David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Model 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:LinearGrowthor DecayExponentialGrowthor DecayLogisticGrowthOvershootandCollapseOscillationDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Logistic Growth - Review• When does logistic growth occur?– In an exponential system that is constrained so that the reservoir achieves a maximum level that is sustainable by the system (e.g. limited by a Carrying Capacity)• What is the rate equation for logistic growth?• What is the solution for the rate equation for logistic growth?dR(t)dt== k(t) * R(t), wherek =G{ 1 –R(t)C}R(t) =C1 + Ae-Gtwhere A =C – R0R0• Does a logistic growth system contain any feedback?– Logistic growth systems have closed loops that provide reinforcing and counteracting feedback in varying amounts at different times in the simulationDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Overshoot and Collapse - Review• When does overshoot and collapse behavior occur?– Overshoot and collapse behavior occurs whenever one reservoir depends on another non-renewable reservoir for survival• What are the rate equations for this type of system?dP(t)dt= {B – [1 - ]} * P(t) R(t)R0dR(t)dt= -C * P(t)• Does a system that produces overshoot and collapse behavior contain any feedback?– Systems that produce overshoot and collapse behavior have closed loops that provide reinforcing and counteracting feedback in varying amounts at different times in the simulation, the key feedback being a counteracting feedback that connects the reservoir that depends upon the non-renewable reservoir for survivalDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation - Example• Consider the following example of a system:– Our free range moose farmer realizes that as the moose population grows in his tract of the land, the abundance of foliage will decrease– As such, the farmer decides to put together a more realistic model of the situation, tracking both the number of moosepresent, and the foliage present that is required to sustain the moose• Let’s begin by identifying the model elements needed:Step 1: Identify the reservoir(s)– Now we need to track moose and foliage :David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation - ExampleStep 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 Moosedying:– But we also need to have a way for Foliage to grow and to be consumed by the Moose:David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation - ExampleStep 3: Identify the converter(s) that determine the rates of inflow and outflow :– In this system, we’re going to assume that the Birthprocess proceeds at a rate which is a function of the amount of Foliage available, and the Consumption of Foliage is a function of the Moose population size:– In effect, in this model, Death and Growth processes are operating in a constant fashionDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation - ExampleStep 4: Define relationships between system elements with connectors:– Draw in the linkages between elements:– You can now see how each reservoir has a process that depends on the other reservoir to determine its function, as reflected in the difference equations for this system:M(t+∆t) = M(t) + ({[B * F(t)] - D} * ∆t)F(t+∆t) = F(t) + ({G -[C * M(t)]} * ∆t)MBFDGCDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation - Example• Once we run the model, we’ll see something like:David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation - Example• This is a classic example of a system, where consumer and resource populations are present:– As the available amount of resource increases, so too do the number of consumers that live off that resource– Then, as the number of consumers increase, the available resource is consumed, thereby leading to reduction of the available amount of the resource–As the resource decreases, the number of consumers also decrease– This in turn leads to a rebound in the quantity of the resource• The main feature of an oscillating system is the presence of a strong counteracting feedback loop that forces the system to oscillate around a set of equilibrium conditionsDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation – System Features, Diagrams, and Equations• An oscillating system has the following features:1. The systems has at least two interdependent reservoirs, where one reservoir consumes the other (e.g. Consumer and Resource)2. The Consumer and Resource reservoirs each oscillate around an equilibrium:David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation – System Features, Diagrams, and Equations3. The further one reservoir is from its equilibriumvalue, the more influence the other reservoir exerts on it to move it back towards equilibrium• Here is the generic diagram for a simplified oscillating system:•The key connectionsare between the two reservoirs and rates associated with processes that affect the other reservoirDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation – System Features, Diagrams, and Equations• Starting with the generic difference equation:R(t+∆t) = R(t) + [Inflows – Outflows] * ∆twe can substitute in the expressions for the processes in our oscillating system for each of the two reservoirs:• Performing the usual reshuffling of R(t) and ∆t, and taking the derivative of the resulting rate equations:dC(t)dt= {G * R(t)} - DdR(t)dt= W – {Q * C(t)}C(t+∆t) = C(t) + ({[G * R(t)] - D} * ∆t)R(t+∆t) = R(t) + ({W -[Q * C(t)]} * ∆t)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Oscillation – System Features, Diagrams, and Equations• We can see how the counteracting feedback will behave by examining the rate equations:dC(t)dt= {G * R(t)} - DdR(t)dt= W – {Q * C(t)}• When R(t) is large, thendC(t)dtwill be positive (C grows)• When C(t) gets large enough,dR(t)dtwill become negative(thus R will begin to shrink in size)•As R(t) shrinks, thendC(t)dtwill get smaller, and then willeventually become negative (thus C shrink in