BSC 417517 Environmental Modeling

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BSC 417/517 Environmental Modeling

• Predator-Prey Oscillations on the Kaibab Plateau

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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

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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 well-documented system
  the snowshoe hare-lynx system in Canada
• Time series available on number of lynx pelts
  purchased by the Hudson Bay Co.

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Snowshoe Hare-Lynx System
7
6
5
4
3
2
Hares
Lynx
1
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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
  predator-prey 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

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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

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Initial Model Equilibrium Conditions
2000
50
4000
2000
0.0
0.5
800
0.0
40
5
1.0
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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

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Predator Kill Rate Functional Response

• Number of deer killed per predator per year is 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

Kill rate
Kill rate
Type II
Type I
Prey density
Prey density
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Predator Kill Rate Graphical Function
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Predator Birth Rate Response

• Net birth rate is dependent on kill rate higher
  kill rate gt 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 highimplications for
  potential oscillation?

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Predator Birth Rate Graphical Function
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Initial Model Results Verify Equilibrium
Conditions
Initial predator density 50
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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 doesnt such unstable behavior
  typically occur in nature?

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Initial Model Results - Nonequilibrium Initial
Prey Density
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Natural Predator-Prey Systems

• Predators dont 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

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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, butcomplexity 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

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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

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Type III Functional Response
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Revised Model Results
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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

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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

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State Space (Phase Plane) Diagram
Point attractor
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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?

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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

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Influence of Random Variation

• System shows sustained oscillation over long time
  scales, with periodicity of ca. 10 years
• Reference mode has been generated

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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

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Model With Selective Removal of Predators
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New Equations
predator_kills IF(TIMEgtstart_year) THEN
(predator_population-maximum_acceptable_predators
) ELSE 0 start_year 1920 maximum_acceptable_pr
edators 55
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Simulation Results With Selective Removal of
Predators
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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

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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