BSC 417/517
Environmental Modeling
Predator-Prey Oscillations
on the Kaibab Plateau
The Predator-Prey Relationship
• Predator-prey relationships have always
occupied a special place in ecology
• Ideal topic for systems dynamics
• Examine interaction between deer and
predators on Kaibab Plateau
• Learn about possible behavior of predator
and prey populations if predators had not
been removed in the early 1900s
Deer and Predators on Kaibab Plateau
• Information on deer population irruption is
not reliable
• Data on predators is even more sketchy
• Gain insight into predator prey relationship
on the Plateau from a more welldocumented system: the snowshoe harelynx system in Canada
• Time series available on number of lynx
pelts purchased by the Hudson Bay Co.
Snowshoe Hare-Lynx System
7
6
5
4
2
1
Lynx
Hares
3
Snowshoe Hare-Lynx System
• Records show peak in number of lynx pelts every 9•
•
•
•
10 years
Data suggest that populations have oscillated in a
cyclical manner for over 100 years
Data are viewed as a classical example of predatorprey interaction
Oscillations are not related to seasonal or other
obvious annual changes
Best examples of predator-prey oscillations in
mammal populations show periodicity of 3-4 or 9-10
years
Reference Mode for Kaibab
Deer-Predator System
• Use hare-lynx example to draw a reference mode
•
•
•
•
for deer-predator relationship
Should the oscillations be sustained, damped, or
growing?
Intuition says sustained, but many other types of
behavior have been observed
For sake of simplicity, go with sustained oscillation
with 9-10 year periodicity as reference mode
Peaks in predator (cougar) populations should lag
behind peaks in deer population by a few years
Initial Model – Equilibrium Conditions
predator population
deer population
net deer births
predation
2000
4000
50
2000
deer net birth rate
~
area in 1000 acres
0.5
predators net birth rate
800
5
~
40
deer killed per predator per y r
1.0
f raction f orage needs met
~
0.0
deer density
predator net births
0.0
Model Structure
• Ignore biomass impact of deer growth
• Assume ample forage is present by setting
fraction forage needs met equal to 1.0
• Predator stock is dependent on deer density
vis-à-vis deer density-dependent kill rate
and kill-rate dependent net birth rate
Predator Kill Rate Functional Response
• Number of deer killed per predator per year is
Type I
Prey density
Kill rate
Kill rate
60 if there are more than 10 deer/1000
acres… ~1 kill/week = satiation limit
• Shape of graphical function reflects a
combination of “Type I” and “Type II”
functional response
Type II
Prey density
deer killer per predator per yr
Predator Kill Rate Graphical Function
70
60
50
40
30
20
10
0
0
2
4
6
deer density
8
10
Predator Birth Rate Response
• Net birth rate is dependent on kill rate: higher kill
•
•
•
•
rate => higher net birth rate
Maximum net birth rate = 0.45/yr
Cougars start to breed young (2-3 years age)
Breed every 2 years with an average of 3 kittens
Maximum net birth rate for predators and prey are
comparable and relatively high…implications for
potential oscillation?
Predator Birth Rate Graphical Function
predator net birth rate
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
10
20
30
40
50
60
deer killed per predator per yr
70
80
Initial Model Results Verify
Equilibrium Conditions
2: deer population
1: predator population
1:
2:
150
10000
Initial predator density = 50
1:
2:
75
5000
1
1:
2:
1
1
0
0
1900.00
Page 1
1
2
2
2
2
1908.00
1916.00
Y ears
Untitled
1932.00
1924.00
10:13 PM Mon, Nov 04, 2002
Initial Model Results Nonequilibrium Initial Prey Density
• Set initial predator density at 45
• System displays unstable behavior (as
illustrated by 30 vs. 50 year simulation)
• Predators virtually annihilate prey after ca.
25 year, which lead to ensuing unstable
behavior
• Question: why doesn’t such unstable
behavior typically occur in nature?
Initial Model Results Nonequilibrium Initial Prey Density
1: predator population
1:
2:
2: deer population
200
9000
2
1:
2:
100
4500
2
1
1
2
1
1
1:
2:
0
0
2
1900.00
Page 1
1908.00
1916.00
Y ears
Untitled
1924.00
1932.00
7:21 AM Thu, Nov 04, 2004
Natural Predator-Prey Systems
• Predators don’t normally hunt prey to zero
• Rather, select individuals from prey population
that are easiest to catch (young, old, weak)
• Minimum threshold concept: prey density limit
below which predators would no longer find it
profitable to hunt the prey and would switch to
different prey
• Threshold is determined by availability of prey
hiding places (refuge) and prey social behavior
Revising The Model
• Should we revise the model to take into account the
threshold concept, effect of prey refuge, and prey
social behavior?
