Assignment 1: Modelling the Ecology of the Kaibab Plateau TE R15 January 2015 Report Submission Due on Monday, 17 February at 9AM No longer than 10 pages, not including appendices Compiling the report, including exercises 7 and 8, should take about 10 hoursReport only needs to include: oProblem definition and context: 1-2 pages oModel content and dynamic hypothesis: 1-2 pages; equations can be listed in an appendix oAnswers to Exercise 7: 2-3 pages oAnswers to Exercise 8: 4-5 pages, with graphs and equations where needed oSee the mark sheet at the end of this document for complete information on what is expected in the submission Formatting: oSingle-spaced with a blank line between paragraphs oClearly indicate exercise numbers and answers oDo not include vertical data tables Submit two files:oReport as Word documentoVensim model with modifications and policy experiments oUpload to IVLE in Student Submissions > 1. Kaibab Report folder An Introduction to Systems Thinking & Modelling Kaibab Plateau Exercise TE R152 / 116 (Blank Page) Kaibab Plateau Exercise TE R153 / 116 Table of Contents 1.0Introduction .......................................................................................................................................................... 7 2.0Creating a Model ................................................................................................................................................... 9 2.1Purpose of Modelling ...................................................................................................................................... 9 2.2Modelling Process and Terminology ............................................................................................................. 11 2.3Modelling Conventions ................................................................................................................................. 15 3.0Modelling the Kaibab Plateau ............................................................................................................................. 16 3.1Exercise 1: Deer Births .................................................................................................................................. 17 How Do I Calculate Units? ..................................................................................................................... 20 3.2Exercise 2: Deer Deaths ................................................................................................................................ 23 3.3Exercise 3: Deer Killed by Mountain Lions .................................................................................................... 28 3.4Exercise 4: Lions and Lion Hunting ................................................................................................................ 35 3.5Exercise 5: Creating Vegetation Consumption.............................................................................................. 41 What Are Limiting Factors? ................................................................................................................... 44 3.6Exercise 6: Creating Vegetation Supply ........................................................................................................ 49 How Much Does Data Impact Results? ................................................................................................. 56 3.7Exercise 7: Validating the Model .................................................................................................................. 57 3.8Exercise 8: Using the Model to Examine Policies .......................................................................................... 59 4.0Using Vensim and Vensim Resources ................................................................................................................. 62 4.1Starting Vensim and Entering the Model Settings ........................................................................................ 63 Should I Save the Model? ..................................................................................................................... 65 4.2Drawing Causal Loop Diagrams ..................................................................................................................... 66 4.3Drawing Stock-and-Flow Diagrams ............................................................................................................... 68 4.4Adding Loop Polarities .................................................................................................................................. 71 4.5Displaying Initial Causes ................................................................................................................................ 73 4.6Entering Numeric Equations ......................................................................................................................... 74 4.7Entering Text Equations ................................................................................................................................ 76 4.8Naming Runs ................................................................................................................................................. 78 Kaibab Plateau Exercise TE R154 / 116 4.9Selecting Runs ............................................................................................................................................... 79 4.10Exporting Equations, Tables, and Data ......................................................................................................... 80 4.11Understanding Sensitivity Analysis ............................................................................................................... 81 4.12Performing Sensitivity Analysis via Sim Setup .............................................................................................. 83 4.13Changing Time Settings ................................................................................................................................. 84 4.14Understanding Lookup Functions ................................................................................................................. 85 4.15Creating Lookup Functions............................................................................................................................ 86 4.16Exporting Lookup Graphs .............................................................................................................................. 87 4.17Creating STEP Functions ................................................................................................................................ 90 4.18Adding Shadow Variables.............................................................................................................................. 92 4.19Deleting Shadow Variables ........................................................................................................................... 93 4.20Understanding Warnings .............................................................................................................................. 94 4.21Finalising a Model ......................................................................................................................................... 95 4.22Graphing Multiple Variables ......................................................................................................................... 96 4.23Creating Custom Graphs ............................................................................................................................... 97 4.24Colouring Arrows and Variables .................................................................................................................... 99 5.0Goodmans Diagrams ........................................................................................................................................ 101 5.1Goodmans Causal Loop Diagram ............................................................................................................... 101 5.2Goodmans Stock-and-Flow Diagram .......................................................................................................... 102 5.3Goodmans Model Behaviour ..................................................................................................................... 103 6.0Model Reference Material ................................................................................................................................ 104 6.1Causal Loop Diagram ................................................................................................................................... 104 6.2Stock-and-Flow Diagram ............................................................................................................................. 105 6.3Vensim Equations ........................................................................................................................................ 106 7.0Report Elements ................................................................................................................................................ 110 7.1Report Checklist .......................................................................................................................................... 110 7.2Written Report Mark Sheet ........................................................................................................................ 114 8.0Bibliography ...................................................................................................................................................... 116 Kaibab Plateau Exercise TE R155 / 116 List of Figures Figure 1 Map of the continental USA; the tip of the arrow marks the Kaibab Plateau................................................ 7 Figure 2 Deer population from 1906 to 1939............................................................................................................... 8 Figure 3 Reference mode with different stages highlighted ....................................................................................... 11 Figure 4 Policy intervention in 1930 ............................................................................................................................ 16 Figure 5 Causal loop diagram for deer births ............................................................................................................... 17 Figure 6 The simulation file name field ........................................................................................................................ 20 Figure 7 The Deer Births stock-and-flow diagram ....................................................................................................... 22 Figure 8 Causal loop diagram for deer births ............................................................................................................... 23 Figure 9 The Deer population stock-and-flow diagram ............................................................................................... 26 Figure 10 Causal loop diagram for deer ....................................................................................................................... 28 Figure 11 Lookup graph for deer density and deer killed per lion ............................................................................... 31 Figure 12 The Deer population stock-and-flow diagram ............................................................................................. 34 Figure 13 Causal loop diagram for lions ....................................................................................................................... 35 Figure 14 Stock-and-flow diagram for Deer and Lions ................................................................................................. 40 Figure 15 Causal loop diagram for deer, lions, and vegetation ................................................................................... 41 Figure 16 Changes in deer average lifetime caused by vegetation availability ........................................................... 43 Figure 17 Deer population from 1906 to 1939 ............................................................................................................ 45 Figure 18 Smooth S-curve function ............................................................................................................................ 46 Figure 19 Stock-and-flow diagram for Deer, Lions, and constant Vegetation ............................................................. 48 Figure 20 Causal loop diagram for deer, lions, and vegetation ................................................................................... 50 Figure 21 Lookup graph for vegetation available per deer .......................................................................................... 52 Figure 22 Lookup for vegetation regeneration time .................................................................................................... 54 Figure 23 Stock-and-flow diagram for Deer, Lions, and Vegetation ............................................................................ 55 Figure 24 Reference mode ........................................................................................................................................... 57 Figure 25 Deer population from 1906 to 1939 ............................................................................................................ 61 Figure 26 Vensim Interface .......................................................................................................................................... 63 Figure 27 Model settings .............................................................................................................................................. 64 Figure 28 Sample causal loop diagram ........................................................................................................................ 66 Figure 29 Variables, stocks, and flows in Vensim ........................................................................................................ 68 Figure 30 Sample stock-and-flow diagram ................................................................................................................... 68 Figure 31 Sample stock-and-flow diagram with arrowhead highlighted ..................................................................... 71 Figure 32 Polarity selections highlighted ..................................................................................................................... 71 Figure 33 Vensim toolbar with Equations box highlighted ........................................................................................ 74 Figure 34 Equation editor with Units, Equations, and Comments fields highlighted ............................................ 74 Kaibab Plateau Exercise TE R156 / 116 Figure 35 Equations editor with Variables field highlighted ...................................................................................... 76 Figure 36 Control Panel ................................................................................................................................................ 79 Figure 37 Export options .............................................................................................................................................. 80 Figure 38 Sim Setup options ........................................................................................................................................ 83 Figure 39 Lookup options ............................................................................................................................................. 86 Figure 40 STEP function selections highlighted ........................................................................................................... 91 Figure 41 Options to delete shadow variables ............................................................................................................ 93 Figure 42 Stock-and-flow diagram with two variables selected .................................................................................. 96 Figure 43 Selecting a new graph .................................................................................................................................. 97 Figure 44 Fields needed for custom graph................................................................................................................... 98 Figure 45 Toolbar with colour selection box highlighted............................................................................................. 99 Figure 46 Arrow colour selection highlighted ............................................................................................................ 100 Figure 47 Causal loop diagram ................................................................................................................................... 101 Figure 48 Stock-and-Flow Diagram ............................................................................................................................ 102 Figure 49 Growth and decline of deer on the Kaibab Plateau ................................................................................... 103 Kaibab Plateau Exercise TE R157 / 116 1.0Introduction The Kaibab Plateau is a large, flat area of about 800,000 acres (3,238 km2) located on the northern rim of the Grand Canyon in the United States of America. The plateau has an elevation of about 6,000 feet (1,829 meters) and is bounded by steep slopes and cliffs. The wildlife living there includes rabbits, deer, mountain lions, wolves, coyotes, and bobcats. Figure 1 Map of the continental USA; the tip of the arrow marks the Kaibab Plateau 1 In 1907, President Theodore Roosevelt created the Grand Canyon National Game Preserve, which includes the Kaibab Plateau. Deer hunting was prohibited and a bounty was established to encourage the hunting of mountain lions and other predators of deer. From 1906 to 1931, nearly 800 mountain lions were trapped or shot. As a result, the deer population grew quite rapidly, from about 4,000 in 1906 to nearly 100,000 in 1924. As the population grew, Forest Service officials warned that the deer would exhaust the vegetation supply on the plateau. One observer wrote: Never before have I seen such deplorable conditions 30,000 to 40,000 deer were on the verge of starvation. Over the winters of 1924 and 1925, nearly 60% of the deer population died of starvation, and continued to decline over the next fifteen years, finally stabilizing at about 10,000 in 1939.

