Model Structures and Behavior Patterns

Transcription

1 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

2 Logistic 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? dr(t) = = k(t) * R(t), where k = G{ 1 R(t) C } What is the solution for the rate equation for logistic growth? R(t) = C 1 + Ae -Gt where A = C R 0 R 0 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 simulation

3 Overshoot 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) R(t) = {B [1 - ]} * P(t) R 0 dr(t) = -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 nonrenewable reservoir for survival

4 Oscillation - 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 moose present, 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 :

5 Oscillation - 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: But we also need to have a way for Foliage to grow and to be consumed by the Moose:

6 Oscillation - Example Step 3: Identify the converter(s) that determine the rates of inflow and outflow : In this system, we re going to assume that the Birth process 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 fashion

7 Oscillation - Example Step 4: Define relationships between system elements with connectors: Draw in the linkages between elements: B M D G F C 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)

8 Oscillation - Example Once we run the model, we ll see something like:

9 Oscillation - 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 conditions

10 Oscillation 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:

11 Oscillation System Features, Diagrams, and Equations 3. The further one reservoir is from its equilibrium value, 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 connections are between the two reservoirs and rates associated with processes that affect the other reservoir

12 Oscillation System Features, Diagrams, and Equations Starting with the generic difference equation: R(t+ t) = R(t) + [Inflows Outflows] * t we can substitute in the expressions for the processes in our oscillating system for each of the two reservoirs: C(t+ t) = C(t) + ({[G * R(t)] - D} * t) R(t+ t) = R(t) + ({W -[Q * C(t)]} * t) Performing the usual reshuffling of R(t) and t, and taking the derivative of the resulting rate equations: dc(t) = {G * R(t)} - D dr(t) = W {Q * C(t)}

13 Oscillation System Features, Diagrams, and Equations We can see how the counteracting feedback will behave by examining the rate equations: dc(t) = {G * R(t)} - D dr(t) = W {Q * C(t)} When R(t) is large, then dc(t) will be positive (C grows) When C(t) gets large enough, dr(t) will become negative (thus R will begin to shrink in size) As R(t) shrinks, then dc(t) will get smaller, and then will eventually become negative (thus C shrink in size) This counteracting feedback will produce the cyclic behavior (oscillation)

14 Oscillation System Features, Diagrams, and Equations To find the equilibrium condition for each of the reservoirs, we can set the individual rate equations equal to zero, and solve: dc(t) = {G * R(t)} - D dr(t) = W {Q * C(t)} {G * R(t)} D = 0 W {Q * C(t)} = 0 {G * R(t)} = D {Q * C(t)} = W R(t) = D G C(t) = -W -Q Thus, provided you can determine the values of D and G or W and Q, you can find the values at which each of the reservoirs will equilibrate in an oscillating system = W Q

15 Oscillation System Features, Diagrams, and Equations To find when an oscillating system reaches a steady state, determine the conditions under which the rate of change of both reservoirs equal zero, i.e.: dc(t) = 0 dr(t) = 0 This occurs when both reservoirs simultaneously are at their equilibrium condition, thus: R = D G C = W Q If only one of the two reservoirs reaches its equilibrium state, it will quickly leave it as the other will force it away from equilibrium according to the counteracting feedback previously described

16 Oscillation - Review When does oscillation behavior occur? Oscillation behavior occurs whenever a pair of interdependent reservoirs are connected in such a way that strong counteracting feedback loop that forces the system to oscillate around a set of equilibrium conditions What are the rate equations for this type of system? dc(t) = {G * R(t)} - D dr(t) = W {Q * C(t)} Does a system that produces oscillation behavior contain any feedback? Oscillation behavior occurs because of the presence of a counteracting feedback loop that keeps the system cycling back and forth, with each reservoir gaining and losing its contents at different points in the cycle

17 System Behavior General Thoughts We ve now examined five common behaviors, and we have used a similar approach to characterize each: Each has a generic model structure (arrangement of model elements) that needs to be present for that behavior to occur Each has mathematical expression(s) for the difference equation(s) for its reservoir(s) that is a function of model structure By deriving the rate equation(s) from the difference equations(s) we can see fundamentally why the system behaves the way it does, by interpreting how the rates of change will change through time, and thus how the reservoir(s) contents will change We can look at plots versus time to see that same behavior We can interpret the cause and effect connections between elements to see when feedbacks produce the behavior

18 System Behavior General Thoughts For the most part, the models we will build can be interpreted to be made up of one or more of these building blocks, and thus exhibit one or more of these behaviors The five behaviors we have examined so far lets us see how each functions in isolation, but the building blocks can be combined, and thus the behaviors can be combined i.e. imagine a system that follows a logistic growth curve in the long run, but oscillates as it progresses along the logistic curve, or any number of other possible combinations Resulting behavior should still be a function of system structure, but its often more complicated than picking out one of these five behaviors on its own