D-4286

A SYSTEM DYNAMICS APPROACH TO RABBITS

(S-shaped growth)

Terri Duhon

Massachusetts Institute of Technology

System Dynamics Group

System Dynamics in Education Project

April 1, 1992

Copyright © 1992 by MIT

Permission granted to copy for non-commercial educational purposes

2 D-4286

A System Dynamics Approach to RabbitsTerri Duhon1

A b s t r a c t

S-shaped growth characterizes the behavior of a system with alimiting factor that eventually causes the system’s exponential growth toapproach zero. S-shaped growth commonly occurs in many differentsystems, and is an important mode of behavior in many system dynamicsmodels. A population living in a limited area, and the number of peoplethat have been infected with an epidemic both exhibit S-shaped growth.

A System Dynamics Approach To Rabbits studies S-shaped growthby examining the behavior of a rabbit population living in ten fenced-inacres. To study the behavior of the rabbit system, the paper will firstdistinguish the different elements that comprise the system, and willthen make assumptions about the relationships between the elements.Next, using the relationships, the paper will present the structure of therabbit system.

A STELLA2 model will follow from the structure. The model willillustrate the system’s dynamic behavior. S-shaped growth will beevident in the illustration.

1Terri Duhon is an undergraduate sophomore at MIT, majoring in mathematics. She is amember of the System Dynamics in Education Project under the direction of Prof. JayForrester. Terri has been a member for six months.

2Systems Thinking Educational Learning Laboratory with Animation

D-4286 3

A SYSTEM DYNAMICS APPROACH TO RABBITS

I Introduction.............................................................................................................4II The Rabbit System and It’s Components ...................................................4

Population-Births Relationship.............................................................4Positive Feedback - Causal Loop Diagram.............................5Positive Feedback - Model............................................................6Positive Feedback - Population Behavior..............................7

Population-Deaths Relationship............................................................8Negative Feedback - Causal Loop Diagram............................8Negative Feedback - Model...........................................................8Negative Feedback - Population Behavior .............................1 0

Loop Dominance and Dynamic Equilibrium.........................................1 0The Rabbit System Behavior....................................................................1 2

III The Limiting Factor ..........................................................................................1 3The Missing Element ...................................................................................1 3The Effect of the Limited Area..............................................................1 3Relationship Between Density and Average Lifetime..................1 3

Second Negative Feedback - Causal Loop Diagram............1 4Second Negative Feedback - Model ...........................................1 6

IV Entire Model..........................................................................................................1 7Population Hypothesis................................................................................1 7Population Behavior- S-Shaped Growth.............................................1 7

Positive Feedback Loop Dominance ..........................................1 8Point of Inflection...........................................................................1 8Approach to Dynamic Equilibrium.............................................1 9

V Conclusion...............................................................................................................1 9VI Appendix.................................................................................................................2 0VII Notes ......................................................................................................................2 2

4 D-4286

I Introduction

A System Dynamics Approach to Rabbits is a partial-introduction tosystem dynamics, a field which studies the behavior of a system throughthe system’s structure. It is assumed that the reader has already beenintroduced to a few basic concepts in system dynamics. An understandingof causal loops--negative and positive feedback loops--is helpful forreading this paper. The reader does not need to completely understandthese terms, a familiarity is sufficient because in this paper, they areexplained using concrete examples.

A System Dynamics Approach to Rabbits studies the structure of arabbit population living in ten acres and introduces S-shaped growth andtwo related terms, loop dominance and dynamic equilibrium. A SystemDynamics Approach to Rabbits should give an understanding of thispartial-introductory level and generate interest in learning andunderstanding the more complex levels and applications of systemdynamics.

II The Rabbit System and It’s Components

Ten rabbits, five females and five males, are placed within anenclosed ten acre field, cut off from predators and unable to escape. Tostudy this system, we are given several important pieces of information.Ideal living conditions for rabbits are about ten rabbits per acre of land.And under ideal conditions, rabbits live an average of seven years. Also,every three months, females have a litter of five babies. Using thesefacts, we will be able to study the resulting change in population. To begin, we must distinguish the different elements of the systemand then make assumptions about the relationships between the elements.The most obvious elements are the rabbit population, births and deaths.

