BSC 417/517 Environmental Modeling

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BSC 417/517 Environmental Modeling

• The Kaibab Deer Herd

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Goal of Chapter 16

• Illustrate the steps of modeling discussed in
  Chapter 15
• Illustrate iterative nature of modeling process
• Learn to appreciate many decisions required to
  build a model
• Do exercises which verify, apply, and improve the
  model

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Getting Acquainted With the System

• Kaibab Plateau is located within the Kaibab
  National Forest, located north of the Colorado
  River in north-central Arizona
• Approximately 60 miles long (N-S) and 45 miles
  wide at its widest point
• One of the largest and best-defined block
  plateaus in the world
• Vegetation types change with elevation and
  include shrubs, sagebrush, grasslands,
  pinion-juniper, Ponderosa pine, and spruce-fir

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Kaibab NationalForest
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The Kaibab Plateau
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Kaibab Plateau Deer Herd

• Kaibab plateau deer herd consists of Rocky
  Mountain mule deer
• Pinion-juniper woodlands provide winter range
  summer range includes pine and spruce-fir forests
• Deer mate in Nov/Dec fawns arrive in Jun/Jul
  deer achieve maturity _at_ ca. 1.5 yr

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Rocky Mountain Mule Deer
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Kaibab Plateau Deer Herd

• Data on deer population size prior to 1900 is
  sparse Rasmussen (1941) estimated total size of
  3000-4000 deer
• Plateau was home to several predators including
  coyotes, bobcats, mountain lions, and wolves,
  which kept deer populations under control
• Starting at turn of the century, predators were
  systematically removed by hunting and trapping
• During 1907-1923, predator kills were estimated
  at 3000 coyotes, 674 lions, 120 bobcats, and 11
  wolves

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Kaibab Plateau Deer Herd

• Deer population grew rapidly during decimation of
  predators in the early 1900s (irruption)
• Rasmussen (1941) estimated deer population at ca.
  100,000 in 1924
• Reconnaissance party reported that forage
  conditions were deplorable
• No new growth of apsen
• White fir, typically eaten unless under stress of
  food shortage, were often found skirted
• Condition of deer was also found to be deplorable

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Kaibab Plateau Deer Herd

• Major deer die-off occurred during winters of the
  years 1924-1928
• Government hunters were deployed in 1928 to
  reduce the size of the deer population
• But, paradoxically, predator control measures
  continued

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Kaibab Plateau Deer Herd

• The year 1930 was a good year for plant growth,
  and deer herd began to recover and stabilize
• By 1932, deer population was estimated at 14,000
  and the range was in reasonable conditions
• Forest service game reports declared that the
  number of deer appeared to be about right for
  the range

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Be Specific About the Problem

• Develop model to gain insight into causes behind
  the deer population irrupution and measures
  that could have been used to prevent it
• Starting point come up with a reference mode,
  i.e. a target pattern for the systems behavior
• In this case, were dealing with the classical
  overshoot pattern discussed earlier in the
  course

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Reference Mode
Pop. peaks at ca. 100,000
Return to pseudo-stability with government
hunting or return of predators
Initial pop. ca. 4000
Rapid growth after removal of predators
1900
1910
1920
1930
1940
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Notes on Reference Mode

• Sketch is not a compilation of precise estimates
  in terms of deer population or timing of events
• Simply a rough depiction of a likely pattern of
  behavior based on accounts of various observers
• Leads to initial modeling goal of a simulating
  deer population which remains stable during the
  initial years, and grows rapidly when predators
  are removed from the system
• Population should peak at something like 100,000
  and then decline rapidly due to starvation

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Specific Goals of Modeling Exercise

1. Gain understanding of forces that led to the
   overshoot and collapse
2. Explore number of predators on the plateau as a
   relevant policy variable which could be
   manipulated in order to achieve a stable deer
   population

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Construct An Initial Stock-and-Flow Diagram

• As a first step, construct a stock-and-flow
  diagram which can reproduce the reference mode
• Could attempt a predator-prey model, since were
  dealing with deer population which is regulated
  (at least originally) by predation
• However, this would lead to complexities that go
  beyond the goal of the current modeling exercise
  (see Chapter 18)

