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MATH3104: Populations@Play
Anthony J. Richardson
Outline
• Investigate predator-prey interaction– without density dependence of prey– with density dependence of prey– with predator satiation and density
dependence of prey
• Focus on cycles and equilibria
Predator-Prey Interaction• Many big predators but most small
• Predators often liked more than prey???
• Questions– How many predators can prey sustain?– How much predation can the prey tolerate?– Are there cycles?
Volterra Predator-Prey Model
• 1st predator-prey model in ecology by Vito Volterra in 1920s
• Prey grow exponentially in absence of predator; predators die exponentially in absence of prey
• No density dependence
• Predators contact prey by “random collision”
Volterra Predator-Prey Model
2111 NaNrN
dt
dN
2212 dNNbaN
dt
dN
1N
2Nr
ab
Number of prey
Number of predators
Intrinsic rate of prey increase
Coefficient relating prey capture to predator-prey collisions
Coefficient relating prey capture to predator births
d Death rate of predator
Volterra Predator-Prey Model
• What does this model tell us?
• What are equilibria of these equations?
• What does your answer mean?
• Let’s plot system of equations and see what it looks like
0 500 1000 1500 2000 2500 3000 35000
20
40
60
80
100
120
140
Prey abundance
Pre
da
tor
ab
un
da
nc
e
nhat
nhat/4nhat/2
Volterra Predator-Prey Model
• 4 main lessons1. Predators can control exponential prey
growth
2. Any prey population can support a predator population
3. Formula for equilibrium abundances give estimate of average abundances through time
4. Predator-prey interaction has tendency to oscillate
Adding Density Dependence to Prey
• How big does K of prey have to be to support the predator?
211
11 )1( NaN
K
NrN
dt
dN
2212 dNNbaN
dt
dN
ba
dN 1ˆ
)1(ˆ2 baK
d
a
rN
Volterra Predator-Prey with Density Dependence
• Let’s plot system and see what it looks like• Density dependence has large effect on dynamics• Value of K critical for coexistence
0 200 400 600 800 1000 1200 1400 1600 1800 20000
10
20
30
40
50
60
70
80
Prey abundance
Pre
da
tor
ab
un
da
nc
e
ba
dK ba
dK
ba
dK
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Prey abundance
Pre
da
tor
ab
un
da
nc
e
0 200 400 600 800 1000 1200 1400 1600 1800 20000
20
40
60
80
100
120
140
160
180
Prey abundance
Pre
da
tor
ab
un
da
nc
e
No Predators
Stable, but no cycles
Stable focus, spiral cycles
Volterra Model with Predator Satiation and Density Dependence
• Term for feeding is
• Can use non-linear feeding term – e.g.
2121 )( NaNNaN
Linear feeding term
)1(1
c
aN
ec
What does this feeding term look like?
Volterra Model with Predator Satiation and Density Dependence
21
11 )1()1(
1
NecK
NrN
dt
dN c
aN
222 )1(
1
dNNebcdt
dN c
aN
0 500 1000 1500 2000 2500 3000 3500 40000
50
100
150
200
250
Prey abundance
Pre
da
tor
ab
un
da
nc
e
0 500 1000 1500 2000 2500 30000
20
40
60
80
100
120
140
160
180
200
Prey abundance
Pre
da
tor
ab
un
da
nc
e
0 500 1000 1500 2000 2500 30000
10
20
30
40
50
60
70
80
90
100
Prey abundance
Pre
da
tor
ab
un
da
nc
e
K = 1000, Stable node K = 2000, Stable focus K = 4000, unstable focus surrounded by limit cycle
What does this system of equations look like?