1 From maps.google.com. Kaibab Plateau Exercise TE R158 / 116 Figure 2 Deer population from 1906 to 1939 2 For more information on the plateau, read the article on Wikipedia at https://en.wikipedia.org/wiki/Kaibab_Plateau.

[it is important to bear in mind that the idea of feedback control systems used for any purpose let alone for policy analysis is a relatively new phenomenon.. Norbert Weiner (1894-1964), Cybernetics (1948),The Human Use of Human Beings (1950),God and Golem, Inc. (1964). W.Ross Ashby (1903-1972)Design for a Brain (1952);Introduction to Cybernetics (1955),Stafford Beer (1926-2002),Cybernetics and Management (1959),Dcision and Control (1966),Platform for Change (1975)

2 The Kaibab Plateau model was originally published in Goodman (1974). The version used here is from Goodman (1989), Exercise 15 Kaibab Plateau Model, pg. 377. Kaibab Plateau Exercise TE R159 / 116 2.0Creating a Model 2.1Purpose of Modelling There are two very important points to remember regarding system dynamics modelling: 1.Model problems, not systems This means that you should have a specific, well-defined problem, normally framed as a question, to investigate. You should also have a reference mode (also called a behaviour-over-time graph) to reproduce before creating a model. Without these, you may be unsure of what variables to include and exclude, resulting in a model that is ever expanding, including more and more variables to account for problematic behaviour. This leads to the second point. 2.In general, a small, simple model is better than a large, complex one This means that a model that reproduces the problematic behaviour in a compact, understandable form is better than one that reproduces the behaviour in an expansive, complex one. This may be contrary to what you have learned, where complex models are viewed as better because their complexity is seen to capture the nuances of a complex problem more fully. However, in system dynamics, simplicity and parsimony are seen as norms when developing models; systems can be complex, but you should simplify whenever possible and only add complexity until you have approximated the reference mode. The problem of the deer population on the Kaibab Plateau is an ideal exercise3 with which to start modelling systems for the following reasons:1.Policy a.It is a problem based on a historical scenario that involved a policy interventionb.The full implications of the policy intervention were unanticipated c.The limiting factor changes as a consequence of the policy intervention 2.Design a.There is empirical information with which to investigate the problem at hand b.The system in which the problem occurred started in a state of equilibrium; i.e., variables were in a state of natural balance, without outside interference[Ecosystems often converge to that state and can be quite resilient]. c.There is an reference mode that provides a clear example the relationship between structure and behaviour4

3 The model is based on the work of Michael Goodman and Donella Meadows, from Roberts, et al, pg. 337-352. The exercise has been adapted by Professor John Richardson, Corbett Hix, and Rehan Ali. Please do not reproduce or share this document without permission. 4 See Section 2.2, Modelling Process and Terminology, for definitions of unfamiliar terms. Kaibab Plateau Exercise TE R1510 / 116 d.There is a long time horizon.[Different methologies for different time horizons] e.There is an unambiguous dynamic hypothesis [What is a dynamic hypothesis] f.The model is of manageable size with three stocks, and is an exemplar for your own projects g.The model boundaries are relatively straightforward[What is a model boundary?} 3.Concepts[Who are stakeholders;what are there values; intersecting ecological and political systems] a.The Issues involve stakeholders with conflicting values.b.The issue population growth that exceeds carrying capacity and is therefore unsustainable is generic and can be applied to multiple problems, such as those faced by humans on Earth This assignment will provide you with practice in: [ Replicating amodel(How does learn about a model). Documenting a modelAnalysing a model Experimenting with and reporting on policy analysis This document takes you through the process of building a model of the problem of the deer population on the Kaibab Plateau using Vensim, a modelling application.Section 2 gives you a basic background in modelling.Section 3 contains the steps you need to build the model, and the questions that you need to answer in the report you will submit. oSections 3.1 to 3.6 guide you through the process of constructing the model, step-by-step; we will go through these in class as a group. oIn Section 3.7 you validate the model and in Section 3.8 you experiment with various scenarios using the model.oIn the report, you only need to include answers to the questions in Sections 3.7 and 3.8.Section 5 contains steps on how to use Vensim.oSince we will be going through this in class, you will probably only need to use Section 5 as a reference in case you forget how to do something in Vensim.Section 7.1, Report Checklist, contains the elements needed in a complete system dynamics report.Section 7.2, Written Report Mark Sheet, contains the mark sheet that will be used to grade your report. There are hyperlinks between sections 3 and 4, so use them to move between different parts of the text. In Section 4, an arrow () indicates the hyperlinks back to section 3. This symbol marks best practice when modelling, so keep an eye out for it. Kaibab Plateau Exercise TE R1511 / 116 2.2Modelling Process and Terminology This section gives an overview of the thinking process to follow when investigating a problem using system dynamics modelling.1.Before modelling a problem, it is best to have a reference mode or behaviour-over-time graph; Figure 3 (below, with the stages of dynamic behaviour highlighted) is an example of one. A reference mode is a visual representation of the problematic behaviour exhibited by a key variable over a fixed period of time, called the time horizon. In this exercise, the problematic behaviour is exhibited by the deer population over a period of 33 years, from 1906 to 1939. When creating a model, one goal is to reproduce the reference mode because this shows that the model has validity and be can be used to run further simulations to explore other solutions to the problem you are trying to resolve.5(Whats wrong; how does it work; how can we fix things; howcan we do better) Figure 3 Reference mode with different stages highlighted 2.Along with the reference mode, system dynamics modelling also requires that the problem be expressed as a question to investigate; this is called the problem definition. The problem definition for this exercise can be put as:What explains the rapid growth, decline, and stabilisation of the deer population from 1906 to 1939?

5 See Sterman, Chapter 21, for more information. HistoricalLimit GrowthCollapse New Kaibab Plateau Exercise TE R1512 / 116 Though the problem definition here is simple, it raises a number of questions that need to be answered: What is the problem? What caused the problem?What would be a desirable end-state or goal?How can that goal be attained? 3.The problem definition is answered by a dynamic hypothesis, which states what caused the problematic behaviour; for example: The policy allowing the hunting of mountain lions removed a factor (mountain lions) that limited the natural growth of the deer population, leading to the overpopulation of deer and their subsequent death through mass starvation. In answering the problem definition, the dynamic hypothesis makes explicit the relationship between structure and behaviour, which is a cornerstone of the modelling process in system dynamics. The way variables are linked6generates what is seen in the reference mode; i.e., the trajectory of important variables over time, in this the deer population. 4.In order to contextualise the problem, we need a state-of-knowledge assessment, which provides background information, in a summarised, narrative form. Section 1.0, Introduction, is an example of a basic state-of-knowledge assessment. A state of knowledge assessment differs from a typical literature review in that the assessment only focuses on literature that is directly relevant to the problem being investigated. The assessment is often summarised in a set of propositions, called a propositional inventory, discussing variables and the causal relationships between them. We will look at a more complete assessment in the next assignment.

