System dynamics math representation

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

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

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.

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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rra
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o
P
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1

1

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2

1
3

1
4

1
5

1

0
.
7
5

1

0
.
5
1

0
.
2
5

1
1

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1

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1

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1

1

1

2

3

4

5

6

7

8

9

1
0

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(d
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0
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2

4

6

8

1

1
0

1
2

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Introduction


System Dynamics describe systems in terms
of state variables (stocks) and their rates of
change with respect to time (flows).
I
n
t
e
r
s

M
o
n
e
y
i
B
a
n
k

State
I
n
t
e
r
s

Rate of change
P
e
r
c
n
t
a
g
e

Mathematical Representation
I
n
t
e
r
s

M
o
n
e
y
i
B
a
n
k

Interest= Interest rate*Money in Bank

I
n
t
e
r
s
a

x
d
x
/
t

r


dx
= r x or x = r x
dt
where :
r = 0.15
x o = 100

In General
S
t
o
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 




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.”

Another Example
(initial = 1000)
P
o
p
u
ln
a
t
i
o
B
i(
r
tB
h
s
)

B
if(
rtf
ta
h
c
i)
o
n

(0.03)

(
P
)

D
e
a
tD
h
s
(
)

A
v(
en
re
as
g
lp
ia
f)
s

(65 years)

A Two Stock Model

(0.0005)

(0.04)
Rso
aIw
briu
iett
ttch
N
n
e
a
Pc
riF
ern
dt(
aia
ta
o)
n
o

P
ri
e
d
a
t
o
n
F
rn
a
c
t(
ib
o
)

(3200)
R
a
b
i(
tR
s
)
R
ah
b(
irI
tt)
B
s

R)
at(
bD
ie
ta
D
h
s

C
o
n
ts
a
c
(
N
)

F
o
x
e
s
(
F
)
F)
o
x
B
is
rO
t(
h
Er
fng
ici
ctn
e
y
o
f
u
pb
ren
eit
dto
as
tr
d
a
f(
o
x
e
s
)

(0.2)

(20)

F)
oT
x
D
e
a
t(
h
s

No
atrn
taa
uc
rai
lh
d
ei
f
t
af(
bo
sd
ec
n)
c
o

(0.2)

Another Population Model
(0.03)

(0.005)

(1000)

C
u
r
e
n
t

B
if(
rtf
ta
h
c
i)
o
n

E
P
D
f

D
e(
a)
tc
h
fr
r
t
i
o
n

8
6
4
2
0

P
o
p
u
ln
a
t
i
o
(
P
)

B
i(
r)
tB
h
s

(10000)

D
e
a
tD
h
s
(
)
E
fp
et
co
tu
o
fa
ln
i
dte
ea
n(
sh
iv
te
y)
o
r
d
s

P
o
p
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ln
a
t
i
o
d
e
n
s
iE
t)
y
(

A
r
e
a
(
A
)

(3)

N
o
rd
m
a
l
i
z
e
d
e
n
s
i)
t
y
(
N
P
oy
p
u
ln
a
td
ie
o
s
i
t
n
o
r)
m
a
l
(
n

E
P
D
f

0

2
X

4

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

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

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

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)

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

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

1
.
5

A Closer Look

1

0
.
5
0

0

The slope is increasing
f ‘(x) is increasing

1
X

f ’’(x)>0

A Closer Look

The slope is
decreasing
f ‘(x) is decreasing

f ’’(x)<0

System dynamics math representation

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