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
Graph for Population
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
Time (Period)
10
11
12
13
Population : Current
14
15
Individuals
Graph for Aggregate production
1,000
900
800
700
600
0
2
4
6
8
Aggregate production : Baserampprop
10
12
14
Time (Year)
16
18
20
22
24
Dollars
Introduction

System Dynamics describe systems in terms
of state variables (stocks) and their rates of
change with respect to time (flows).
Interest
Rate of change
Interest
Percentage
Money in
Bank
State
Mathematical Representation
Interest
Money in
Bank
Interest= Interest rate*Money in Bank
Interest rate
x
dx/dt
r

dx
 r x or x  r x
dt
where :
r  0.15
x o  100
In General
Stock
Outflow
Inflow
X
dx 
 x  net flow  inflow - outflow
dt
 dx
x change in x 


comes from

t
change in t 
 dt
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)
Births (B)
Birth fraction (f)
(0.03)
Population
(P)
Deaths (D)
Average life span (s)
(65 years)
A Two Stock Model
(0.0005)
(0.04)
Rabbit Net Increase without
Predation Fraction (a)
(3200)
Predation
Fraction (b)
Rabbits (R)
Rabbit Births (I)
Rabbit Deaths (D)
Contacts (N)
Foxes (F)
Fox Deaths (T)
Fox Births (O)
Efficiency of turning
predated rabbits into
foxes (e)
(0.2)
(20)
Natural death fraction in
absence of food (c)
(0.2)
Another Population Model
(0.03)
Birth fraction (f)
Births (B)
(1000)
Population
(P)
(10000)
Death fraction (r)
Current
Deaths (D)
Effect of population
density over deaths (e)
Population
density (E)
Area (A)
(0.005)
Normalized
density (N)
Population density
normal (n)
(3)
EPDD f
EPDD f
8
6
4
2
0
0
2
-X-
4
How to Describe a Graphical
Function?
Current
Ef
Ef
y (effect of…)
Current
2
1.5
1
0.5
0
0
1
-X-
2
2
1.5
1
0.5
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
x o  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)
A Closer
Current
Look
Ef
2
1.5
1
0.5
0
f(2)=2
f(0)=0
f(1)=1
0
1
-X-
2
A Closer
Current
Look
Ef
2
1.5
1
0.5
0
Slope is
positive
f ’(x) is
positive
f ’(x)>0
0
1
-X-
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