modelLinearInteraction

Linear model of interaction between two stocks

Extends from Interfaces.Basics.GenericFlow (Flow Template with two FlowPorts), Interfaces.Basics.Interaction4SO (Output connectors for interaction elements reporting rates), Icons.Interaction (Icon for interactional flow classes).

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

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

Connectors

Type	Name	Description
Connectors.FlowPort	portA (from GenericFlow)	Flow from/to Stock A
Connectors.FlowPort	portB (from GenericFlow)	Flow to/from Stock B
RealOutput_B	y_B (from Interaction4SO)	Rate for flow to and from B (positive value indicates inflow)
RealOutput_B	y1_B (from Interaction4SO)	Rate for flow to and from B (positive value indicates inflow)
RealOutput_A	y1_A (from Interaction4SO)	Rate for flow to and from A (positive value indicates inflow)
RealOutput_A	y_A (from Interaction4SO)	Rate for flow to and from A (positive value indicates inflow)
Input_Connector	dataIn	Continuous inputs for the component

Name	Description
TypeA	Type selector for stock A (default = Unspecified)
TypeB	Type selector for stock B (default = Unspecified)
Input_Connector	Data bus for inputs

Revisions

Input_Connector defined as encapsulated expandable connector in v2.1.0.

Values for optional parameters changed to unspecified in v2.1.0.

Adapted types and added divisors for compliance with unit checking and convenient unit conversions in v2.2.