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Design of First Model

• Allow number of predators to be specified by the
  user, and set number of deer killed per predator
  per unit at a constant value
• Note that for simplicity, all predators are
  treated equally, i.e. coyotes, bobcats, and
  mountain lions are combined into an aggregate
  category, or functional group

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Design of First Model

• Initial deer population 4000 (spread out over
  800 thousand acres 5 deer per thousand acres)
• Net birth rate 50 based on favorable range
  conditions
• Net birth rate comes from assumptions that
• Half the deer population is female
• 2/3 of females are fertile at any given time
• Average litter size 1.6
• Average deer life span 15 yr gt average death
  rate 1/15 0.0667 0.07
• With the above assumptions
• Net birth rate 0.5 (2/3) 1.6 0.07
  0.47 0.5

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Design of First Model

• Number of deer killed per predator per yr is set
  at 40 based on the following assumptions
• All predators can be measured by the equivalent
  number of cougars (mountain lions)
• 75 of cougar diet is mule deer
• Average cougar requires one kill per week
• With the above assumptions
• Deer kills/predator/yr 0.75 52 38 40

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Design of First Model

• Number of predators (in cougar equivalents) in
  the original ecosystem is unknown
• To get started, set equal to value that, when
  multiplied by the assumed number of deer killed
  per predator per year (40), produces a number of
  deer deaths equal to the number of net deer
  births per year at the start of the simulation
  (0.5 4000 2000)
• In other words, set the initial number of
  predators equal to 2000/40 50 cougar equivalents

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First Model
deer_killed_per_predator_per_year
GRAPH(TIME) (0.00, 40.0), (485, 40.0), (970,
40.0), (1455, 40.0), (1940, 40.0) number_of_predat
ors GRAPH(TIME) (1900, 50.0), (1910, 50.0),
(1920, 0.00), (1930, 0.00), (1940, 0.00)
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Results of First Model

• Deer population is constant for first 10 yr
• Grows exponentially after predators are removed
  during 1910-1920, reaching 10X the initial
  population by 1920
• Population goes off-scale around 1920 and never
  comes back
• Simulation clearly fails to reproduce the
  reference mode

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A Second Model With Forage

• Next version of the model will keep track of the
  forage requirements and the available forage on
  the plateau
• Proceed with assumption that total forage
  requirement is 1 MT dry biomass/deer/yr
• Estimate is based on Vallentine (1990)s
  suggestion that mule deer require ca. 23 of an
  animal unit equivalent (AUE) dry matter
  consumed by a 1000-pound non-lactating cow (ca.
  12 kg dry biomass/d)
• 0.23 12 kg/d 365d/yr 1007 kg/yr 1000 kg/yr

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Second Model

• Assume that plateau produces vast excess of plant
  matter each year
• With all plants combined into a single category
  (valid?), forage production is set at 40,000
  MT/yr 10X deer requirement
• Forage availability ratio forage
  production/forage required

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Second Model

• As long as forage availability ratio gt 1,
  fraction of forage needs met is 100
• If forage availability lt 1, then fraction of
  forage needs met forage availability
• Fraction of forage needs met influences net birth
  rate according to a graph function

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Second Model

• Net birth rate 0.5 when deer are meeting 100
  of their forage needs
• As fraction of forage needs met decreases, net
  birth rate declines, and falls to zero when deer
  are meeting only half of their forage needs
• Net birth rate reaches -40/yr if deer
  meet 30 or less of their forage needs

Note the relationship depicted here is a
plausible guess only, as little info is
available on deer birth and death rates undef
difficult conditions
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Second Model
forage_availability_ratio forage_production/fora
ge_required fr_forage_needs_met
MIN(1,forage_availability_ratio)
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Second Model Results

• Deer population remains constant until predator
  removal starts, then increases rapidly to ca.
  80,000
• As population grows, fraction of forage needs met
  decreases rapidly to 0.5, which causes net birth
  rate to go to zero, which in turn stops growth gt
  constant population for the remainder of the
  simulation

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Second Model Results
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Second Model Results

• Results are closer to reference mode than the
  first model, but simulation does not reproduce
  the major die-off that occurred during the late
  1920s
• Sensitivity analysis reveals that lack of die-off
  is not caused by an erroneous value for the
  forage required per deer per year general
  pattern remains the same with values of 0.75,
  1.0, and 1.25 MT dry biomass/deer/yr
• Failure to reproduce die-off is likely due
  tolack of change in forage biomass?

