2)
a12<-sum(Par*Pach)/sum(Par^2); a12
a21<-sum(Par*Pach)/sum(Pach^2); a21
Simulate the population dynamic using differential model system
for the period of 30 years. The initial densities are N01=200 and
N02=10.
An invasive ant species is spreading and may replace a native ant
species as both have similar niches. The following parameters are
know for the native (1) and invasive (2) species.
r1 = 0.2
K1 = 200
α12 = 1.1
r2 = 0.9
K2 = 300
α21 = 0.7
N1
N2
K2
K1
21
2
α
K
12
1
α
K
182
300
200 429
How to achieve stable coexistence?
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<-200;N2<-10
param<-c(r1=0.2,r2=0.9,a12=1.1,a21=0.7,K1=200,K2=300)
time<-seq(0,30,0.1)
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)
N1<-200;N2<-10
param<-c(r1=0.2,r2=0.9,a12=0.5,a21=0.7,K1=200,K2=300)
time<-seq(0,30,0.1)
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)