6 See Section 4.2, Drawing Causal Loop, for information on how variables are labelled. Structure BehaviourKaibab Plateau Exercise TE R1513 / 116 5.From the state-of-knowledge assessment, compile a list of stakeholders and parties involved in the problem. Deciding who a stakeholder is depends on vantage point and judgement, but it is basically a party whose points-of-view and values matter in the policy process. Some of the following may not be considered stakeholders because of their lack of volition, but they are nonetheless involved in the problem: DeerEnvironmentalists Forest Service officials Hunters Mountain lions Policy makers (Stakeholders are often found on model boundaries but can also be within the model;what are boundaries?) 6.Policy makers are a sub-set of stakeholders who are able to bring about change within the boundaries of the problem by accessing and changing leverage points, which are variables or sectors in a model that can directly be influenced and where policies can be implemented to have maximum impact. In this problem, hunting of lions is a leverage point.[How do we identify leverage points] 7.Once youve identified the leverage points, you should think of policy interventions that contributed to the problem. In this case, Forest Service officials (policy makers) provided incentives to hunters to kill mountain lions but not deer.[Most often, the are interventions on model boundaries] 8.The problem also demonstrates the principle of loop dominance, which is when a feedback loop, in this case deer growth, causes the problematic behaviour seen in the reference mode. 9.You also must determine what the model boundaries are. In short, these are limits that help to determine the problematic behaviour of key variables. For example, the years that we are focusing on, 1906 to 1939, are one set of temporal boundaries. Another is the size of the plateau, 800,000 acres, which is a physical boundary. [Endogenous vs. exogenous.Systmem dynamics emphasizes endogenous] 10.In this exercise, we will create three major sectors. Each sector includes variables that comprise the sector or form a sub-sector. A sector can be thought of as a subsystem within a model that is important enough to note on the macro-level. Variables and sub-sectors can be thought of elements that exist on a micro-level within that macro-level view.In this exercise, the three sectors that are the most important and that exist on a macro-level are.[The human body as an example, digestion, cardiovascular, neurological] Deer Lions Vegetation Kaibab Plateau Exercise TE R1514 / 116 The reason that we focus on the major sectors is that when building a model, it is best to begin with a simple model and then expand it over the course of several steps rather than create and run an entire model at once. Determining the major sectors is the first step in this process. [The model may be your creation, but do you understand your creation, not necessarily] After identifying the major sectors, focus on one sector and identify the variables and sub-sectors that exist within that sector. This means that our focus is always on one part of a model that we can construct with confidence before worrying about the overall model. In this exercise, a major sub-sector, bounty hunting, will be added to the Lions sector. While modelling, think about other variables that might be included in the model. The first sector we will focus on is Deer. Start with a small model focusing on a few of the factors that influenced the size of the deer population, and then add complexity in order to replicate the ecology of the plateau. This will result in a model that generates the behaviour shown in Figure 2. Each new sector builds on the previous one, so be sure to understand how variables interact within the existing system and generate feedback before adding more sectors and variables. When moving from one sector to another, save each model under a different name so that a previous version is available if needed. 11.A deliberate, step-by-step model-building strategy is usually a good one to follow so that each variable can be verified before adding a new one. Be sure of why a variable is being added, and what it adds to the model as a whole in order to answer the question posed in the problem definition. Always think about what is being constructed; a model tells a story, and that story should be conveyed clearly to the audience. When adding a variable, also consider whether variables are endogenous (originate within the sector) or exogenous (originate outside the sector); the boundary between endogenous and exogenous can shift depending on what we are focusing on in the model. 12.Running a model, especially as a beginning modeller, almost always produces unanticipated results. If a model is complex, comprising two or more stocks (see Section 4.1, Starting Vensim and Entering the Model Settings, for a definition), and model runs produce unanticipated results, determining the source of the results can be difficult if we dont properly document the construction of the model. If unanticipated results appear at any stage, stop modelling and try to determine why the results are different from what was expected by checking the model, sector-by-sector and variable-by-variable. When doing this, it is often helpful to view the numerical results and graphs your model is generating. After gaining confidence in the model through testing various scenarios, use it to assess proposed strategies and devise new ones.Kaibab Plateau Exercise TE R1515 / 116 2.3Modelling Conventions There are a few conventions that are used in this document and in Vensim to visually designate different types of content.Adhering to these conventions will be helpful in maintaining consistency in your own work and sharing it with others.[MODELING IS A PRACTICE.YOU MASTER A PRACTICE BY PRACTICING] Bold Used to indicate all model variables in this document to make them easier to read and find within the text. Bolding is also used to indicate buttons and functions in Vensim. Dont worry; there is no overlap between these two uses of bolding so this shouldnt be confusing. You dont need to bold any text in Vensim. Bolded variables are further designated by type: oALL UPPER CASE LETTERS A stock; for example DEER or LIONS7 oCapital Letter Initials All other variables; for example Deer Births Colours Colours are used for arrows (and text, if needed) to visually distinguish sectors and indicate constants. See Section 4.24, Colouring Arrows and Variables, for more information. Naming The list of equations in Vensim (See Section 4.10, Exporting Equations, Tables, and Data, for information on viewing the list of equations) is always arranged in alphabetical order. This means that related variables may appear apart from one another, making it difficult to see which variables belong together. For this reason, it is useful to keep the same first word for all related variables; for example, all variables in the deer sector start with Deer, all the variables in the lion sector start with Lion, etc. So when we look at the list of equations, all the deer equations are grouped together, making it easy to review and check them.[As you are creating your first model,print out equations early.This will feel for how Vensim produces documentation and will allow you to document more USEFULLY.That is so you understand what you have created and then so you can share it with others.] AND NOW, BACK TO THE KAIBAB PLATEAU..

7 See Section 4.2, Drawing Causal Loop, for information on how variables are labelled. Kaibab Plateau Exercise TE R1516 / 116 3.0Modelling the Kaibab Plateau Imagine it is 1930 and you are an official from the National Forest Service interested in the deer population on the Kaibab Plateau. Figure 4 Policy intervention in 1930 To examine the population problem, we have decided to build a simulation model. Our main concerns are the growth and rapid decline of the deer population from 1900 to 1930, and the future of the population from 1930 to 1950. Therefore, the time frame for the model we are building is 50 years, from 1900 to 1950, which covers the time horizon for the problematic behaviour shown in Figure 2 and replicated above in Figure 4. As a first step in creating a model to replicate the reference mode, we will create the deer population sector.[IN ADDITION TO THE PRINCIPLES DISCUSSED UP TO THIS POINT,THIS ILLUSTRATE ANOTHER SYSTEM DYNAMICS FUNDAMENTAL WE DONT UNDERTAKE TO SOLVE A PROBLEM UNTIL WE UDERSTAND THE PROBLEM.BUT WHAT DOES IT MEAN TO UNDERSTAND THE PROBLEM?IT MEANS THAT WE CAN GENERATE THE PROBLEM (THAT IS, WE CAN BUILD A SYSTEM DYNAMICSMODEL THAT REPRODUCES THE REFERENCE MODE; I.E.THE PROBLMATIC BEHAVIOR. ALSO NOT ALL PROBLEMS LEND THEMSELVES TO SYSTEM DYNAMICS MODELING. You are here Kaibab Plateau Exercise TE R1517 / 116 3.1Exercise 1: Deer Births The deer population serves as a basic population model with births and deaths flowing into and out of a stock named DEER; this is the total number of deer in the population at any given time. Step 1.Start by drawing a causal loop diagram showing the relationship between DEER, deer births, and deer birth rate. See Section 4.1, Starting Vensim and Entering the Model Settings, for information on starting a model in Vensim. You do not need to enter any initial values for causal loop diagram. Step 2.If DEER is the total number of deer on the plateau, naturally the size of the population is increased by deer births, so add that variable. Is the loop reinforcing (positive) or balancing (negative)?8

See Section 4.2, Drawing Causal Loop Diagrams for information on adding polarities. Step 3.Add a new variable, deer birth rate. What the link polarity between it and deer births. Step 4.Save the model with a name such as KaibabCL-Deer. Figure 5 Causal loop diagram for deer births The next step is to translate the causal loop diagram into a stock-and-flow diagram that explicitly identifies which variables are stocks (levels) and which are flows (rates).9

1.See Section 4.1, Starting Vensim and Entering the Model Settings for information on stocks, flows, and variables in Vensim.

8 Causal loop diagrams and notation are discussed in brief in Section 4.2, Drawing Causal Loop, and in more detail in Sterman, Chapter 5.9 Terminology has changed over the years, so be aware that both are used. Vensim, however, uses the older terminology, level and rate, consistently. DeerDeer Births++Deer PopulationGrowthDeer Birth Rate+Kaibab Plateau Exercise TE R1518 / 116 2.See Sections 4.2, Drawing Causal Loop Diagrams, and 4.3, Drawing Stock-and-Flow Diagrams, for information on the differences between causal loop and stock-and-flow diagrams. Step 5.Create a new model and use the following for the initial values: INITIAL TIME = 1900 Units: Year FINAL TIME = 1950 Units: Year THEN CREATE A STOCK AND FLOW (OR LEVEL AND RATE DIAGRAM) WE ARE STILL SORTING OUT THE TERMINLOGY It is a good idea to keep the causal loop diagram in mind or in front of you as you create the stock-and-flow diagram. Step 6.Add DEER and Deer Births to the stock-and-flow diagram. What units are they measured in? What are the polarities of the links between them? DEER is a stock variable. We know this because it is an accumulation of deer at any given moment in time.Deer Births is a flow variable that is used as an input to a stock, in this case DEER. Discipline yourself to add link polarities as you draw the links between variables. You can place the polarity on the handle of the link rather than the arrow to reduce clutter. Step 7.Since a model is used to simulate reality and we have a defined time horizon, add a new variable, Deer Initial.[WHY DO WE NEED INITIAL VALUES BECAUSE OF THE WAY VENSIM COMPUTES] Deer Initial is an initial variable, which is a type of constant. This initial tells Vensim the total number of deer that existed at the start of the run, in this case, 1900. You dont need to change its value unless you want the model to run under different conditions. Even though an initial is a type of constant, Vensim treats initials differently, so See Section 4.5, Displaying Initial Causes, for more information. According to our state-of-knowledge assessment, what is the initial value for DEER?10 What units is it measured in? Now that we have a few variables and some data, we can think about how variables are linked by formulating equations for two new variables, Deer Births and Deer Birth Rate. The number of deer births per year equals the