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Second Model Results - Contd
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Third Model Forage Production and Consumption

• Simulate growth and decay of biomass using a
  simple S-shaped growth model (check-out Ford
  Chapter 6 to get reacquainted with S-shaped
  growth models)
• Production of new plant biomass is dependent on
  ratio of current biomass to a maximum biomass of
  400,000 MT
• First-order decay of standing biomass

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Third Model Biomass Sector
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Third Model Biomass Sector
addition_to_standing_biomass new_growth-forage_c
onsumption decay standing_biomassbio_decay_rate
bio_decay_rate 0.1 bio_productivity
intrinsic_bio_productivityprod_mult_from_fullness
forage_consumption forage_requiredfr_forage_ne
eds_met fullness_fraction standing_biomass/max_b
iomass intrinsic_bio_productivity
0.4 max_biomass 400000 new_growth
standing_biomassbio_productivity prod_mult_from_f
ullness GRAPH(fullness_fraction) (0.00, 1.00),
(0.2, 1.00), (0.4, 0.9), (0.6, 0.6), (0.8, 0.2),
(1.00, 0.00)
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Third Model Biomass Sector Simulation
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Third Model Full Version
Kaibab Deer Herd Third Model
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Third Model Results

• Deer population increases to 80,000 by 1920,
  after which net birth rate falls to slightly
  below zero
• Small decrease in deer population occurs during
  the 1920s and 1930, but not as dramatic as was
  observed
• Alteration of annual forage rate per deer does
  not change outcome
• Model still fails to reproduce reference mode

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Fourth Model Deer May Consume Older Biomass

• Deer prefer new growth, but under stressed (i.e.
  starvation) conditions will consume older biomass
  (gt skirting)
• As deer population becomes large, keep track of
  additional consumption requirements which arise
  when the fraction of forage needs met by new
  growth falls below 1
• Assume 25 of standing older biomass is available
  to deer, and that the nutritional value of the
  old biomass is only 25 of that of new growth
• New drainage flow must be added to depict loss of
  standing biomass through consumption of older
  growth

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Fourth Model Key Equations
additional_con_required forage_required-forage_c
onsumption stand_bio_available
standing_biomassfr_standing_available fr_standing
_available 0.25 old_biomass_availability_ratio
stand_bio_available/MAX(1,additional_con_require
d) old_biomass_consumption additional_con_requir
edfr_additional_needs_met fr_additional_needs_met
MIN(1,old_biomass_availability_ratio) equivalen
t_fraction_needs_met MIN(1,fr_forage_needs_metf
r_additional_needs_metold_biomass_nutritional_fac
tor) old_biomass_nutritional_factor 0.25
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Fourth Model Density-Dependent Predator Kill Rate
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Fourth Model Full Version
Kaibab Deer Herd Fourth Model
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Fourth Model Results

• Deer population peaks at ca. 115,000 in the early
  1920s, then declines rapidly
• Net birth rate falls to zero in 1921, and reaches
  0.25 by the end of the 1920s and remains there
  for the remainder of the simulation period
• The desired overshoot pattern has been achieved!

