STELLA and Calculus

STELLA numerically simulates the solutions to systems of differential

equations.

STELLA INTEGRATES!

Calculus Example 1STELLA Diagram

Stock X

Flow 1

Constant a

STELLA Equations: Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt

INIT Stock_X = 100Flow_1 = Constant_aConstant_a = 10

Flow is constant.

Example 1 Graph

7:28 AM Tue, Apr 10, 2001

0.00 3.00 6.00 9.00 12.00

Time

1:

1:

1:

100.00

200.00

300.00

1: Stock X

1

1

1

1

Graph 1 (Untitled)

Question?

• Does the graph have an equation?

• “Obviously”X = 10t + 100

Differential Equation 1

• Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt

• (Stock_X(t) - Stock_X(t - dt))/dt = Flow_1• Flow_1 = Constant_a

• (X(t) - X(t - dt))/dt = a• let dt 0

• dX/dt = a (a differential equation)

• Solution to DE: X = at + X(0)

Calculus Example 2STELLA Diagram

Stock X

Flow 1

Constant a

STELLA Equations:Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt

INIT Stock_X = 100Flow_1 = Constant_a*Stock_XConstant_a = .1

Flow is proportional to X.

Example 2 Graph

7:34 AM Tue, Apr 10, 2001

0.00 3.00 6.00 9.00 12.00

Time

1:

1:

1:

100.00

250.00

400.00

1: Stock X

1

1

1

1

Graph 1 (Untitled)

Differential Equation 2

• Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt

• (Stock_X(t) - Stock_X(t - dt))/dt = Flow_1• Flow_1 = Constant_a*Stock_X

• (X(t) - X(t - dt))/dt = aX(t-dt)

• dX/dt = aX (differential equation)

• Solution to DE:

X = X(0) exp(at)

Calculus Example 3STELLA Diagram

Two flows, an inflow and an outflow.

Population

Births

Birth Fraction

Deaths

STELLA Equations 3

• Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dt• INIT Stock_X = 1000

• Flow_1 = Constant_a*Stock_X(t-dt)• Flow_2 = Constant_b

• Constant_a = .11

• Constant_b = 100

Example 3 Graph

4:17 PM Wed, Feb 08, 2006

Population

Page 1

0.00 3.00 6.00 9.00 12.00

Time

1:

1:

1:

0

1000

2000

1: Population

1 1 11

Differential Equation 3

• Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dt• (Stock_X(t) - Stock_X(t - dt))/dt = Flow_1 - Flow_2

• Flow_1 = Constant_a*Stock_X(t-dt)

• Flow_2 = Constant_b

• dX/dt = aX - b

Solution to DE:?

Calculus Example 4STELLA Diagram

Two flows, an inflow and an outflow.

Population

Births Deaths

Death Fraction

Example 4 Graph

3:54 PM Wed, Feb 09, 2005

Population

Page 10.00 25.00 50.00 75.00 100.00

Time

1:

1:

1:

0

600

1200

1: Population

11 1 1

Differential Equation 4

• Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dt• (Stock_X(t) - Stock_X(t - dt))/dt = Flow_1 - Flow_2

• Flow_1 = Constant_a

• Flow_2 = Constant_b*

• dX/dt = a - bX

Solution to DE:?

Calculus Example 5STELLA Diagram

Stock X

Flow 1

Constant a

Stock Y

Flow 2

Constant b Constant c

Flow 3

Two stocks, X and Y, and three flows.

The outflow from Stock X is the inflow to Stock Y.

STELLA Equations 5

• Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dt• INIT Stock_X = 100

• Flow_1 = Constant_a*Stock_X

• Flow_2 = Constant_b*Stock_X*Stock_Y

• Stock_Y(t) = Stock_Y(t - dt) + (Flow_2 - Flow_3) * dt• INIT Stock_Y = 100

• Flow_2 = Constant_b*Stock_X*Stock_Y

• Flow_3 = Constant_c*Stock_Y

• Constant_a = .2

• Constant_b = .001

• Constant_c = .01

Example 5 Graph

7:35 AM Tue, Apr 10, 2001

0.00 12.50 25.00 37.50 50.00

Time

1:

1:

1:

2:

2:

2:

0.00

100.00

200.00

100.00

300.00

500.00

1: Stock X 2: Stock Y

1

1

1 1

2

2 2

2

Graph 1 (Untitled)

Differential Equations 5

• (Stock_X(t) - Stock_X(t - dt))/dt = Flow_1 - Flow_2

• (Stock_Y(t) - Stock_Y(t - dt))/dt = Flow_2 - Flow_3

• dX/dt = aX - bXY

• dY/dt = bXY - cY

(A pair of differential equations)

How many differential equations do we need?

Differential Equations 6

• dX/dt = -aXY

• dY/dt = aXY- bY

• dZ/dt = bY

STELLA Diagram 6

Stock X Stock Y Stock Z

Flow 1 Flow 2

Constant a Constant b

STELLA Equations 6

• Stock_X(t) = Stock_X(t - dt) + (- Flow_1) * dt• Flow_1 = Constant_a*Stock_X*Stock_Y /dt

• Stock_Y(t) = Stock_Y(t - dt) + (Flow_1 - Flow_2) * dt• Flow_1 = Constant_a*Stock_X*Stock_Y

• Flow_2 = Constant_b*Stock_Y

• Stock_Z(t) = Stock_Z(t - dt) + (Flow_2) * dt• Flow_2 = Constant_b*Stock_Y