Population-Births Relationship

The relationship between the rabbit population and the rabbit birthsis characterized by reinforcement. The rabbit population produces acertain number of babies each year. The new babies are then added to thepopulation and soon they begin to reproduce. The increased population inturn produces a larger number of babies in the next year. Thus, thepopulation reinforces itself.

The reinforcement process is continual. It does not happen onceeach year. Rabbit babies are born and added to the rabbit population allthe time.

D-4286 5

Positive Feedback - Causal Loop Diagram

Continuous reinforcement as described above constitutes a positivefeedback loop3. The population and the births reinforce each other. Theloop is shown below in Fig 1.

Fig 1 - The causal loop diagram of the positive feedback loop involves the population and the births. It isa self reinforcing loop.

3For further explanation of positive feedback loops please reference Chapter 5 of the STELLA IIUser’s Guide, Hanover, NH 03755: High Performance Systems, ©1990)

6 D-4286

Positive Feedback - Model

A different representation of the positive feedback loop is themodel in Fig 2. The three parts of the model are the rabbit population, thebirths, and the birth fraction. The birth fraction represents the fractionof reproducing rabbits times the number of babies each reproducing rabbithas per year. When the birth fraction is multiplied by the rabbitpopulation, the number calculated is the number of baby rabbits per year.

rabbit population

rabbit births

birth fraction

Fig 2 - The positive feedback loop is more explicit when modeled on STELLA II4. The model allows usto use explicit numbers and equations to determine the population behavior.

The rabbit population and the birth rate are connected in a circulararrangement in the model, illustrating the positive feedback loop. Thebirths are calculated by multiplying the birth fraction and the rabbitpopulation. The number of new babies then flows into the total population.In turn, the population increase causes an increase in the number of newbabies and so on.

initial rabbit population = 10 rabbits

rabbit births = rabbit population * birth fraction = rabbits / year

birth fraction = percentage of reproducing rabbits* number of baby rabbits per female per year

birth fraction = 0.5 * 20(rabbits / rabbit) / year =10 / year

Fig 3 - These are the equations and units for the positive feedback loop model in Fig 2.

The birth fraction is a constant in the model. A constant birthfraction is a simplification of the rabbit system. Each female rabbit doesnot give birth to the same number of babies each year. Thus, to make the

4STELLA II V.1.02, High Performance Systems; Hanover, NH O3755; © 1990

STELLA II V.1.02 will be used throughout the paper to model the rabbit system.

D-4286 7

model as realistic as possible, the numbers that we use to calculate thebirth fraction are averages. The average number of reproducing femalesis half of the population, and the average number of babies each femalehas per year is twenty; therefore, the birth fraction is tenfold per year.

Positive Feedback - Population Behavior

When isolated, the positive feedback loop in the rabbit systemcauses the population to increase without bound. The graph belowillustrates the exponential increase in both population and births. Thegraph in Fig 4 and the equations in Fig 3 show that as the populationincreases, the number of births increases by ten times the populationincrease.

0.00 0.50 1.00 1.50 2.00Years 9:42 AM 2/27/92Page 3

1 :

1 :

1 :

2 :

2 :

2 :

0.00

1800.00

3600.00

1: rabbit population 2: births >42<

1

12

Fig 4 - This is the graph of the rabbit population and births when affected by the isolated positivefeedback loop. The behavior of both is exponential increase.

8 D-4286

Population-Deaths Relationship

The relationship between rabbit population and rabbit deaths ischaracterized by decline. Every year a certain number of deaths in therabbit population occurs. Once a rabbit dies, it is no longer included aspart of the population; therefore, the population decreases with eachdeath. At the same time as the population decreases, the number ofrabbits dying also decreases because there are a smaller amount ofrabbits left to die.

The approach of the rabbit population to zero is a continual process.It does not happen once each year. Rabbits die and are subtracted form thepopulation all the time.

Negative Feedback - Causal Loop Diagram

The relationship described above is a negative feedback loop5.Negative feedback loops work to adjust their subsystem towardsequilibrium. For the negative feedback loop in Fig 5, equilibrium occurswhen the rabbit population is zero. The work by the negative feedbackloop to reach equilibrium is a continual process.