• Perhaps expand deer population to multiple stocks to
simulate deer age structure, and then allow predators
to concentrate on young and old deer
• Sounds good, but…complexity would increase
dramatically in face of limited data…
• Better to consider if combined effect of these factors
could be taken into account within existing, simple
model structure
Revised Model
• Try using a different functional response for
density-dependent kill rate which incorporates
the concept of threshold prey density
• No kills if deer density falls below 2 deer per
1000 acres, e.g. because of the ability of deer to
find safe refuge when overall density is low
• S-shaped function response corresponds to
“Type III” functional response
deer killed per predator per yr
Type III Functional Response
70
60
50
40
30
20
10
0
0
2
4
6
deer density
8
10
12
Revised Model Results
1: predator population
1:
2:
2: deer population
100
8000
1
1
1
1:
2:
50
4000
1
2
2
2
2
1:
2:
0
0
1900.00
Page 1
1910.00
1920.00
Y ears
Untitled
1930.00
1940.00
11:12 PM Mon, Nov 04, 2002
Revised Model Results
• Initial predator population is set at 100
• Large predator population causes an initial
decline in deer population, but predator
population declines quickly
• Damped oscillatory behavior ensues with
periodicity of ca. 10 years
• Result essentially corresponds to the
original reference mode
Further Interpretation
• The initial “dynamic hypothesis” was that the
cougar and deer populations could interact to
produce stable cycles with a period similar to
the classic 9-10 year cycle observed in other
mammalian predator-prey systems
• Requirement for a Type III functional
response to produce stable behavior can be
interpreted as an indication of the importance
of prey refuge or threshold levels
State Space (Phase Plane) Diagram
predator popul… v . deer population: 1 100
Point attractor
50
0
0
Page 1
4000
deer population
Untitled
8000
11:26 PM Mon, Nov 04, 2002
Patterns of Oscillation
• Previous simulations show possibility for
both damped and growing oscillations,
depending on the nature of the predator
functional response
• What about potential for sustained
oscillation, as state in the reference mode?
• Could random disturbances lead to
persistent cycles?
Influence of Random Variation
• Introduce randomness into the deer net birth
rate via the following equations
• net birth rate = 0.5 + random factor
• random factor = random(-0.2,0.2,123)
• The random factor allows net birth rate to vary
randomly from a low of 0.3 to a high of 0.7
• The value 123 is a “seed” for the random
number generator
Influence of Random Variation
1: predator population
1:
2:
• System shows
sustained oscillation
over long time scales,
with periodicity of
ca. 10 years
• Reference mode has
been generated
2: deer population
100
8000
1
1
1
1:
2:
50
4000
2
2
1:
2:
1
2
2
0
0
1900.00
1920.00
1940.00
Y ears
Page 1
1960.00
1980.00
11:38 PM Mon, Nov 04, 2002
Untitled
1: predator population
1:
2:
100
8000
1:
2:
50
4000
2: deer population
2
2
1
2
1
1
1
1:
2:
0
0
1900.00
Page 1
2
1960.00
2020.00
Y ears
Untitled
2080.00
2140.00
11:43 PM Mon, Nov 04, 2002
Policy Test: Selective Removal of
Predators
• Results of revised model with random variation in
deer birth rate suggests that stable predator-prey
interactions would have been possible if the
predators had not been removed from the Kaibab
Plateau
• Although predator population averages 50,
substantially higher numbers occur in some years,
which could pose problem for ranchers livestock
• Test influence of allowing hunters to kill some
predators to protect live stock
Model With Selective Removal
of Predators
start y ear
maximum acceptable predators
net deer births
deer population
predator kills
predation
predator population
deer net birth rate
area in 1000 acres
predators net birth rate
predator net births
~
~
deer killed per predator per y r
random f actor
deer density
New Equations
predator_kills = IF(TIME>start_year) THEN
(predator_population-maximum_acceptable_predators) ELSE 0
start_year = 1920
maximum_acceptable_predators = 55
Simulation Results With
Selective Removal of Predators
1:
2:
3:
100
8000
10
1:
2:
3:
50
4000
5
1:
2:
3:
0
0
0
1
2
1
1
2
1930.00
2
1
3
3
3
1900.00
Page 1
3: predator kills
2: deer population
1: predator population
1960.00
Y ears
Untitled
2
3
2020.00
1990.00
9:33 AM Thu, Nov 04, 2004
Interpretation
• Results suggest that it might have been possible to
reduce peak values of predator population without
destroying the stability of the predator-prey
system
• However, managers in early 1900s had essentially
no knowledge of predator-prey dynamics
• Even today, other factors besides predator-prey
population dynamics are know to be important in
governing response of the system…
Current Interpretation of the
Hare-Lynx Predator Prey System
• Krebs et al. (Bioscience 2001) (see PDF on web-site)
conclude that Lotka and Volterra were only partly
correct when the concluded that the snowshoe hare
cycle was the product of a predator-prey oscillation
• Missed critical point that the cycle can only be
understood by considering three trophic levels rather
than just two
• Hare cycle is produced by interaction between
predation and food supplies
• Dependence on food supply ripples across many
species of predators and prey in boreal forest