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
G
rra
aln
p
h
fu
o
P
p
t
i
o
1
1
1
1
1
1
1
1
1
2
1
3
1
4
1
5
1
0
.
7
5
1
0
.
5
1
0
.
2
5
1
1
0
1
0
1
1
1
1
1
2
3
4
5
6
7
8
9
1
0
T
ie
m
(d
P
r
i)
o
Pu
or
pe
un
li:t
a
tn
o
C
1
1
1
1
1
1
1
1
1
1
Iu
n
d
ia
v
l
s
Gc
rran
aet
pp
hr
fro
od
A
gi
tu
o
1
1
,
0
1
1
9
0
1
1
8
0
1
1
1
1
7
0
1
1
1
1
1
6
0
0
2
4
6
8
1
1
0
1
2
1
4
1
6
1
8
2
0
2
T
i()
m
e
Y
e
a
r
Anp
gdsr
rpB
eue
gt:ro
aca
triap
eo
om
1
1
1
1
1
1
1
D
o
ls
a
r
2
4
5. 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
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)
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)
10. 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)
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