10 See Figure 4. Kaibab Plateau Exercise TE R1519 / 116 population of DEER multiplied by the number of births per deer per year. We will assume that each female deer gives birth to one deer per year. Since half the deer population is female, this requires a new variable, Deer Birth Rate, which equals 0.5 deer born per deer per year. When writing equations, a good rule of thumb is to enter the simplest values first in the order of constants > initial values> STOCKS> FLOWS. As you enter equations, Vensim will auto-populate values for auxiliaries and flows, which tend to have more complex equations. Even though these have auto-populated values, you must check them AND YOU MOST TYPICALLY WRITE THE EQUATIONS.3.For each equation, be sure to add the units of measurement and to include comments such as the source of the values or equations you are using. In some cases, the source will not be hard data, but your intuitive estimate of an order-of-magnitude value, based on interviews or other means of estimation.If you are doing this, be sure to include the rationale for your estimate. This will be of great value when you are documenting results. 4.Check the syntax of each equation as you write it; this will save you problems later on.5.See Sections 4.6, Entering Numeric Equations, and 4.7, Entering Text Equations, for more information on entering values and equations. Step 8.Write the equations using the following values.Constant Deer Birth Rate = 0.5 Units: Deer/Deer/Year Initial Deer Initial = 4000 Units: Deer Flow(Rate) Deer Births = DEER*Deer Birth Rate Units: Deer/Year Stock (LEVEL) DEER = INTEG (Deer Births-Deer Deaths, Deer Initial) Units: Deer For Deer Birth Rate, 0.5 represents our best estimate of rate under normal conditions, which is that one-half of the deer population consists of fecund females and that each female foals one new deer each year. Kaibab Plateau Exercise TE R1520 / 116 How Do I Calculate Units? So far, we have created a stock, DEER, a flow, Deer Births, and a fraction, Deer Birth Rate. Lets take a moment to compare the units for each of these: 6.DEER This is a stock that is calculated as the number of deer in existence at any given point in time, so its units are simply Deer.[IF YOU TAKE A SNAPSHOT OF THE MODEL THE LEVELS ARE ALWAYS PRESENT] 7.Deer Births This is a flow or rate, that is, the number of new deer that are added to the total number of deer in any given year; Year is the unit of measurement of time that we selected when we set up the model. Therefore, the units are Deer/Year. 8.Deer Birth Rate This is a fraction used by Deer Births to calculate the number of new deer, so the units for this are Deer/Deer/Year, that is, the fraction of deer that are born per deer per year.REMEMBER HOW WE TOOK GENDER INTO ACCOUNT.HOW DID WE DO IT?

As variables are added, be sure to think about the units for each variable and enter them in in the equations box. This can help to make sure that you are using comparable variables that can be balanced by the model. Now that we have created a basic stock-and-flow diagram and populated it with values and equations, we can perform a simulation run. Step 9.In the Simulation results file name field above the modelling area, enter Baseline Ex1p9 DBR=0p5 and click Simulate; this gives a baseline or reference run. In the complete model, the baseline run will replicate the reference mode, but at this point it will simply generate data on the deer population.See Section 4.8, Naming Runs, for information on naming runs. Figure 6 The simulation file name field Step 10.Vensim generates a variety of graphs. Select a variable whose graph we want to see, for example, DEER, and click the Graph button in the toolbar to the left of the modelling area.11

The graph appears in a separate window above the modelling area, at the top of the sketchpad. The selected variable, DEER, is called the workbench variable.

