System Dynamics Math Representation

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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M
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B
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k
Interest= Interest rate*Money in Bank
I
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s
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
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O
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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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(0.03)
(
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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)
C
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t
B
if(
rtf
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h
c
i)
o
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(10000)
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(3)
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m
a
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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