Fig 5 - The causal loop diagram for the negative feedback loop involves the rabbit population and deaths.This loop works to reach equilibrium.

Negative Feedback - Model

A different type of representation of the negative feedback loop isthe model in Fig 6. The three parts of the model are the rabbit population,

5For further explanation of negative feedback loops please reference Chapter 5 of the STELLA IIUser’s Guide, Hanover, NH 03755: High Performance Systems, ©1990)

D-4286 9

deaths, and average lifetime. The average lifetime is used to calculatethe number of deaths in the rabbit system.

The model connects the rabbit population and the deaths in a circulararrangement, illustrating the negative feedback loop. This loop works toreach equilibrium by continually draining the number of deaths from thepopulation. The deaths are calculated by dividing the population by theaverage lifetime. When run in isolation, population and deaths eventuallyreach zero.

~

average lifetime

rabbit population

deaths

Fig 6 - This is the negative feedback loop modeled on STELLA II. The model allows us to use explicitequations and numbers to plot the behavior of the population over time.

initial rabbit population = 10 rabbits

rabbit deaths = rabbit population / average lifetime = rabbits / year

average lifetime = 7 years

Fig 7 - Here are the equations and the units for the negative feedback loop modeled in Fig 6.

In constructing this model, we assume that the average lifetime ofrabbits is always seven years. A constant average lifetime of seven yearsoccurs under ideal living conditions; therefore, we assume that livingconditions for the rabbits will always be ideal. With this assumption, wehave simplified the actual rabbit system.

10 D-4286

Negative Feedback - Population Behavior

The isolated negative feedback loop in Fig 6 causes the population toslowly decrease. Both the deaths and the population are graphed below.As the population dies off, the number of deaths slowly decreases. Thebehavior of the rabbit population, and the deaths is exponential decline.

0.00 5.00 10.00 15.00 20.00Years 9:46 AM 2/27/92Page 5

1 :

1 :

1 :

2 :

2 :

2 :

0.00

5.00

10.00

1: rabbit population 2: deaths >14<

1

1

1

122

2 2

Fig 8 - This is the graph of the population and deaths when affected by the isolated negative feedbackloop in Fig 6. Their behavior is exponential decline.

Loop Dominance and Dynamic Equilibrium

When a positive and negative feedback loop are used together, asshown in Fig 9, the strongest loop is the dominant one. In Fig 9, for oneloop to be dominant, it must must have a greater effect on the population.The rate of births by the positive feedback loop and of deaths by thenegative feedback loop are compared. Their comparison with each otherdetermines dominance.

D-4286 11

~

average lifetime

births

rabbit population

deaths

birth fraction

Fig 9 - The model above is the rabbit system as we see it now. The population behavior depends uponthe dominance of the positive and negative feedback loops.

Positive feedback loop dominance occurs when the rabbit birthsare greater than the deaths. The population behavior is then exponentialincrease. If, on the other hand, the rabbit deaths are greater than thebirths, the negative feedback loop dominates. The population behavior isthen exponential decrease. The population behavior for Fig 9 is similar tothe behavior of the isolated dominating loop.

When the birth rate equals the death rate, neither loop dominates,and the population is in dynamic equilibrium. It is dynamic because thepopulation continues to change the individual rabbits although the numberof rabbits remains the same. For example, if each year two rabbits dieand two rabbits are born, with an initial number of ten rabbits, thepopulation after one year still has ten rabbits, but two of the rabbits aredi f ferent .

12 D-4286

0.00 0.50 1.00 1.50 2.00Years 9:38 AM 2/27/92Page 2

1 :

1 :

1 :

0.00

10.00

20.00

1: rabbit population 1 >31<

1 1 1 1

Fig 10 - The graph is of the population in dynamic equilibrium when the births and the deaths are equal.The population is constantly ten rabbits.

The Rabbit System Behavior

Under ideal conditions in the rabbit system, births are greater thandeaths. Births are ten times the population each year, while the deathsare only one-seventh times the population each year, Fig 11. When we runthe model in Fig 9, and look at the population behavior, we see the numberof rabbits increasing without bound. Hence the phrase, “multiplying likerabbi ts.”

birth fraction = 10 / year = 10 rabbits per rabbit / year

birth fraction * rabbit population = 10 * rabbit population = births

average lifetime = 7 years

rabbit population / average lifetime = 1/7 * rabbit population = deaths

Fig 11 - These two equations show the difference between the number of births and the number of deathsin the rabbit population under ideal conditions. The births easily outnumber the deaths.