11 See Appendix I for information on how to create custom graphs. Kaibab Plateau Exercise TE R1521 / 116 9.See Section 4.22, Graphing Multiple Variables, for information on viewing multiple variables on one graph. 10.Note that aside from graph, there is another button, Causes Strip. Whereas Graph shows you a single variable by default, Causes Strip shows you graphs for the workbench variable as well as the flows into and out of it (if it is a level) and the values of variables that cause it to change over time. We will use this in the next exercise. In addition to graphs, Vensim also generates data organised into tables. Step 11.Select a variable, for example, DEER, and click the Table button, below the Graph button. The table also appears in a separate window above the modelling area. Table shows data in a horizontal table. Table Time, below Table, shows data in a vertical table.In general, you will find the Table Time display to be useful because cause it shows patterns of change, over time, more clearly than the Table display. The time step at which data is recorded depends on the value selected in Model Settings when you created a new model.See Section 4.10, Exporting Equations, Tables, and Data, for information on exporting information from Vensim. Get into the habit of viewing both tabulations and graphs for all key variables. Now that we have an initial run, we will perform some sensitivity analysis.11.For information on sensitivity analysis, see Section 4.11, Understanding Sensitivity Analysis. Step 12.Since Deer Birth Rate is a variable for which we theorised a value, we will try a few runs with different values. Run the model under new run names, per the new values being tested for Deer Birth Rate: 0.3 Ex1p12 DBR=0p3 0.7 Ex1p12 DBR=0p7 INSERT Instead of simulating by permanently changing a value (in this case, Deer Birth Rate), you can use a feature called Sim Setup, in which you temporarily change the value; see Section 4.12, Performing Sensitivity Analysis via Sim Setup, for information on how to use this tool. Step 13.Compare the results from the three runs via graphs and tables. Kaibab Plateau Exercise TE R1522 / 116 How do these results differ?For information on viewing and comparing multiple runs, see Section 4.9, Selecting Runs. Note that as we graph more variables, the colour for each variable changes; the last run will always appear as the blue line, so do not compare runs by looking at the colour of a variable. Instead, use the label, which is the label you specified when you defined the run. This is one of many reasons why labelling runs clearly is important. Step 14.View the data via Table Time. How does this compare with the reference mode? Step 15.Open the Control panel and move the following runs to the Available Info column.Ex1p12 DBR=0p3 Ex1p12 DBR=0p7 Now Vensim is only using the run with Deer Birth Rate = 0.5. This is the normal run that we want to use for the remainder of this exercise. Step 16.Save the model. We have completed part of the stock-and-flow model. We will build on this model in the following sections. Figure 7 The Deer Births stock-and-flow diagram DEERDeer BirthsSDEER HERD GROWTHDEERGROWTHDeer Birth RateSDeer InitialSKaibab Plateau Exercise TE R1523 / 116 3.2Exercise 2: Deer Deaths In this exercise, we will focus on the other half of the deer sector: deer deaths. Adding deaths to the model gives us a more realistic deer population sector with deaths balancing out births. Step 1.Return to your causal loop diagram and add the relationship between DEER and Deer Deaths. Is this loop reinforcing or balancing? Figure 8 Causal loop diagram for deer births Step 2.Save the causal loop diagram under a new name and open the stock-and-flow diagram. Step 3.Save your stock-and-flow model under a new name. That way, you have a saved model from each exercise so that you can go back to previous version of the model if needed.This is important because as your model becomes more complex, the probability of errors increases, so Its good to be able to return to an error-free model if you get into trouble. Step 4.Now add Deer Deaths to the stock-and-flow model. Be sure to add the polarity (+ or ) of variables and loops, and the name of each causal loop. In the deer birth loop, births were represented by a fixed value of 0.5. Based on the birth loop, what is missing in the death loop? It can be theorised as something like fractional death rate of deer, which is a decimal representation of the fraction of deer that die each year. However, we dont have this information in the state-of-knowledge assessment and it is hard to theorise what this value could be. Instead, lets go with something more intuitive: the average lifetime of DeerDeer Births++Deer PopulationGrowthDeer Birth Rate+Deer Deaths+-Deer PopulationDeclineKaibab Plateau Exercise TE R1524 / 116 deer, called Deer Average Lifetime, since that is something on which information exists. Let us assume that Deer Average Lifetime is 5 years; this is called a normal value. Normal values are used in situations where we assume ceteris paribus. The inverse of Deer Average Lifetime determines the percentage of deer that die each year so we could also use a crude death rate, which is mathematically equivalent. When two representations of a constant are mathematically equivalent, think about when youre presenting your model to a non-technical audience, so choose a representation that makes more intuitive sense. Step 5.In the Vensim toolbar, go to Windows > Control Panel > Datasets and transfer any datasets in Loaded Info to Available Info and delete them. Remember that you can use runs for comparison purposes, so be sure to label them clearly. Step 6.Add Deer Deaths and Deer Average Lifetime to the stock-and-flow diagram. What is the equation for Deer Deaths? Step 7.Add it to the model.Deer Average Lifetime = 5 Units: Years Deer Deaths = DEER / Deer Average Lifetime Units: Deer/Year Step 8.Label the run Baseline Ex2p8 ALD=5 and run the model. Step 9.Select DEER and view its graph. Also view the Causes Strip graph. The Causes Strip graph displays graphs for the selected variable and its input and output variables so we can directly see causes and effects in one view. Step 10.View a data table of the values. Since the model takes place over a few decades, we dont need to see annual data; instead, we will look at the values for every 5 years.You can change this in the Settings menu; see 4.13, Changing Time Settings, for more information. Step 11.Run the model with the title Baseline Ex2p11 ALD=5.Kaibab Plateau Exercise TE R1525 / 116 See Section 4.13, Changing Time Settings for information on changing the time step. How does the behaviour of the run differ from that in Step 10? Do these results seem plausible? Why or why not? Step 12.Now, run the model with Deer Birth Rate = 0.2. Label the run Baseline Ex2p12 DBR=0p2. How do the results differ from those in Step 9? Do these results seem plausible? Step 13.Perform a few other experiments and explain the results in brief. Before experimenting: Be sure to have a hypothesis about what you are doing or the adjustment you are making A clear rationale for why you are doing it An expectation of what should happen Document the hypothesis, rationale, and expected result Compare the actual result to the expected result Document the differences and account for any discrepancies Step 14.Now, run the model with Deer Birth Rate = 0.3. Label the run Baseline Ex2p14 DBR=0p3. Step 15.Now do a few additional runs, changing and testing variables and look at the results. Does this replicate the reference mode? Step 16.If you changed the value for Deer Birth Rate via the equation window, dont forget to change it back to 0.5. We also changed the time step to 5, so revert it to the original setting, 1. Step 17.Save the model. This completes the deer sector. We have created a stock (DEER) with an input (Deer Births) and an output (Deer Deaths). This is an example of a basic population stock. Kaibab Plateau Exercise TE R1526 / 116 Figure 9 The Deer population stock-and-flow diagram Note that the deer population has not equilibrated; births still outnumber deaths to a significant degree. We will work on equilibrating the deer population in the next section. Kaibab Plateau Exercise TE R1527 / 116 (Review 1). About the reading in Sterman, Chapter 3.The modeling process.Focus on Tabel 3-1 Why are we spending so much time with the Kaibab Pleateau problem (Review) 1.A real world policy problem the problem is created by a policy intervention 2.The system begins in a state of equilibrium (whether or not it is desirable is a matter of judgement) 3.The full implications of the policy intervention are unanticipated 4. An empirical reference mode, that provides a clear example the relationship between structure and behaviour. 5.A long time horizon. 6.There is an unambiguous dynamic hypothesis -structure and behavior7.The model is of manageable size - 3 stocks it is a good exemplar for your projects 8.The model boundaries are relatively straightforward. 9.The Issues involve stake holders with conflicting values 10.The issue population growth that is unsustainable, that exceeds the carrying capacity of the species niche, is generic 11. The limiting factor changes as a consequence of the policy intervention. 12.The model provides a metaphor for a larger problem, the challenges faced by the human species as a consequence of behaviours that are exceeding the carrying capacity of planet earth. Kaibab Plateau Exercise TE R1528 / 116 3.3Exercise 3: Deer Killed by Mountain Lions Now that we have a basic population stock, we are going to add mountain lions, which are a natural predator of deer. It is useful to think about the story that we are modelling, in this case a story about how lions hunting deer affects the deer population. The rate at which deer are killed by mountain lions is a bit more complicated than natural birth and death rates. How many deer might be killed by mountain lions on the Kaibab Plateau during one year? The answer depends on two assumptions: 1.The number of mountain lions; obviously, the more mountain lions, the more deer that will be killed2.The number of deer; the fewer deer per acre, the more difficult they are to track, so the fewer that will be killed Step 1.Following the first assumption, add deer killed by mountain lions to the causal loop diagram, assuming that the number of mountain lions on the plateau is constant. Figure 10 Causal loop diagram for deer Is the loop reinforcing or balancing? Representing a system as a structure of feedback loops does not require data but it does require clear thinking. When you are documenting your model and explaining it to others, your causal loop diagram and the thinking behind it will be useful, especially as you integrate increasingly complex feedback loops. Step 2.Open your stock-and-flow model from the last exercise and save it under a new name. That way, you have a saved model from each exercise so that you can go back to previous version of the model if needed. Step 3.In the Control Panel, clear the datasets from the last exercise. DeerDeer Births++Deer PopulationGrowthDeer Birth Rate+Deer Deaths+-Deer PopulationDeclineDeer AverageLifetime-Deer Killed byLions+Kaibab Plateau Exercise TE R1529 / 116 Step 4.In the stock-and-flow diagram, add a new variable, Deer Killed by Lions. Which existing variable should this variable connect to? It can connect to Deer Deaths or DEER; is one better than the other? An important principle when modelling is to keep the model consistent. DEER is a stock of the total number of deer in the population. Deer Deaths is the total number of deer that are killed. As such, it makes sense to link Deer Killed by Lions to Deer Deaths instead of DEER so that Deer Deaths comprises the total number of deer deaths, regardless of cause. Step 5.Add the equation. Deer Deaths = (DEER/Deer Average Lifetime)+Deer Killed by Lions Units: Deer/Year Following the second assumption, think about variables associated with that scenario. What do we need to add? The plateau is bounded, so there is a fixed Land Area. We have the total deer population so we can calculate the resulting Deer Density. Since we are talking about predation, we also need to add Lions. Land Area and Deer Density are quantitative variables, so we dont need to add them to the causal loop diagram, but it is necessary to think about their dimensionality (units). Step 6.Add the variables to the stock-and-flow diagram and add the units when linking the variables in the equations editor. Land Area = 800000 Units: Acres Lions = 400 Units: Lions Deer Density = DEER / Land Area Units: Deer/Acre Kaibab Plateau Exercise TE R1530 / 116 Step 7.Add the polarity for each relationship, beginning with the constants. We have created a new feedback loop. Is the loop reinforcing or balancing? What is a good name for the loop? We have one variable left: what is the equation for Deer Killed by Lions? Per assumptions 1 and 2 at the start of this exercise, it depends on the number of mountain lions and the number of deer each lion can kill. How can we link these variables? We need to add a new variable, Deer Killed per Lion, which depends on Deer Density, measured in deer per acre, on the plateau; naturally, the more deer on the plateau, the more deer for lions to eat, so this relationship can be thought of as showing the efficiency of lions in their search for deer to kill. Deer Killed per Lion is the first lookup function (also called a table function) that we will create.CREATIING A LOOKUP FUNCTION See Sections 4.14, Understanding Lookup Functions, 4.15, Creating Lookup Functions, and 4.16, Exporting Lookup Graphs, for information on what lookup functions are and how to use them when modelling.ANOTHER EXAMPLE OF CREATING ELEMENTS OF A MODEL WITH INCOMPLETE INFORMATION Before we create the lookup function, we should think about the information we have. According to the state-of-knowledge assessment, in 1900, there were 4,000 deer on the plateau, which measured about 800,000 acres. Therefore, Deer Density in 1900 was 4,000/800,000, or 0.005 deer per acre. Lets assume that when Deer Density = 0.005 deer per acre, each mountain lion can kill 3 deer per year. How might be arrive at this assumption? We might have historical or biological data, or this is a value that may have come up though conversations with Park Services employees. What we do know from the historical record is that the deer and lion populations were in a state of equilibrium. Note that that the number of lions on the plateau does not enter into this lookup equation, though it does depend on the behaviour of lions. Also if we didnt know the behaviour of lions in the Kaibab Plateau, we could use proxy information on the behaviour of lions elsewhere. As the density of deer on the plateau increases, it becomes easier for the mountain lions to find and kill more deer. So when Deer Density = 0.025 (five times the density in 1900), many deer are unable to find cover, and as a result, each lion can kill 30 deer per year. Eventually, when Deer Density = 0.05 deer per acre (ten times the density in Kaibab Plateau Exercise TE R1531 / 116 1900), the number of deer each lion can kill reaches its maximum value of 60 deer per year. Think of this data as (x, y) values on a graph. These values can be summarised as: 12.(0, 0) the lower bound; if there are no deer on the plateau, no deer can be killed 13.(0.005, 3) the values for 1900 14.(0.025, 30) useful mid-point values for the S-shaped relationship being represented 15.(0.05, 60) the upper bound HOW DO YOU KNOW WHEN A LOOKUP FUNCTION IS CALLED FOR.(CLUES) [1]WHEN YOU ARE MOVING BETWEEN SECTORS AND CHANGING THE DIMENSION OF VARIABLES [2]WHEN IN A CAUSAL LOOP OR STOCKFLOW DIAGRAM YOU HAVE ONE INPUT VARIABLE AND ONE OUTPUT VARIABLE Step 8.Use this data to create a lookup function.The lookup window with data is shown below. Figure 11 Lookup graph for deer density and deer killed per lion Step 9.Two additional points: Kaibab Plateau Exercise TE R1532 / 116 Be sure to label the x and y axes (in the blue boxes) Adjust the minimums and maximums (in the red boxes) REMEMBER TO INCLUDE A COMMENT DESCRIBING THE RATIONALE BEHOND YOUR LOOKUP FUNCTION.ALSO, A LOOKUP FUNCTION LIKE ALL OTHER FUNCTIONS IN SYSTEM DYNAMICS MODELING IMPLIES CAUSALITY Numbers that quantify relationships are very important to modelling. The above data captures the relationships between deer, land, and mountain lions. As the values of Deer Density and Deer Killed per Lion change, the pattern exhibited by the reference mode variable, DEER, changes. Even though it seems simple, make sure that you understand this. The Causes Strip is helpful in understanding this behaviour of cause and effect. Step 10.Now that we have entered lookup data, write the equation for Deer Killed per Lion. The comment box is a good place to explain the logic behind how this equation was constructed.Deer Killed per Lion = WITH LOOKUP (Deer Density, ([(0,0)(0.05,60),(0,0),(0.005,3),(0.025,30),(0.05,60)],(0,0), (0.005,3),(0.025,30),(0.05,60) )) Units: Deer/lion Step 11.Now that we have the equation for Deer Killed per Lion, write the equation for Deer Killed by Lions.Deer Killed by Lions = Deer Killed per Lion*Lions Units: Deer/Year You may ask, What is the difference between these two variables? Deer Killed per Lion is a lookup function whose input is dependent on the area in the plateau, the number of deer, and the number of lions. Deer Killed by Lions is the total number of deer killed by lions. As such, the number of Deer Killed by Lions per year is equal to the number of Lions multiplied by Deer Killed per Lion. At this point we are assuming that there were 400 mountain lions on the plateau in 1900. We will later complicate this further by creating a dynamic mountain lion population and bounty killing. Step 12.Run the new model under the name Baseline Ex3p12. Step 13.View the graphs for Deer Deaths and Deer Killed by Lions. What behaviour does the deer population exhibit? A model will be in equilibrium, as it is here, whenever the inflows and outflows to a stock are equal. Creating a set of assumptions in which a model is in equilibrium can be helpful in understanding the model and performing Kaibab Plateau Exercise TE R1533 / 116 comparative runs. In its present state, the model is in a state of equilibrium because we are working on the assumption that the Kaibab Plateau was in a state of equilibrium between the deer and mountain lion populations before policy makers intervened.EQUILIBRIATING A MODEL HAVING IT IN STATE OF EQUIBRIUIUM INVOLVES TWO CONSIDERATIONS ONE THEORETICAL AND ONE EMPIRICAL.IN THIS CASE BOTH ARE PRESENT.HOWEVER WE CAN ALWAYS CREATE EQUILBRIUM BY HAVING THE INFLOWS OF OUR STOCKS BE EQUAL. Was this a desirable state? Step 14.Run the model with Lions = 600. Name the run Baseline Ex3Step 14 L=600. Before looking at the results, frame a hypothesis about what will happen. What is the rationale behind the hypothesis? Step 15.Now look at the results. What did happen? How do the results differ from the hypothesis? What have we learnt? Step 16.Run the model again with Lions = 200. Name the run Baseline Ex3p16 L=200. Step 17.Frame a hypothesis about what will happen. What is the rationale? Step 18.Now look at the results. What did happen? How do the results differ? What has been learnt? Step 19.Now lets compare some runs. Go to the Control Panel and load all three runs. Note the value of naming the runs to denote their properties. What does the comparison show? Does this replicate the reference mode? Step 20.Save the model. Below is the complete stock-and-flow diagram for the deer sector. Kaibab Plateau Exercise TE R1534 / 116 Figure 12 The Deer population stock-and-flow diagram DEERDeer BirthsS DEER HERD GROWTHDEERGROWTHDeer Birth RateSDeer InitialSDeer DeathsSDEERDECLINELand AreaDeer DensitySODeer Killed perLionSDeer Killed byLionsSSDEER KILLEDBY LIONS

