STELLA numerically simulates the solutions to systems of ...jcmayer/Stella and Calculus.pdfSTELLA...
Transcript of STELLA numerically simulates the solutions to systems of ...jcmayer/Stella and Calculus.pdfSTELLA...
STELLA numerically simulates the solutions to systems of differential
equations.
STELLA INTEGRATES!
STELLA Diagram
Stock X
Flow 1
Constant aSTELLA Equations:Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt
INIT Stock_X = 100Flow_1 = Constant_aConstant_a = 10
Flow is constant.
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Graph 1 (Untitled)
Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt
(Stock_X(t) - Stock_X(t - dt))/dt = Flow_1Flow_1 = Constant_a
(X(t) - X(t - dt))/dt = alet dt 0
dX/dt = a (a differential equation)
Solution to DE: X = at + X(0)
STELLA DiagramStock 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.
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Stock_X(t) = Stock_X(t - dt) + (Flow_1) * dt
(Stock_X(t) - Stock_X(t - dt))/dt = Flow_1Flow_1 = Constant_a*Stock_X
(X(t) - X(t - dt))/dt = aX(t-dt)
dX/dt = aX (differential equation)
Solution to DE:
STELLA Diagram
Two flows, an inflow and an outflow.
Population
Births
Birth Fraction
Deaths
Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dtINIT Stock_X = 1000
Flow_1 = Constant_a*Stock_X(t-dt)
Flow_2 = Constant_b
Constant_a = .11
Constant_b = 100
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Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dt
(Stock_X(t) - Stock_X(t - dt))/dt = Flow_1 - Flow_2Flow_1 = Constant_a*Stock_X(t-dt)
Flow_2 = Constant_b
dX/dt = aX - b
Solution to DE:?
STELLA Diagram
Two flows, an inflow and an outflow.
Population
Births Deaths
Death Fraction
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Stock_X(t) = Stock_X(t - dt) + (Flow_1 - Flow_2) * dt
(Stock_X(t) - Stock_X(t - dt))/dt = Flow_1 - Flow_2Flow_1 = Constant_a
Flow_2 = Constant_b*
dX/dt = a - bX
Solution to DE:?
STELLA 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) * dtINIT 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) * dtINIT 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
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Graph 1 (Untitled)
(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?
Stock X Stock Y Stock Z
Flow 1 Flow 2
Constant a Constant b
Stock_X(t) = Stock_X(t - dt) + (- Flow_1) * dtFlow_1 = Constant_a*Stock_X*Stock_Y /dt
Stock_Y(t) = Stock_Y(t - dt) + (Flow_1 - Flow_2) * dtFlow_1 = Constant_a*Stock_X*Stock_Y
Flow_2 = Constant_b*Stock_Y
Stock_Z(t) = Stock_Z(t - dt) + (Flow_2) * dtFlow_2 = Constant_b*Stock_Y