D-4286 13

III The Limiting FactorThe Missing Element

Initially the system had ten rabbits living in ten acres. But, withthe present model of the system, there will eventually be more rabbits inthe ten acres of land than is possible.

The model has left out a major element in the rabbit system: thelimited area given to the rabbits. This element will affect the number ofrabbits in the system.

The Effect of the Limited Area

Ideal living conditions for rabbits are ten rabbits per acre. Underthese conditions rabbits live for an average of seven years. Livingconditions for rabbits are crowded when there are more than ten rabbitsper acre.

When crowding begins to occur, the average lifetime begins todecrease. There is not enough food per rabbit to keep each rabbit healthy.Some die of starvation, while others are more susceptible to disease.From the close living conditions, disease is spread more easily. Many babyrabbits are stillborn, and others are born sick and don’t live long enough toreproduce.

The limited area does not actually directly affect the averagelifetime of the rabbits, it is the density of rabbits per area, that causesthe change. Above ten rabbits per acre, the higher the density , the lowerthe average lifetime.

Relationship Between Density and Average Lifetime

The relationship between density and average lifetime is illustratedin the graph in Fig 12. As stated above, seven years is the averagelifetime of a rabbit under ideal conditions. An average lifetime of threemonths (0.25 years) or less means that all rabbits that are born, diebefore they reproduce; therefore, the entire rabbit population dies off. Anaverage lifetime of three months or less occurs when the density is twohundred and eighty rabbits per acre and above.

14 D-4286

Fig 12 - The graph illustrates the assumed relationship between the density and the average lifetime. Asthe density increases above ten rabbits per acre, the average lifetime decreases.

average lifetime = years

density = rabbit population / area = rabbits / acre

Fig 13 - These equations illustrate the units in the graph in Fig 12. As rabbits per acre increases, theaverage lifetime in years decreases.

Second Negative Feedback - Causal Loop Diagram

The density and average lifetime work within a second negativefeedback loop whose stable point is a population of zero. When thepopulation is not equal to zero, a combination of elements works to bringthe system back to stability.

D-4286 15

Fig 14 - The causal loop diagram of the negative feedback loop above illustrates the relationship betweenthe new and old elements in the rabbit system.

When the density is above ten rabbits per acre, the average lifetimedecreases, and the number of deaths increases. In turn the populationdecreases causing a lower density than was originally the case. Noweither the density is still above ten rabbits per acre, or it has decreasedenough to be below ten.

If the density is still above ten rabbits per acre, then we follow theloop again. But this time, the average lifetime is not as low as originallybecause the density is not as high. Therefore, the number of deaths is notas high as originally because the average lifetime has increased, and thepopulation has decreased. In turn the density once again decreases.

This process continues until the density is below ten rabbits peracre. Once this happens, the density no longer affects the averagelifetime. And the average lifetime becomes a constant seven years.

Now, the cycle only travels through the rabbit population, the deathsand the average lifetime; the three elements of the first negativefeedback loop. The density continues to change as the population becomeslower and lower, but the density no longer affects the average lifetime.The average lifetime is a constant, and the population continues toapproach zero. Each time the loop is traversed, whether with a changingaverage lifetime, or a constant average lifetime, the population declinesexponentially.

16 D-4286

Second Negative Feedback - Model

The loops described above are modeled in Fig 15. The density and thearea are two new elements which help to create a second negativefeedback loop. The constant in this model is the area. It limits thenumber of rabbits that can live in the system. When the density is belowten rabbits per acre the average lifetime is also a constant.

area

~

average lifetime

rabbit population

deaths

density

Fig 15 - The model above includes the first negative feedback loop, and the negative feedback loop relatedto rabbit density.

Density = rabbit population / area = rabbits / area

area = 10 acres

Fig 16 - These are the new elements in the model. The other elements in Fig 15 have already been givenequations and units in Sec II, Fig 7.

