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Predator-prey equations
Readings
Predator-prey: Chapter 12: Case TJ (2000) An illustrated
guide to theoretical ecology. Oxford University Press,
Oxford.
Multispecies exploitation: Branch TA et al. (2013)
Opportunistic exploitation: an overlooked pathway to
extinction. Trends in Ecology and Evolution. doi:
10.1016/j.tree.2013.03.003
The predators and prey
Simple predator-prey theory
(Lotka-Volterra)
• Prey governed by exponential growth
• Predator deaths are density independent, births
depend upon number of prey eaten
• Prey eaten per predator is proportional to prey
density
Lotka AJ (1925) Elements of physical biology. Williams & Wilkins Co., Baltimore
Volterra V (1926) Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Mem. Acad. Lincei Roma 2:31-113
Lotka-Volterra equations
Intrinsic rate
of increase
Wildebeest
numbers
Lion
numbers
dW
 rW  eWL
dt
Predation efficiency
dL
 mL  eaWL
dt
Natural mortality
Assimilation
efficiency
Equivalent in time steps
Wt 1  Wt  rWt  eWt Lt
Lt 1  Lt  mLt  eaWt Lt
 Lt 1  sLt  eaWLt
Dynamic behavior
These models are either unstable or cyclic
90,000,000
10,000,000
80,000,000
9,000,000
Wildebeest
70,000,000
8,000,000
Lions
60,000,000
7,000,000
6,000,000
50,000,000
5,000,000
40,000,000
4,000,000
30,000,000
3,000,000
20,000,000
2,000,000
10,000,000
1,000,000
0
0
0
50
150
100
Time
200
250
300
Biological unrealism of Lotka-Volterra
• No prey self limitation
• No predator self limitation
• No limit on prey consumption per predator
– This is called the “functional response”
Adding some biological realism
Logistic equation
Survival
Number of
wildebeest
deaths
 Wt 
Wt 1  Wt  rWt  1    Dt
K 

Assimilation: number of lions
Lt 1  sLt  aDt produced per wildebeest death
Dt  Wt 1  exp(hLt )
“Functional response”
Proportion of prey searched
for, found and killed per
year per predator
Lab 5 Predator prey wildebeest lions.xlsx
Dynamic behavior in time
18,000
16,000
1,000,000
14,000
800,000
12,000
Wildebeest numbers
10,000
600,000
8,000
400,000
6,000
Lion numbers
Lion numbers
Wildebeest numbers
1,200,000
4,000
200,000
2,000
0
0
0
50
100
150
200
250
300
Lab 5 Predator prey wildebeest lions.xlsx
Predator-prey phase diagram
20,000
Lions
15,000
10,000
5,000
0
0
500,000
1,000,000
1,500,000
Wildebeest
Lab 5 Predator prey wildebeest lions.xlsx
Predation dynamics
Developing a functional response
Wildebeest
deaths
Dt  Wt 1  exp(hLt )
Fraction wildebeest killed
• Predators do a random
walk, encounter and kill a
fraction of what prey they
encounter.
• Exponential model is used
to correct for the fact that
no prey can be killed and
eaten twice by different
lions.
Proportion killed
by one lion
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50000
100000
150000
200000
Number of lions
Lab 5 Predator prey wildebeest lions.xlsx
Lion kill rate
(one way to think about estimating h)
•
•
•
•
•
•
•
A lion walks 10 km per day
Can see 200 m in either direction
Thus sees all wildebeest covering 4 km2/day
This amounts to 1460 km2/year
Serengeti ecosystem is 90,000 km2
A lion can chase and catch 1 in 1000 animals it sees
Thus one lion kills h = (1460/90,000)/1000 =
0.000016 of the wildebeest population per year
Key assumptions
• Kill rate proportional to prey abundance
• No self regulation of predator
• No predator saturation
The prey isocline
When is prey abundance constant?
Original equation
Equilibrium implies
Wt+1 = Wt = W
Divide by W
Rearrange
Solve for W
 Wt 
Wt 1  Wt  rWt  1    Wt 1  exp  hLt 
K 

 W
W  W  rW  1    W 1  exp  hLt 
K 

 W
1  1  r  1    1  exp  hLt 
K 

 W
r  1    1  exp  hLt 
K 

K
W  r  1  exp  hL 
r
Lab 5 Predator prey wildebeest lions.xlsx
The predator isocline
When is predator abundance constant?
Original equation
Set Lt+1 = Lt = L
Divide by L
Solve for W
Lt 1  Lt s  aWt 1  exp  hLt 
L  Ls  aW 1  exp  hL
a
1  s  W 1  exp  hL
L
1  s

L
W
a 1  exp  hL
Lab 5 Predator prey wildebeest lions.xlsx
Predator
isocline
20,000
Lions
15,000
10,000
5,000
Prey isocline
0
0
500,000
1,000,000
1,500,000
Wildebeest
Lab 5 Predator prey wildebeest lions.xlsx
Multiple prey species
Multispecies equations
Hyperpredation
Channel Islands:
introduced feral pigs
allowed golden eagles
to establish and
increase, greatly
increasing predation
(hyperpredation) on
native foxes. Low fox
numbers allowed
competitively inferior
skunks to flourish.
No pigs
Pigs
Roemer GW et al. (2002) Golden eagles, feral pigs, and insular carnivores: How exotic species turn native predators into prey. PNAS 99:791-796
Opportunistic exploitation
S.H. pelagic whaling:
despite rarity of
blue whales,
whaling continued
on fin whales;
whalers
opportunistically
caught valuable
blue whales when
encountered
Branch TA et al. (2013) Opportunistic exploitation: an overlooked pathway to
extinction. Trends in Ecology and Evolution. doi: 10.1016/j.tree.2013.03.003
Opportunistic exploitation
Branch TA et al. (2013) Opportunistic exploitation: an overlooked pathway to
extinction. Trends in Ecology and Evolution. doi: 10.1016/j.tree.2013.03.003