SKaibab Plateau Exercise TE R1535 / 116 3.4Exercise 4: Lions and Lion Hunting You will have noticed that we are slowly adding complexity to the model in order to reproduce our reference mode. You may ask, How much complexity do we need? This is to some degree a theoretical question, but when modelling there is also an empirical element involved. After all, we are adding values and equations at each step of the way so we are building the model in a coherent, mathematical way, adding exactly the amount of complexity needed to reproduce the reference mode. Once the reference mode has been replicated, you can add additional complexity to explore additional policy options. REMINDER:REPRODUCTING THE REFERENCE MODE IS NOT THE ONLY OR A SUFFICIENT TEST OF VALIDITY.In the previous section, the lion population was constant while the deer population was dynamic. Now, the model will be modified to create a dynamic mountain lion population with the following variables:16.Lion births 17.Lion deaths18.Lions killed by hunters Figure 13 Causal loop diagram for lions Step 1.Open your stock-and-flow model from the last exercise and save it under a new name. That way, you have a saved model from each exercise so that you can go back to previous version of the model if needed. Step 2.In the Control Panel, clear the datasets from the last exercise. Step 3.Instead of a constant, reformulate Lions as a stock, LIONS. Create this as part of a new sector below the DEER sector. You will have to delete the variable Lions first; Vensim does not allow two variables with the Deer Killed byLions+LionsLion Births++Lion PopulationGrowthLion Birth Rate+Lion Deaths+-Lion PopulationDeclineLion AverageLifetime-Lion Hunting++Kaibab Plateau Exercise TE R1536 / 116 same name. LIONS, Lion Births, and Lion Deaths are formulated exactly like the birth and death rates for deer. Step 4.Write equations for Lion Births and Lion Deaths, starting with the constants.Lion Birth Rate = 0.2Units: Lions/Lion/YearLion Average Lifetime = 10Units: Years Lions Initial = 400 Units: Lions Lion Births = LIONS*Lion Birth Rate Units: LIONS/year Lion Deaths = LIONS*Lion Average Lifetime Units: LIONS/year LIONS = INTEG (Lion Births-Lion Deaths, Lions Initial) Units: LIONS Step 5.Run the model with the name Baseline Ex4c.SHIFT IN LOOP DOMINANCE:THIS IS ANOTHER EXAMPLE OF THE RLEATIONSHIP BETWEEN STRUCTURE AND BEHAVIOR What we will see is called a shift in loop dominance, in system dynamics terminology. This is when the activity in one loop, in this case, exponential growth in the deer population, begins to change because of activity in another loop, in this case, the lion population. What has happened is that the deer population is initially dominated by the positive feedback loop of deer births, but as the lion population grows, the deer population becomes dominated by the negative feedback loop, deer deaths, and the deer population begins to decline.NOW WE ARE GOING TO LOOK AT OUR FIRST POLICY INTERVENTION,A VERY SIMPLE ONE.THERE ARE TWO QUALITIES. FIRST, THE INTERVENTION OCCURS AT THE BOUNDARY.SECOND IT OFFERS POLICY OPTIONS,BOTH THE SIZE OF THE INTERVENTION AND WHEN IT OCCURS.So far, there are no predators for lions. Now, we will add a new element, Lions Killed, which is the rate of mountain lions killed by hunters per year. The hunting of lions had always been allowed on the Kaibab Plateau, so this is similar to Deer Killed by Lions because hunters were predators of lions as lions were predators of deer. There are a few different ways to represent hunting depending on the policy that we want to model. One way is as a constant (e.g. x number of lions can be killed per year) and another is as a constant fraction (e.g. 10% of the lion population can be killed each year). For now, let us assume the latter. Kaibab Plateau Exercise TE R1537 / 116 Step 6.Call this Lions Killed Normal, because this was the amount of hunting that took place before the bounty was offered. Assume that in 1900, Lions Killed Normal = 0.1 (10% as a decimal number).Lions Killed Normal = 0.1Units: Lions/Year Lions Killed = See the next step for more information Per the state-of-knowledge assessment, a problem arose in 1906 when a bounty was offered on the hunting of lions, which went above the normal amount of hunting, which was 10% of the lion population. Add bounty hunting to the mix to examine the effects of this policy imposed by the Forest Service. This is the first example of a policy intervention. We will do this in Vensim using a mathematical function called a STEP function (see Section 4.17, Creating STEP Functions, for more information); here are the variables we need: Lions Killed Bounty = 0.2 Units: Lions/Year Lion Bounty Start Time = 1906Units: Years Lion Bounty End Time = 1950Units: Years Lion Deaths = (LIONS/Lions Average Lifetime)+Lions Killed Units: Lions/Year Lions Killed = LIONS*(Lions Killed Normal+STEP(Lions Killed Bounty, Lion Bounty Start Time)-STEP(Lions Killed Bounty, Lion Bounty End Time)) Units: Lions In the state-of-knowledge assessment, it was indicated that the bounty remained in force indefinitely. It could be interesting to examine what might have happened had the bounty been stopped sometime after 1906. This can be tested by choosing a year between 1906 and 1950 for the Bounty End Time. In the days before the bounty was started, Lions Killed (the total number of lions killed) = Lions Killed Normal. After 1906, with the bounty, we have a new formula: Lions Killed = Lions Killed Normal + Lions Killed BountyNORMAL IS A TERM YOU WILL ENCOUNTER FREQUENTLY IN PROFESSOR FORRESTERS MODELS.IT TYPICALLY REFERES TO CONDITIONS IN THE ABSENCE OF INTERVENTION ON THE PART OF THE MODELER. Kaibab Plateau Exercise TE R1538 / 116 Step 7.Add this to the diagram. This gives us a total of 0.1 + 0.2 = 0.3 for the percentage of lions killed each year. Before running the model, answer a few questions about anticipated model behaviour!!. How many mountain lions were born in 1900? How many died? How many were killed by hunters? What does this indicate about the mountain lion population? How many mountain lions were born in 1906? How many died? How many were killed by hunters? What does this indicate about the mountain lion population? What will happen to the deer population beginning in 1906? Step 8.Now that the LIONS sector has been created, link it to the DEER sector. This is done using a new type of variable called a shadow variable, which is discussed in more detail in Section 4.18, Adding Shadow Variables. Step 9.Clear the previous datasets in Control Panel > Datasets > Loaded Info. Step 10.Run the model under the title Baseline Ex4p10 L=400. What behaviour does the deer population exhibit after 1906? Why? What behaviour does Deer Killed by Lions exhibit? What explains this seemingly puzzling behaviour? Use the causes strip graphs to help find the answer. Step 11.Run the model with Lion Bounty Start Time = 1920 under the title Baseline Ex4p11 BST=1920. How does the behaviour in this run compare with the behaviour in the initial run? Kaibab Plateau Exercise TE R1539 / 116 Step 12.Run the model under the title Baseline Ex4p12 L=600 ALD=20setting Deer Average Lifetime = 20 years and Lions Initial = 600. Step 13.Using the causes strip graphs, compare the values of DEER, LIONS, and Deer Killed by Lions. Describe and explain the differences between the two runs. Does this reproduce the reference mode? Step 14.Return the variables to their original values: Lion Bounty Start Time = 1906 Lions Initial = 400 Deer Average Lifetime = 5 Step 15.Save the model. This completes the population stocks-and-flows for deer and lions. We have two similar sectors with an input for births and an output for deaths. Deaths are determined through two calculations, one for normal deaths and the other for deaths through predation. In the case of deer, normal deaths occur because of Deer Average Lifetime while predation deaths occur from mountain lions. In the case of lions, death occurs because of hunting, which is of two types, normal hunting, which kills 10% of the population, and bounty hunting, which kills 20% of the population. Kaibab Plateau Exercise TE R1540 / 116 Figure 14 Stock-and-flow diagram for Deer and Lions NEXT WEEKS READING IS IN STERMAN CHAPTER SIX, STOCKS AND FLOWS.WE WONT BE GIVING MUCH CLASS TIME TO DISCUSSING THIS.THE READING IS NOT TECHNICALLY DIFFICULT.BUT THAT DOESNT MEAN YOU SHOULD JUST SKIM OVER IT.BY MY COUNT THERE ARE 43 DIFFERENT EXAMPLES OF STOCKS.THE GOAL IS TO DEVELOP AND INTUITIVE UNDERSTANDING OF STOCK AND STERMANS CHAPTER GIVES YOU PLENTY OF OPPORTUNITY TO DO THIS.I WOULD SUGGEST THE FOLLOWING: (1)COMPILE A LIST OF THE STOCKS, AND FLOWS FOR EACH BY LOOKING AT THE EXAMPLES. (43) (2)TO REDUCE THE COMPLEXITY AND SIMPLIFY THE PROCESS OF GAINING AN INTUITIVE UNDERSTANDING, GROUP THE STOCKS INTO CATEGORIES. (3) FOR EACH CATEGORY, COME UP WITH SIMILAR EXAMPLES OF STOCKSON YOUR OWN. (4) THIS MIGHT BE AN EXERCISE TO DO WITH A FRIEND. (5)SINCE WE WILL ALSO BE DISCUSSING THE REPLICATION PROJECTSYOU COULD ALSO CHECK THOSE OUT.

