Model Structures and Behavior Patterns • The systems modeler believes that the behavior of the system is a function of the system itself • Translating that idea to the model realm, this means that certain structures of elements should produce certain types of behavior patterns • We are going to look at five common behavior patterns and their associated structures:

Linear Growth or Decay

Exponential Growth or Decay

Logistic Growth

Overshoot and Collapse

Oscillation

David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Linear Growth or Decay - Review • When does linear growth or decay occur? – The difference between all inflows and outflows is a constant at all times

• What is the rate equation for linear growth or decay? =

dR(t) dt

n

=

n

Outflow Inflow Σ Σ i=1 j=1 i

j

= constant = k

• What is the solution for the rate equation for linear growth or decay? – R(t) = R(0) + kt, growth when k > 0 and decay when k < 0

• Does a simple linear system contain any feedback? – No, there are no closed loops David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Exponential Growth or Decay - Review • When does exponential growth or decay occur? – IFF the rate constant is a constant proportion of the size of the reservoir

• What is the rate equation for exp. growth or decay? =

dR(t) dt

= kR(t), where k = Inflow Rate – Outflow Rate

• What is the solution for the rate equation for exponential growth or decay? – R(t) = R(0)ekt, growth when k > 0 and decay when k < 0

• Does an exponential growth/decay system contain any feedback? – Yes, there are closed loops in an exponential system David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example • Consider the following example of a system: – A businessman in the far North decides there is a large, untapped market for free range moose meat, and buys and fences off a large tract of land for raising the moose – The tract of land is capable of supporting a population of 100,000 moose based on the size of the tract – A mating pair of moose are introduced – How will the moose population progress through time?

• Once again, we will try to determine the structure of the system in a series of steps Step 1: Identify the reservoir(s) – Since we are tracking the number of moose: David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example Step 2: Identify the process(es) that will change the contents of the reservoir(s) over time: – We obviously need to have moose being born, and moose dying:

Step 3: Identify the converter(s) that determine the rates of inflow and outflow: – Here is where things differ from an exponential model – We will need rates to regulate the Birth and Death processes, but we also include a Carrying Capacity: David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example Step 4: Define relationships between system elements with connectors: – Once the cause and effect relationships in this system are drawn in with the connectors, the structural difference from an exponential system becomes apparent:

– In particular, Death is defined differently, with Carrying Capacity also regulating this outflow process David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example • What is the difference equation for this system? M(t+∆t) = M(t) + ({Birth - Death} * ∆t)

• We can find the expressions for the Birth and Death processes, taking Carrying Capacity into account: Birth = Birth Rate * M(t) = 1 * M(t) = M(t) Death = Death Rate * M(t) * [M(t) / Carrying Capacity ]

• You can see that as M(t) Æ Carrying Capacity the Death Rate is going to become quite large, regulating the moose population’s growth, because [M(t) / Carrying Capacity ] > 1 • However, early on [M(t) / Carrying Capacity ] will be small, and the system will show exponential growth David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example • Using the following values for the converters: – Birth Rate = 1 birth/capita/year – Death Rate = 0.1 deaths/capita/year – Carrying Capacity = 100,000

we can run the model and see the following output:

David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example • We can interpret the shape of the curve as follows:

1

2

1. Initially, the logistic system behaves like an exponential system 2. Later, the growth levels out to approach a steady state value

• The logistic growth curve is often referred to as an sigmoid curve or s-curve • Logistic growth systems have closed loops that provide reinforcing and counteracting feedback in varying amounts at different times in the simulation David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth - Example • QUESTION: Why does the growth in a logistic system initially behave like an exponential growth system? • Using values from our example, initially Moose at time (t) is a very small value Carrying Capacity

=

2 100,000

to begin with

thus the Outflow Rate is close to zero • Logistic growth occurs whenever an exponential system is constrained so that the reservoir achieves a maximum level that is sustainable by the system (e.g. limited by a Carrying Capacity) David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth – System Features, Diagrams, and Equations • A system that has logistic behavior generally takes the following form:

• It is not strictly necessary to have a rate controlling each of the processes (as in the moose example), all that is required is that some sort of Carrying Capacity be used to define the behavior of the Outflow process in terms of a proportion on the size of the Reservoir David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth – System Features, Diagrams, and Equations • For our generic example, the difference equation is: R(t+∆t) = R(t) + [Inflows – Outflows] * ∆t

• Substituting in the expressions for the processes in a logistic system (symbolizing the Growth Rate with G and Carrying Capacity with C): R(t+∆t) = R(t) + [{G * R(t)} – {G * R(t) *

R(t) } * ∆t C

R(t) } * R(t) * ∆t = R(t) + G{ 1 – C

• Subtracting R(t) from both sides, and dividing by ∆t: R(t+∆t) - R(t) ∆t

= G{ 1 –

R(t) } * R(t) C David Tenenbaum – GEOG 110 – UNC-CH Fall 2005

Logistic Growth – System Features, Diagrams, and Equations • We can now take the derivative with respect to time: lim ∆t Æ 0

R(t+∆t) - R(t) ∆t

=

dR(t) dt

= k(t)R(t)

• The difference between logistic and exponential behavior is that the growth rate is no longer a constant proportion of the size the reservoir, but changes with time: R(t) k = G{ 1 – R(t)

C

}

• When R(t)