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Fourth Model Results

• Forage variables are consistent with
  expectations
• Huge increase in forage requirements and new
  forage consumption in parallel with deer
  population explosion
• Old biomass consumption kicks in a few years
  after start of irruption
• Standing biomass drops to low values

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Fourth Model Results

• A milestone has been achieved in the modeling
  process!
• Model generates the reference mode, at least in
  general terms
• Improvements are possible (note that reference
  mode does not depict total decimation of the deer
  population), but model is ready for sensitivity
  analysis

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Sensitivity Analysis

• Now that the model generates the reference mode,
  it is appropriate to conduct more extensive
  sensitivity analysis
• Purpose of the analysis is to determine if the
  models behavior is strongly influenced by
  changes in the most uncertain parameters
• If the same general pattern emerges in many
  different simulations, then the model is said to
  robust
• Robust models are particularly useful in
  environmental science, where models tend to
  contain numerous highly uncertain parameters

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Model 4 Sensitivity Analysis

• First, vary forage requirement 25
• Peak population sizes vary considerably
• However, general pattern of behavior is identical
• Model is robust with respect to changes in forage
  requirement.

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Model 4 Sensitivity Analysis

• Next, vary old biomass nutritional factor from 0
  to 0.75
• Peak population sizes vary considerably, but
  general pattern of behavior is identical
• Model is robust with respect to changes the old
  biomass nutritional factor

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Model 4 Sensitivity Analysis

• Previous tests are easily implemented with STELLA
  using the built-in sensitivity analysis facility
• May also be important to test sensitivity to
  changes in nonlinear functions
• To illustrate, alter the relationship between
  equivalent needs met and net deer birth rate

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Model 4 Sensitivity Analysis
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Model 4 Sensitivity Analysis
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Model 4 Sensitivity Analysis

• Results of sensitivity analyses indicate that if
  we were trying to accurately predict peak deer
  population, we would not be able to proceed
  without more confidence in certain parameter
  values
• However, our stated purpose was not to predict
  specific numbers, but rather to obtain a general
  understanding of the systems tendency to
  overshoot
• Sensitivity analysis reveals that the same
  general pattern is obtained regardless of the
  particular parameter values or relationship

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Model 4 Sensitivity Analysis Extended

• Conclude sensitivity analysis with a combination
  of changes which stretch the value of several
  parameter beyond what might be considered to be
  plausible estimates
• Changes are designed to reinforce each other by
  increasing the chances that the deer population
  could continue to grow throughout the simulation
  period
• Testing of response to extremes is designed to
  learn the true extent of the models robustness

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Model 4 Sensitivity Analysis Extended

• Changes include
• Double foraging area from 800 to 1600 kA
• Double the initial value of standing biomass from
  300,000 to 600,000 MT
• Double the maximum biomass from 400,000 to
  800,000 MT
• Lower food requirement from 1 to 0.5 MT/yr
• Assume that old biomass has 2X the nutritional
  value compared to the base case (i.e. 0.5 vs.
  0.25)

Effectively Doubles Plateau Size
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Model 4 Sensitivity Analysis Extended - Results
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Model 4 Sensitivity Analysis - Summary

• Extreme testing, together with other sensitivity
  analyses, indicate that the model is very robust
  wrt the tendency to demonstrate overshoot once
  predators are removed
• Have achieved another important milestone in the
  modeling process
• May now proceed with testing the impact of policy
  alternatives

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First Policy Test Predators

• Number of predators was identified as a policy
  variable at the outset of the modeling exercise
• Start with assessment of how changes in the
  number of predators might alter the tendency for
  the deer population to overshoot
• Allow decline in predator population to occur
  less rapidly (drop from 50 to 0 over 20 years
  rather than 10 years)

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First Policy Test Results
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First Policy Test Predators

• Deer population undergoes the same overshoot
  pattern regardless of the decline in predator
  removal rate
• Even if number of predators is returned to 50 in
  1920 after the irruption has begun, the
  population still irrupts because the deer are too
  numerous for the fixed number of predators to
  control
• Obvious conclusion is that the predators should
  never have been removed in the first place
• To more accurately test the ability of predators
  to control the deer population, model would need
  to be expanded to allow the number of predators
  to rise and fall with changes in the deer
  population, predator-prey style (focus of Chapter
  18)

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Second Policy Test Fixed Deer Hunting

• Explore deer hunting as an alternative method of
  controlling the deer population
• Controlled hunting is common in Europe and North
  America
• Add a deer harvest flow to the model to account
  for a policy to harvest a fixed number of deer
  each year after a specified start date