DEERDeer BirthsSDEER HERD GROWTHDEERGROWTHDeer Birth RateSDeer InitialSDeer DeathsSDEERDECLINEDeer AverageLifetimeOLand AreaDeer DensitySODeer Killed perLionSDeer Killed byLionsSSDEER KILLEDBY LIONSLIONSLion BirthsLion Birth RateSLion DeathsLions AverageLifetimeSSLION DECLINESLION GROWTHLions InitialSLions KilledNormalLions KilledBountyLion BountyStart TimeLion BountyEnd Time

SLions KilledSSOSSSKaibab Plateau Exercise TE R1541 / 116 3.5Exercise 5: Creating Vegetation Consumption Like any herbivore, deer depend on vegetation for sustenance. How should the amount of vegetation available per deer be incorporated into the model? One way is to assume a constant vegetation supply, Vegetation, on the plateau; this makes sense since the plateau is a bounded area. Step 1.Add Vegetation as a new variable to the causal loop diagram, perhaps to the right of DEER. Use a different colour, for example, green, to colour the arrows added. Figure 15 Causal loop diagram for deer, lions, and vegetation Is the new relationship positive or negative? Step 2.Open your stock-and-flow model from the last exercise and save it under a new name. That way, you have a saved model from each exercise so that you can go back to previous version of the model if needed. Step 3.In the Control Panel, clear the datasets from the last exercise. Since our goal in building this model is to understand the overall dynamics of the Kaibab Plateau, the ecosystem we create will be somewhat general. If, in our policy analysis, the need to be more specific later arises, we can do so. To measure Vegetation, we need to create a unit. Assume that the normal amount of vegetation grown on 4 acres is one unit. Following this, in 1900 there were 800,000/4 = 200,000 units of vegetation on the plateau. These 200,000 DeerDeer Births++Deer PopulationGrowthDeer Birth Rate+Deer Deaths+-Deer PopulationDeclineDeer AverageLifetime-Deer Killed byLions+LionsLion Births++Lion PopulationGrowthLion Birth Rate+Lion Deaths+-Lion PopulationDeclineLion AverageLifetime-Lion Hunting++Vegetation+Kaibab Plateau Exercise TE R1542 / 116 units of vegetation supported 4,000 deer, meaning that on average, each deer consumed 200,000/4000 = 50 units; this is the calculated value for a new variable: Vegetation Available per Deer. Step 4.Add DEER as a shadow variable. Step 5.Add Vegetation and Vegetation Available per Deer. Vegetation = 200000 Units: Units Vegetation Available per Deer = Vegetation/DEER Units: Units/Deer Although 50 units are what is available to eat per deer under equilibrium conditions, in order to survive and live an average lifetime (5 years), a deer only requires 1 unit. Naturally, as the deer population increases, following the bounty, Vegetation Available per Deer decreases, reducing Deer Average Lifetime. As Vegetation Available per Deer falls below 1 unit, Deer Average Lifetime falls as well. Assume: 19.0 units = 0 years20.0.4 units = 2 years 21.0.6 units = 4 years 22.1 unit = 5 years These values represent the relationship between Vegetation Available per Deer (the independent variable) and Deer Average Lifetime (the dependent variable). Naturally, when no vegetation is available, deer cannot survive. As the vegetation supply increases, Deer Average Lifetime increases, first slowly and then more rapidly. Beyond a certain point (Vegetation Available per Deer = 0.6 in the lookup), the effects of Vegetation Available per Deer on Deer Average Lifetime begin to slow. When Vegetation Available per Deer = 1, the normal Deer Average Lifetime is assumed. If Vegetation Available per Deer exceeds the maximum value of 1, it is assumed that vegetation availability has no further effect on longevity. Step 6.Draw a lookup function showing the relationship between Vegetation Available per Deer and Deer Average Lifetime. Kaibab Plateau Exercise TE R1543 / 116 When there are only given a few values for a lookup, as we have here, it raises the question, What should be done about the discontinuities or values that seem implausible in a lookup function? For example, it might be expected that the graph would be flatter at the lower end and that the graphs S shape might be smoother. It is possible to supplement the data with research on the biology of deer and the relationship between their consumption of vegetation and average lifetime. We should also perform sensitivity analysis to see how variations in the shape of this curve can impact results in the model. Including sensitivity analysis on this and other lookup functions in the report you will submit at the end of this exercise is entirely appropriate. Figure 16 Changes in deer average lifetime caused by vegetation availability Kaibab Plateau Exercise TE R1544 / 116 What Are Limiting Factors? This is a good time to discuss the idea of a limiting factor. We have already been exposed to limiting factors, but have not labelled them as such. Limiting factors play an important role in systems where behaviour is determined by the dominance of one or more positive feedback loops. In the Kaibab Plateau exercise, the deer population grew until it encountered its limiting factors:23.Lions The first limiting factor, which created a naturally equilibrated population level for the deer population, prior to the introduction of the lion hunting bounty. 24.Vegetation The second limiting factor, which came into play after the removal (or reduction) of the first limiting factor and the limits on the system (the plateau is a bounded space). As with stakeholders, it is useful to think of the limiting factors that exist in a model when we are creating it because no system in which growth in generated by a positive feedback (the growth of a population, a cancer, or a stock of capital that is being reinvested), can grow forever; all systems are bounded, and therefore have limiting factors. Limiting factors can also help to determine what the leverage points are so that we can think clearly about where to apply changes in a system. Step 7.Now write the equation for the relationship between Deer Average Lifetime and Vegetation Available per Deer as a lookup function.Deer Average Lifetime = WITH LOOKUP (Vegetation Available per Deer, ([(0,0)-(1,5)],(0,0),(0.4,2),(0.6,4),(1,5) )) Units: Years Without actually running the model, what growth pattern will be generated in the deer population? Why? This lookup function shows that changes in Deer Average Lifetime are caused by changes in the amount of Vegetation Available per Deer. Before considering this lookup function, it is helpful to recall the equation for Deer Deaths. Excluding Deer Killed by Lions, the equation is Deer Deaths = DEER / Deer Average Lifetime. Recall that in Exercise 2 in Deer Deaths (excluding lions and possible vegetation-supply shortfalls), Deer Average Lifetime was defined as 5 years. Think of this as the biologically determined lifetime of deer, on average, living on the Kaibab Plateau. Therefore, 1/5th (20% or 0.2) of the deer die each year. We could obtain a more precise number by consulting naturalists or publications knowledgeable about the Kaibab Plateau, but 1/5th is a reasonable Kaibab Plateau Exercise TE R1545 / 116 approximation. This is another example of a normal, that is, a value entered under the assumption of all things being equal. When might all other things not be equal? They may be unequal, for example, during a shortage in the vegetation supply. At this point, it is useful to introduce a new concept, that of carrying capacity. Carrying capacity is the maximum size of a given population that an ecosystem (or ecological niche) can sustain. It is determined by the limiting factor or factors, which are different for different species. For example, the amount of vegetation determines the carrying capacity of the Kaibab Plateau ecosystem for deer. The size of the deer population determines the carrying capacity of the ecosystem for lions, and so forth. What we will see in the reference mode is the decimation of the deer population as the population exceeds the carrying capacity of the land. The reference mode shows an alternative scenario for the carrying capacity of the land under the title of Probable Capacity if Herd Reduced in 1918, as seen below. Figure 17 Deer population from 1906 to 1939 What if the value of the independent variable exceeds the upper bound (X-max value) specified in a lookup function? In Vensim, the function continues to output the highest dependent variable value specified, no matter how large the independent variable value becomes. The model that we are building is not intended to deal with the effects of overconsumption of vegetation on Deer Average Lifetime, so we do not need to include it in the model. Remember that a model should focus on the problem at hand and not try to replicate an ecosystem in all its details. Obviously we use the most precise data that we have. However if we dont have precise data, we can ask the question: how much difference does the absence of precise data make? What about problems introduced by the lack of precision in the representation of the S curve? There can be no question that the simple representation of the S curve between Vegetation Available per Deer and Deer Average Lifetime is only a crude approximation. A more accurate representation is found in Figure 18; note that the shape of Kaibab Plateau Exercise TE R1546 / 116 the curve is more important than the specific numbers. The most accurate line would be a smooth S curve with no discontinuities. Figure 18 Smooth S-curve function Now, instead of focusing on the simple representation of the S curve, perform some sensitivity analysis and look at the different outcomes generated by these representations to see if the differences are relevant to the problem and conclusions that we are using the model to analyse. Before moving on, it is useful to note that some data points and values in a lookup graph are more important than others. In this case, it is the lower bound of the independent variable and the upper bound of the dependent variable. When creating a lookup function, begin by defining the upper and lower bounds. Assume that when Deer Average Lifetime = 5, 20% of the deer population dies. What percentage of the deer population dies when Deer Average Lifetime = 4? 3? 2? 1?12

