The document discusses how to mathematically represent system dynamics models using differential equations. It explains that system dynamics describe systems using state variables (stocks) and their rates of change (flows), which can be expressed as differential equations. These equations can be linear or nonlinear. The document provides several examples of system dynamics models translated into systems of differential equations. It also discusses how to interpret graphical functions and their derivatives to determine behaviors.

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
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5. Introduction

System Dynamics describe systems in terms
of state variables (stocks) and their rates of
change with respect to time (flows).
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State
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Rate of change
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6. Mathematical Representation
I
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Interest= Interest rate*Money in Bank
I
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a
x
d
x
/
t
r

dx
= r x or x = r x
dt
where :
r = 0.15
x o = 100

7. In General
S
t
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c
k
O
u
t
f
l
o
w
I
n
f
l
o
w
X
dx 
= x = net flow = inflow - outflow
dt
 dx
∆x changein x 

 dt comes from ∆t = changein t 




8. In General
dx 
= x = net flow = inflow - outflow
dt



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)
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if(
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(65 years)

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

11. Another Population Model
(0.03)
(0.005)
(1000)
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B
if(
rtf
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c
i)
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(10000)
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n
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D
f
0
2
X
4

12. How to Describe a Graphical
Function?
C
u
r
e
n
t
E
f
E
f
y (effect of…)
C
u
r
e
n
t
2
2
1
.
5
1
.
5
1
1
0
.
5
0
.
5
0
0
1
X
2
0
0
x (some ratio)
1
X
2

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

14. Can We Do the Opposite?
dx
=y
dt
dy
k
c
= − x− y
dt
m
m
where :
k / m = 64
c / m = 0.2
xo = 4.5
y o = −0.45

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. C A Closer Look
u
r
e
n
t
E
f
2
f(2)=2
1
.
5
f(0)=0
1
f(1)=1
0
.
5
0
0
1
2

17. C A Closer Look
u
r
e
n
t
E
f
2
Slope is
positive
1
.
5
f ’(x) is
positive
1
0
.
5
f ’(x)>0
0
0
1
2

18. 1
.
5
A Closer Look
1
0
.
5
0
0
The slope is increasing
f ‘(x) is increasing
1
X
f ’’(x)>0

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