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Second Policty Test Revised Animal Sector
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Second Policty Test Results
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Second Policy Test Fixed Deer Hunting

• Harvest amounts of 1000-4000 are not sufficient
  to prevent the irruption
• If keep increasing harvest amount, find that a
  value of 4700 delays the irruption by ca. 15
  years, but it still occurs
• Tempting to increase harvest amount even
  furtherbut find that a value of 4704 leads
  ultimately to crash of the population after 1930
• Searching for the ideal harvest amount is
  futile even if the ideal harvest amount could be
  identified, the slightest disturbance in any of
  the model variables would reveal that the
  equilibrium is not a stable one

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Third Policy TestVariable Deer Hunting

• Need a better policy for hunting, e.g. one which
  incorporates information (feedback) on the size
  of the deer population
• Modify model to make harvest amount dependent on
  deer population by setting harvest equal to a
  fixed fraction of the population
• Set harvest fraction equal to 0.5 to match the
  maximum net birth rate
• Start hunting in 1915

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Third Policy TestVariable Deer Hunting

• Deer harvest increases quickly around 1915, and
  loss of deer through hunting is twice as great as
  the losses to predation during the previous
  decade
• Deer harvest then declines and the system reaches
  equilibrium
• Deer population remains at around 10,000 for the
  rest of simulation
• Standing biomass is maintained at value near its
  starting level

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Third Policty Test Results(Start Hunting in 1915)
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Third Policy TestVariable Deer Hunting

• If start hunting only 3 years later (1918),
  equilibrium deer population is ca. 40,000 and
  standing biomass declines gradually to a new
  equilibrium, with 20-30 less biomass than at the
  start of the simulation
• If delay start of hunting to 1920 too late!!!
• Deer population is already starting to decline
  due to food resource depletion, and hunting only
  hastens the population crash
• Standing biomass does not recover
• Obviously, hunting control must be implemented
  before signs of severe overbrowsing are evident

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Third Policty Test Results(Start Hunting in 1918)
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Third Policty Test Results(Start Hunting in 1920)
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Additional Policy Tests

• Ford identifies five additional policy tests that
  it would make sense to evaluate
• Impact of lags in measuring deer population and
  discrepancies between target and actual harvest
• Expand hunting policy to make desired deer
  population size an explicit policy variable
• Impact of variable weather on the overall system,
  e.g. wrt biomass productivity, biomass decay
  rate, and deer net birth rate
• Allow hunting policy to be sensitive to the
  amount of standing biomass, so as to prevent
  overbrowsing and associated population irruption
• Alter hunting policy to include distinction
  between hunting of male vs. female deer (bucks
  vs. does)

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What About Excluded Variables?

• Easy to identify many variables and processes
  that are excluded by the high level of
  aggregation
• Influence of seasonality, snowfall
• Distinctions between different types of predators
• Distinctions between different types of
  vegetation
• Impact of cattle and sheep on range forage
  conditions
• Should these things bother us?
• Does the simulation provide wrong answers
  because such variables and processes were not
  included?
• Can the model ever be big enough to deliver the
  right answer?

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What About Excluded Variables?

• Keep in mind that computer simulation is not a
  magic path to the right answer
• Modeling of environmental systems should be
  viewed as a method to gain improved understanding
  of the dynamics of the system
• Inclusion of additional variables and processes
  should be done with caution once a working model
  has been obtained doing so may lead to confusion
  rather than illumination!

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Post Script

• Was removal of predators really responsible for
  Kaibab Plateau deer population irruption?
• Caughley (1970) concludes that habitat alteration
  by fire and grazing played a major role
• Many confounding factors were likely involved
• Botkin (1990) notes that the focus by prominent
  naturalists (e.g. Aldo Leopold) on the role of
  predators reveals their paradigm of a highly
  ordered nature in which predators play an
  essential role

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Post Script

• Take home message for students of modeling
  constructing and testing a model of the Kaibab
  deer herd based on the impact of predator removal
  does not make the story true
• Although model is internally consistent, other
  models could be developed to account for the
  population irruption