Recall that the Deer Birth Rate = 0.5. For what value of Deer Average Lifetime will Deer Birth Rate and Deer Deaths be equal? What value of Vegetation Available per Deer results in the value of Deer Average Lifetime determined in Q4?

12 Respectively: 0.25, 0.33, 0.5, and 1.0. Kaibab Plateau Exercise TE R1547 / 116 What size deer population is required to produce this value for Vegetation Available per Deer? What does this imply? Step 8.In the Control Panel, clear the datasets and do a new run Baseline Ex5p8 V=200,000. Record the results, focusing in particular on DEER.See Section 4.20, Understanding Warnings, for information on warnings that may appear. Step 9.Run the model again, setting Vegetation = 300,000. Label the run Baseline Ex5p9 V=300,000. Step 10.Run the model with Vegetation = 100,000. Label the run Baseline Ex5p10 V=100,000. Step 11.Now use the causes strip graph to compare the three runs. What explains the pattern that each of the runs exhibits? What explains the differences between the three runs? Does this replicate the reference mode? Step 12.Save the model. Now that we have a partial model of the vegetation sector, we will complete it in the next exercise. Kaibab Plateau Exercise TE R1348 / 116 Figure 19 Stock-and-flow diagram for Deer, Lions, and constant VegetationDEERDeer BirthsS DEER HERD GROWTHDEERGROWTHDeer Birth RateSDeer InitialSDeer DeathsSDEERDECLINEDeer AverageLifetimeOLand AreaDeer DensitySODeer Killed perLionSDeer Killed byLionsSSDEER KILLEDBY LIONSLIONSLion BirthsLion Birth RateSLion DeathsLions AverageLifetimeSSLION DECLINESLION GROWTHLions InitialSLions KilledNormalLions KilledBountyLion BountyStart TimeLion BountyEnd Time

SVegetationAvailable per Deer

OLions KilledSSOSSSSVegetationSKaibab Plateau Exercise TE R1349 / 116 3.6Exercise 6: Creating Vegetation Supply The model discussed in the last section generates S-shaped growth in the deer population beginning in 1906. The discrepancy between model behaviour in the last section and the reference mode in Figure 2 indicates that something is still missing from the model. What could this be? This is the most difficult part of the vegetation model to understand, so lets look at a causal loop diagram first to get the overall picture. Note that the major feedback loops are colour-coded and have loop polarities indicated.In the last section, we assumed a constant vegetation supply of 200,000 units. Now, lets examine the supply in more detail. Vegetation is not a population in the way that we normally think of a population, but it is living and so can be formulated as a population stock influenced by two principal flows, similar to what we have already made:25.Vegetation Growth Similar to births 26.Vegetation Consumption Similar to deaths DeerDeer Births++Deer PopulationGrowthDeer Birth Rate+Deer Deaths+-Deer PopulationDeclineDeer AverageLifetime-Deer Killed byLions+LionsLion Births++Lion PopulationGrowthLion Birth Rate+Lion Deaths+-Lion PopulationDeclineLion AverageLifetime-Lion Hunting++VegetationVegetationConsumed per Deer++Vegetation GrowthVegetationConsumption-+-+Vegetation GrowthVegetationConsumedKaibab Plateau Exercise TE R1350 / 116 Figure 20 Causal loop diagram for deer, lions, and vegetation Lets consider Vegetation Consumption. Who is eating vegetation? Based on the state-of-knowledge assessment, the amount consumed each year is determined by the number of DEER and a new variable, the amount of Vegetation Consumed per Deer; this is a variation on the variable we created in the previous section, Vegetation Available per Deer. Why do we have two different variables? In reality, the amount of vegetation available and the amount of vegetation consumed are two different things, the latter being a subset of the former. So naturally, the amount of Vegetation Consumed per Deer is influenced by the Vegetation Available per Deer. Remember that in the scenario we are modelling, as the deer population grew, the number of deer exceeded the equilibrium rate for the deer-to-vegetation ratio, meaning that Vegetation Available per Deer and Vegetation Consumed per Deer both decreased as the deer population increased, leading to mass starvation. As with deer and lions, lets also assume a new variable, Vegetation Initial, to show how much vegetation was available on the plateau at the start of the model. Assume that in 1900, under normal conditions, the plateau supported 196,000 units of vegetation instead of the maximum, 200,000. Step 1.Open your stock-and-flow model from the last exercise and save it under a new name so you have a saved model from each exercise so that you can go back to previous version of the model if needed. Step 2.In the Control Panel, clear the datasets from the last exercise. Step 3.Add the new variables.Vegetation Initial = 196000 Units: Units VEGETATION = INTEG (Vegetation Growth-Vegetation Consumption, Vegetation Initial) Units: Units Step 4.Draw a lookup function showing the relationship between Vegetation Available per Deer and Vegetation Consumed per Deer. Step 5.Change the link to Deer Average Lifetime from Vegetation Available per Deer to Vegetation Consumed per Deer. For Vegetation Consumed per Deer, assume:27.When vegetation is freely available, defined as being greater than 2.0 units, the maximum a deer can eat is 1.0 unit per year, giving us the standard Deer Average Lifetime of 5 years.Kaibab Plateau Exercise TE R1351 / 116 28.When vegetation available is less than 2.0 units, grazing becomes more difficult and the amount each deer consumes falls slowly.29.When vegetation available reaches 1.0 unit, each deer consumes 0.8 units.30.When vegetation available falls below 1.0 unit, the amount each deer consumes falls more rapidly. Note that the data provided is more limited than for previous lookups. How should we decide on ranges for the independent and dependent variables? How should we decide on the shape of the curve and the number of data points used? This is a case where inserting variables in the New fields to smoothen the shape may be helpful. Here is h