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Information about System dynamics math representation

System Dynamics Math Representation

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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 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

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 u 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

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