Mathematical Representation of System Dynamics Models

1. Mathematical Representation of System Dynamics Models Vedat Diker George Richardson Luis Luna

2. Our Today’s Objectives • Translate a system dynamics model to a system of differential equations • Build a system dynamics model from a system of differential equations

3. Introduction • Many phenomena can be expressed by equations which involve the rates of change of quantities (position, population, principal, quality…) that describe the state of the phenomena.

4. Introduction • The state of the system is characterized by state variables, which describe the system. • The rates of change are expressed with respect to time

5. Introduction • System Dynamics describe systems in terms of state variables (stocks) and their rates of change with respect to time (flows). State Rate of change

6. Mathematical Representation Interest=Interest rate*Money in Bank

7. In General X

8. In General • This equation that describes a rate of change is a differential equation. • The rate of change is represented by a derivative. • You can use any letter, not just “x.”

9. Another Example (initial = 1000) (0.03) (65 years)

10. A Two Stock Model (0.0005) (0.04) (3200) (20) (0.2) (0.2)

11. Another Population Model (0.03) (0.005) (1000) (10000) (3)

12. How to Describe a Graphical Function? y (effect of…) x (some ratio)

13. In summary f ’(x)>0 Þf(x) f ’(x)<0 Þf(x) f ’’(x)>0 Þf(x) f ’’(x)<0 Þf(x)

14. Can We Do the Opposite?

15. Final ideas • Any System Dynamics model can be expressed as a system of differential equations • The differential equations can be linear or non-linear (linear and non-linear systems) • We can have 1 or more differential equations (order of the system)

16. A Closer Look f(2)=2 f(0)=0 f(1)=1

17. A Closer Look Slope is positive f ’(x) is positive f ’(x)>0

18. A Closer Look The slope is increasing f ‘(x) is increasing f ’’(x)>0

19. A Closer Look The slope is decreasing f ‘(x) is decreasing f ’’(x)<0