JanoschekPositive

Janoschek's (S-shaped) growth curve with positive slope

Diagram

Information

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Many natural growth processes can be very closely described by s-shaped parametric functions (e.g., logistic curve). A. Janoschek [19] has proposed a growth curve that is determined by four parameters in 1957, which is very similar to the Weibull growth curve.

The function is given by the following equation:

y = l - (l-beta) * Exp(-k*u^delta)

Where l is the upper bound, β is the lower bound, and δ is a parameter that determines the rate of growth (e.g., steepness of the curve). Given the point of inflection (x_ref,y_ref) the parameter k can be obtained in closed form, which is made use of in this implementation.

The following animation shows a growth curve with the following parameters:

Parameter	Value
`upperBound`	2
`lowerBound`	0.1
`(x_ref, y_ref)`	(1,1)
`growthRate`	1, ... ,10

Notes

The lowerBound must be smaller than the upperBound. Also y_ref needs to be in the range [lowerBound, upperBound], while x_ref has to be a positive number larger than zero.

By default (strict = true) the component will not generate events.

Parameters (10)

convertInput	Value: false Type: Boolean Description: = true, if the input value is a rate or time variable that needs to be converted
inputIsRate	Value: false Type: Boolean Description: = true, if the input is given in units of time, otherwise it is assumed to be a rate
convertOutput	Value: false Type: Boolean Description: = true, if the output value is a rate or time variable that needs to be converted
outputIsRate	Value: false Type: Boolean Description: = true, if the output is given in units of time, otherwise it is assumed to be a rate
lowerBound	Value: 0 Type: Real Description: Lower asymptote for u <= 0 (lowerBound < upperBound)
upperBound	Value: 2 Type: Real Description: Upper asymptote for u -> infinity
growthRate	Value: 2 Type: Real Description: Determines the gradient of the curve
x_ref	Value: 1 Type: InputType Description: x-value (i.e., u) for point of inflection(x_ref > 0)
y_ref	Value: 1 Type: Real Description: y-value for point of inflection(lowerBound < y_ref < upperBound)
strict	Value: true Type: Boolean Description: = true, if strict limits with noEvent(..)

Connectors (4)

	y	Type: RealOutput Description: Output signal
	u	Type: RealInput Description: Input
	u_c	Type: RealOutput Description: Input after conversion
	y_c	Type: RealOutput Description: Output before conversion

Components (7)

	modelSettings	Type: ModelSettings Description: Setting important global variables within a model
	rateConversionIn	Type: RateConversion Description: Convert the input (rate) to another time base
	noConversionIn	Type: PassThrough Description: Use input as is
	timeConversionIn	Type: TimeConversion Description: Convert the input to another time base
	noConversionOut	Type: PassThrough Description: Use output as is
	rateConversionOut	Type: RateConversion Description: Convert the output (rate) to 1/s
	timeConversionOut	Type: TimeConversion Description: Convert the output to seconds