LinearInteraction

Linear model of interaction between two stocks

Diagram

Information

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The LinearInteraction component allows to model the interaction between two stocks A (portA) and B (portB) as linear equations:

Note: Capital letters were chosen to represent the stocks (state variables) connected at portA and portB in the formula above. Also dot notation is used for a stock's rate of flow—its first derivative with respect to time.

Coefficient	Unit	Description
`a_0`	base units for flow to A (`OutputType_A`)	Rate of growth for stock A
`b_0`	base units for flow to B (`OutputType_B`)	Rate of growth for stock B
`a_A`	1/s	Fractional rate of growth for stock A (e.g., the self-coupling coefficient for A)
`b_B`	1/s	Fractional rate of growth for stock B (e.g., the self-coupling coefficient for B)
`a_B`	base units for flow to A (`OutputType_A`) per base units of B (`TypeB`)	Rate of growth for stock A per stock B (e.g., the coupling-cofficient for B towards A)
`b_A`	base units for flow to B (`OutputType_B`) per base units of A (`TypeA`)	Rate of growth for stock B per stock A (e.g., the coupling-coefficient for A towards B)

The coefficients can be given as parameters or as variables via the expandable connector dataIn.

Examples

A linear harmonic oscillator can be modeled as a special case of linear interactions where the rates for the mutual influences (e.g., fractionalGrowthRate_B_per_A and fractionalGrowthRate_A_per_B) are negative. In that case, positive stock A will decrease stock B and vice versa.

The Linear Interaction can be used to model Strogatz' famous model for the dynamics of a love affair (conspicuously between "Romeo" and "Juliet") as shown in →LoveHateDynamics [16].

Notes

The type selectors OutputType_A and OutputType_B will be used to conveniently set the types, e.g., units, of the parameters a_0,a_B and b_0, b_A.

The coupling coefficient a_B might have rather convoluted units, e.g., base units of A/(base units of B.s). For convenient modeling, a_B will be divided by refB, which allows to use displayUnit should unit conversion be required. Since TypeB by default is set to Unspecified in most cases nothing needs to be done.

This will, conversely, also apply to b_A.

Parameters (11)

hasConstantGrowthRates	Value: false Type: Boolean Description: = true, if the linear growth rates are given by constant parameters
hasConstantFractionalGrowthRates	Value: false Type: Boolean Description: = true, if the fractional growth rates are given by constant parameters
hasConstantCouplingCoefficients	Value: false Type: Boolean Description: = true, if the induced growth rates are given by constant parameters
a_0	Value: unspecified Type: OutputType_A Description: Constant rate of growth for stock A [OutputType_A]
b_0	Value: unspecified Type: OutputType_B Description: Constant rate of growth for stock B [OutputType_B]
a_A	Value: unspecified Type: Rate (1/s) Description: Constant fractional rate of growth for stock A (self-coupling)
b_B	Value: unspecified Type: Rate (1/s) Description: Constant fractional rate of growth for stock B (self-coupling)
a_B	Value: unspecified Type: OutputType_A Description: Constant rate of growth for stock A per reference units of B (coupling B to A)
refB	Value: 1 Type: TypeB Description: Reference value for B, i.e., divisor for a_B
b_A	Value: unspecified Type: OutputType_B Description: Constant rate of growth for stock B per reference units of A (coupling A to B)
refA	Value: 1 Type: TypeA Description: Reference value for A, i.e., divisor for b_A

Connectors (13)

	portA	Type: FlowPort Description: Flow from/to Stock A
	portB	Type: FlowPort Description: Flow to/from Stock B
	y_B	Type: RealOutput_B Description: Rate for flow to and from B (positive value indicates inflow)
	y1_B	Type: RealOutput_B Description: Rate for flow to and from B (positive value indicates inflow)
	y1_A	Type: RealOutput_A Description: Rate for flow to and from A (positive value indicates inflow)
	y_A	Type: RealOutput_A Description: Rate for flow to and from A (positive value indicates inflow)
	dataIn	Type: Input_Connector Description: Continuous inputs for the component
	u_growthRate_A	Type: RealOutput Description: Output signal
	u_fractionalGrowthRate_A	Type: RealOutput Description: Output signal
	u_growthRate_A_per_B	Type: RealOutput Description: Output signal
	u_growthRate_B_per_A	Type: RealOutput Description: Output signal
	u_fractionalGrowthRate_B	Type: RealOutput Description: Output signal
	u_growthRate_B	Type: RealOutput Description: Output signal

Components (20)

	parGrowthRateA	Type: ConstantConverter Description: Constant growth rate for A (optional)
	parFractionalRateA	Type: ConstantConverter Description: Constant fractional growth rate for A (optional)
	parCrossFractionalRateA	Type: ConstantConverter Description: Constany growth rate for A per B (optional)
	parGrowthRateB	Type: ConstantConverter Description: Constant growth rate for B (optional)
	parFractionalRateB	Type: ConstantConverter Description: Constant fractional growth rate for B (optional)
	parCrossFractionalRateB	Type: ConstantConverter Description: Constant growth rate for B per A (optional)
	unitMultiplierA	Type: Gain Description: The factor should have units 1/[base units of B]
	unitMultiplierB	Type: Gain Description: The factor should have units of 1/[base units of A]
	growingA1	Type: ExogenousChange Description: Linear growth
	growingA2	Type: ExponentialChange Description: Exponential growth for A
	growingA3	Type: ExogenousChange Description: Cross-fractional growth rate for A per B
	stockB	Type: FlowPortSensor Description: Amount in stock B
	stockA	Type: FlowPortSensor Description: Amount in stock A
	inducedGrowthA	Type: Product_2 Description: Rate of growth induced by stock B
	totalRateA	Type: Add_3 Description: Total of three inputs
	growingB1	Type: ExogenousChange Description: Linear growth
	growingB2	Type: ExponentialChange Description: Exponential growth for B
	growingB3	Type: ExogenousChange Description: Cross-fractional growth rate for B per A
	inducedGrowthB	Type: Product_2 Description: Rate of growth induced by stock A
	totalRateB	Type: Add_3 Description: Total of three inputs