D-4286 17

IV Entire Model

Combining all of the loops that have been discussed, we create acomplete model of the rabbit system. We know the population behavior ofeach isolated loop for our system. Using loop dominance, we should beable to determine the population behavior for the combination of all threeloops.

area

~

average lifetime

births

rabbit population

deaths

densitybirth fraction

Fig 17 - This is the finished model for the rabbit system. All of the major elements have been included,along with their assumed relationships with each other. Any element in the real rabbit system that has notbeen included has been assumed to have a negligible influence on the population behavior.

Population Hypothesis

Let’s hypothesize before we run the model. Initially the rabbits arenot crowded; there is only one rabbit per acre. They will be able toreproduce quickly because few rabbits die. But, once the populationreaches one hundred or more, a density of ten rabbits per acre or more, theaverage lifetime should begin decreasing due to the crowded conditions.

The population will continue to grow because the average lifetimemust decrease to one-tenth of a year before the deaths will equal thebirths. This may take a while, but once the births and deaths are equal,the population will be in dynamic equilibrium.

Population Behavior- S-Shaped Growth

When we run our model, we see that the population behavior is S-shaped growth. Does the graph correlate with the hypothesis?

18 D-4286

0.00 0.50 1.00 1.50 2.00Years 12:04 PM 4/17/92Page 6

1 :

1 :

1 :

10.00

1805.00

3600.00

1: rabbit population >15<

1

1

1 1

Fig 18 - The population behavior is shown above. This type of curve is referred to as S-shaped growth.

Yes! The hypothesis was correct! The graph in Fig 18 is exactlywhat was just said. Using more technical terms the graph is explainedbelow.

Positive Feedback Loop Dominance

Initially, the positive feedback loop is dominant, because the densityis low and the average lifetime is high. The exponential growth of thepopulation is illustrated on the graph. But, as the density increases aboveten rabbits per acre, the average lifetime begins to decrease. Thepopulation continues to increase, but the positive feedback loop dominanceslowly decreases.

Point of Inflection

Right after the first half year, the curve changes from growth, toapproach to equilibrium, and the point is called the point of inflection. Asthe average lifetime continues to decrease, the number of deathsapproaches the number of births, and the increase in population becomessmaller.

D-4286 19

Approach to Dynamic Equilibrium

After the point of inflection, the population curve begins to approachequilibrium. Around the end of the first year, the population reachesdynamic equilibrium. At this point neither loop is dominant. Thepopulation is now stable at three thousand six hundred rabbits living inten acres. Overcrowding, lack of food, and disease, are large enough tocompensate for the high number of births each year.

V Conclusion

When presented with a system, the first thing to do is todistinguish the different elements within the system, and determine therelationships between them. Using the relationships, one can determinethe system’s structure, and using the structure, make a model of thesystem. The model can illustrate the behavior and the system can bestudied.

When a model was first created for the rabbit system, theillustrated behavior was not possible within the given ten acres. It wasdiscovered that the limited area, an essential element in the rabbitsystem, had been overlooked. The area was then modeled into theappropriate place in our system. The final model created is not an exact replication of reality. Firstof all, the number of rabbits in the system will not be exactly 3600. Thisnumber is an average number of rabbits in the system once it is inequil ibrium.

Also, the relationships between the elements had to be assumed, andaverage numbers had to be used. The relationships between the elementsin the rabbit system cannot be given an exact equation, because theequations used are derived from assumptions and averages, and they comeas close to reality as possible, given the level of complexity.

20 D-4286

VI Appendix

area

~

average lifetime

rabbit population

deaths

density

births

birth fraction

STOCK:rabbit_population(t) = rabbit_population(t - dt) + (births - deaths) * dt

DOCUMENT: Initial rabbit population = 10 rabbits

INFLOWS:births = birth_fraction*rabbit_population

DOCUMENT: births = [(percentage of reproducing rabbits)*(number of baby rabbits eachreproducing rabbit has per year)]*(total number of rabbits)

OUTFLOWS:deaths = rabbit_population/average_lifetime

DOCUMENT: deaths = rabbits /year

CONVERTERS:area = 10 acres

DOCUMENT: The rabbits in our system only have ten acres of land in which to live.

