BSC 417/517 Environmental Modeling The Kaibab Deer Herd.

BSC 417/517 Environmental Modeling

The Kaibab Deer Herd

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

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

Kaibab NationalForest

The Kaibab Plateau

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 @ ca. 1.5 yr

Rocky Mountain Mule Deer


• 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


• 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


• 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…


• 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”

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 system’s behavior

• In this case, we’re dealing with the classical “overshoot” pattern discussed earlier in the course

Reference Mode

1900 1910 1920 1930 1940

Initial pop.ca. 4000

Rapid growth afterremoval of predators

Pop. peaks at ca. 100,000

Return to pseudo-stabilitywith government hunting orreturn of predators

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

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

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 we’re 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)

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”

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 => 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


• 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


• 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

First Modeldeer population

net births

net birth rate

deer density

area

deaths from predation

~

deer killed per

predator per year

~

number of predators

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_predators = GRAPH(TIME)

(1900, 50.0), (1910, 50.0), (1920, 0.00), (1930, 0.00), (1940, 0.00)

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

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

Second Model

• As long as forage availability ratio > 1, fraction of forage needs met is 100%

• If forage availability < 1, then fraction of forage needs met = forage availability

• Fraction of forage needs met influences net birth rate according to a graph function

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

-0.4

-0.2

0

0.2

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0.6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fraction Forage Needs Met

Net

Bir

th R

ate

Note: the relationship depicted here is a “plausible guess” only, as little info is available on deer birth and death rates undef difficult conditions

Second Modeldeer population

net births

~

net birth ratedeer density

area


~

deer killed per

predator per year

~

number of predators

forage required

forage required per deer per yr

forage availability ratio

fr forage needs met

forage production

forage_availability_ratio = forage_production/forage_requiredfr_forage_needs_met = MIN(1,forage_availability_ratio)

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 => constant population for the remainder of the simulation


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• 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 to…lack of change in forage biomass?

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

Third Model: Biomass Sector

forage consumption

standing

biomass

decay

bio decay rate

max biomassfullness fraction

~

prod mult from

fullness

addition to standing biomass

new growth

intrinsic bio productivity

bio productivity

Third Model: Biomass Sectoraddition_to_standing_biomass = new_growth-forage_consumption

decay = standing_biomass*bio_decay_rate

bio_decay_rate = 0.1

bio_productivity = intrinsic_bio_productivity*prod_mult_from_fullness

forage_consumption = forage_required*fr_forage_needs_met

fullness_fraction = standing_biomass/max_biomass

intrinsic_bio_productivity = 0.4

max_biomass = 400000

new_growth = standing_biomass*bio_productivity

prod_mult_from_fullness = 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)

0

0.2

0.40.6

0.8

1

0 0.2 0.4 0.6 0.8 1

fullness

prod

mul

t

Third Model: Biomass Sector Simulation

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Biomass sector module

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Third Model: Full Version

Kaibab Deer Herd Third Model

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

Fourth Model: Deer May Consume Older Biomass

• Deer prefer new growth, but under stressed (i.e. starvation) conditions will consume older biomass (=> “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

Fourth Model: Key Equationsadditional_con_required = forage_required-forage_consumption

stand_bio_available = standing_biomass*fr_standing_available

fr_standing_available = 0.25

old_biomass_availability_ratio = stand_bio_available/MAX(1,additional_con_required)

old_biomass_consumption = additional_con_required*fr_additional_needs_met

fr_additional_needs_met = MIN(1,old_biomass_availability_ratio)

equivalent_fraction_needs_met = MIN(1,fr_forage_needs_met+fr_additional_needs_met*old_biomass_nutritional_factor)

old_biomass_nutritional_factor = 0.25

Fourth Model: Density-Dependent Predator Kill Rate

0

10

20

30

40

0 1 2 3 4

number of deer per 1000 acres

kill

s p

er p

red

ato

r p

er y

r

Fourth Model: Full Version

Kaibab Deer Herd Fourth Model

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!


• 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


• 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

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 model’s 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

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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• 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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• 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


-0.6

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• 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 system’s tendency to overshoot

• Sensitivity analysis reveals that the same general pattern is obtained regardless of the particular parameter values or relationship

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 model’s robustness

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)

{EffectivelyDoublesPlateau Size

Model 4 Sensitivity Analysis Extended - Results

0

100,000

200,000

300,000

400,000

500,000

600,000

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r

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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

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)

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)

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

Second Policty Test: Revised Animal Sector

deer population

net births

~net birth rate

deer density

area


~

deer killed per

predator per year

~number of predators

equivalent fraction needs metold biomass nutritional factor

fr additional needs met

deer harvest

harvest amount harvest start year

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 further…but 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

Third Policy Test:Variable 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


• 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

Third Policty Test Results(Start Hunting in 1915)

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• 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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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)

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”?

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!

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

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

BSC 417/517 Environmental Modeling The Kaibab Deer Herd.

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