1. BSC 417/517 Environmental Modeling
The Kaibab Deer Herd

2. 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

3. 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

4. Kaibab National Forest

5. The Kaibab Plateau

6. 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 ca. 1.5 yr

7. Rocky Mountain Mule Deer

8. Kaibab Plateau Deer Herd
Data on deer population size prior to 1900 is sparse; Rasmussen (1941) estimated total size of 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 , predator kills were estimated at 3000 coyotes, 674 lions, 120 bobcats, and 11 wolves

9. 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

10. Kaibab Plateau Deer Herd
Major deer die-off occurred during winters of the years
Government hunters were deployed in 1928 to reduce the size of the deer population
But, paradoxically, predator “control measures” continued…

11. 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”

12. 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

13. Reference Mode 1900 1910 1920 1930 1940 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

14. 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

15. Specific Goals of Modeling Exercise
Gain understanding of forces that led to the overshoot and collapse
Explore number of predators on the plateau as a relevant “policy variable” which could be manipulated in order to achieve a stable deer population

16. 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)

17. 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”

18. 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 = ~ 0.07
With the above assumptions:
Net birth rate = [0.5 × (2/3) × 1.6] – 0.07 = 0.47 ~ 0.5

19. 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

20. 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

21. 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_predators = GRAPH(TIME)
(1900, 50.0), (1910, 50.0), (1920, 0.00), (1930, 0.00), (1940, 0.00)

22. Results of First Model Deer population is constant for first 10 yr
Grows exponentially after predators are removed during , 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

23. 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

24. 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

25. 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

26. 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 %/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

27. Second Model
forage_availability_ratio = forage_production/forage_required
fr_forage_needs_met = MIN(1,forage_availability_ratio)

28. 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

29. Second Model Results

30. 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 to…lack of change in forage biomass?

31. Second Model Results - Contd

32. 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

33. Third Model: Biomass Sector

34. Third Model: Biomass Sector
addition_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 =
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)

35. Third Model: Biomass Sector Simulation

36. Third Model: Full Version
Kaibab Deer Herd Third Model

37. 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

38. 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

39. Fourth Model: Key Equations
additional_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

40. Fourth Model: Density-Dependent Predator Kill Rate

41. Fourth Model: Full Version
Kaibab Deer Herd Fourth Model

42. 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!

43. 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

44. 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

45. 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

46. 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.

47. 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

48. 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

49. Model 4 Sensitivity Analysis

50. Model 4 Sensitivity Analysis

51. 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 system’s tendency to overshoot
Sensitivity analysis reveals that the same general pattern is obtained regardless of the particular parameter values or relationship

52. 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

53. 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

54. Model 4 Sensitivity Analysis Extended - Results

55. 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

56. 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)

57. First Policy Test Results

58. 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)

59. 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

60. Second Policty Test: Revised Animal Sector

61. Second Policty Test Results

62. Second Policy Test: Fixed Deer Hunting
Harvest amounts of 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

63. 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

64. Third Policy Test: Variable 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

65. Third Policty Test Results (Start Hunting in 1915)

66. Third Policy Test: Variable 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

67. Third Policty Test Results (Start Hunting in 1918)

68. Third Policty Test Results (Start Hunting in 1920)

69. 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)

70. 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”?

71. 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!

72. 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

73. 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