birth_fraction = 10

DOCUMENT: Birth fraction = (percentage of reproducing rabbits) * (number of babies eachreproducing rabbit has per year)

density = rabbit_population/area

DOCUMENT: Density = rabbits/acre

D-4286 21

average_lifetime = GRAPH(density)(0.00, 7.00), (41.7, 2.38), (83.3, 1.29), (125, 0.805), (167, 0.525), (208, 0.385), (250, 0.21),(292, 0.175), (333, 0.14), (375, 0.07), (417, 0.00), (458, 0.00), (500, 0.00)

DOCUMENT: Under ideal conditions, rabbits live an average of seven years. Under crowdedconditions, the average lifetime declines.

22 D-4286

VII Notes

In writing this paper, I came across a few problems that could becalled generic. That is, when first encountering system dynamics, alearner may have the same problems. Thus, it may be helpful to recognizethese problems before they become too confusing.

One of my main problems was that I used the time units of quartersinstead of years throughout the paper. Quarters seemed the easiest timeunit to use because female rabbits reproduced every quarter, and it onlytook one quarter for baby rabbits to begin reproducing. The only reasonquarters seemed easier was because of the modeling technique I wasusing.

I thought that it would be easier to construct my model aroundquarters so that my equations would add the correct number of babies tothe population each quarter. After some deliberation, I discovered thatthe inflow to the population is not added according to the time units thatthe equations are based on, they are added according to how many timesthe model is compiled per each time unit (Delta Time).

Also, most people do not think in terms of quarter years. It isinappropriate to use ideas that are unfamiliar to people who will belearning about your model. Therefore; once I realized that it didn't matterwhat time units I based my equations upon, I decided that the best timeunits to use would be years because that is the unit that most people arefamiliar with.

Another major problem I came across was assumptions versus facts.My first drafts contained ambiguities between an assumed or simplifiedrelationship, and the factual relationship. It is very important to statethat a relationship has been assumed or simplified, in order to justify theresults of the model created.

Also to keep confusion down, the modeler should label the elementsof the structure in the simplest manner possible. For example, theaverage lifetime in the rabbit system is the average lifetime. Initially, Icalled it the death fraction to resemble the birth fraction. The deathfraction was the reciprocal of the average lifetime. But, if the deathfraction was the reciprocal of the average lifetime, why not just call itthe average lifetime?

These were my main problems is writing my first paper in systemdynamics. Because I am a beginner is the field, my mistakes may be thesame as other beginner mistakes, and I hope that people can learn frommine before they make them themselves.

© 2013 Creative Learning Exchange www.clexchange.org  Working in K-12 education to develop Systems Citizens

Permission granted for photocopying and electronic distribution for non-commercial educational purposes with attribution    

Connection to Characteristics of Complex Systems Project Lesson Titles: A System Dynamics Approach to Rabbits (S-shaped Growth) Understanding Oscillations in Simple Systems

Overview: The Oscillations curriculum features two fundamental behavior modes (logistic growth and oscillation) to illustrate the idea that a system’s dynamic behavior is a consequence of its structure. For the enthusiastic learner, these two lessons provide in-depth coverage of each of these behavior modes and thus complement the Oscillations curriculum.

Related Characteristic(s) of Complex Systems: The cause of the problem is within the system.

Ideas and Examples for Connecting to the Characteristic: Mathematics is a tool for understanding system structure, but making connections between elements of equations and their real-world meanings can be difficult for students. These papers present two fundamental ideas from calculus in a way that many people find intuitive and accessible. These articles can also help an educator become more comfortable with the STELLA® software environment. Models are presented in step-by-step fashion to facilitate the skill of representing system elements in the stock-flow language of system dynamics. Teachers willing to engage with these supplementary materials will likely be able to effectively use the Oscillations curriculum to the fullest extent.

Resource(s) Oscillations curriculum is available from the Creative Learning Exchange http://clexchange.org/curriculum/complexsystems/oscillation/ Mathematics Education: A Way Forward (source of above diagram) http://www.edutopia.org/blog/mathematics-real-world-curriculum-david-wees

Instructional materials for learning system dynamics: http://clexchange.org/training/gettingstarted.asp