LogisticGrowth

Logistic growth model

Diagram

Information

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LogisticGrowth describes the growth of some population that is limited by the availability of some finite resource. While the population starts to grow exponentially at first at a given fractional rate of growth (either given by the constant parameter r or by the time-variant input u[1]), its growth rate will continously diminish until the population reaches its sustainable level (either given by the constant parameter K or the time-variant input u[2]), which is called the carrying capacity.

The rate of inflow to a connected stock is given by the so called Verhulst equation:

dx/dt = r x ( 1 - x/K )

The diagram below shows the s-shaped growth for a population x for different rates of growth:

LogisticGrowth for Different Rates r (x0 = 0.1)

Notes

By default it is assumed, that the rate given is a continously compounding rate (isCCR = true). If this is not case, the rate will be converted using the →ForceOfInterest converter.

While the component refers to growth it can easily be seen from the equation above, that for negative rates or for K < x the inflow to the stock can become negative. This is allowed and the modeler has to take care that the input or parameter values make sense for the process being modeled.

Acknowledgements

The logistic growth equation is originally due to the Belgian mathematican Pierre-François Verhulst (1804 - 1849).

Parameters (4)

r	Value: unspecified Type: Rate (1/s) Description: Maximum fractional growth rate
K	Value: unspecified Type: CapacityType (#) Description: Carrying capacity or maximum sustainable amount for the connected stock (>0)
isCCR	Value: true Type: Boolean Description: = true, if the maximum fractional rate given is assumed to be a continuously compounding rate else the rate will be converted
hasConstantInputs	Value: false Type: Boolean Description: = true, if constant parameters aure to be used instead of the real inputs u[1] and u[2]

Connectors (5)

	y	Type: RealOutput Description: Rate information
	y1	Type: RealOutput Description: Rate information
	y2	Type: RealOutput Description: Rate information
	massPort	Type: FlowPort Description: Inflow to or outflow from connected stock
	u	Type: RealMultiInput[2] Description: [1] maximum fractional growth rate [2] carrying capacity

Components (12)

	population	Type: FlowPortSensor Description: Current level of the connected stock
	cloud	Type: Cloud Description: System boundary
	growing	Type: Switching Description: Process of logistic growth
	fractionalGrowthRate	Type: Product_2 Description: Actual fractional rate of growth
	relativePopulation	Type: Division_Guarded Description: Population as percentage of carrying capacity
	residualCapacity	Type: Gap Description: Percentage of the carrying capacity left for growth
	one	Type: ConstantConverter Description: 100 percent
	growthRate	Type: Product_2 Description: Rate of growth
	carryingCapacity	Type: ConstantConverter Description: Constant carrying capacity (optional)
	maxFractionalGrowthRate	Type: ConstantConverter Description: Constant maximum rate of growth (optional)
	convertedRate	Type: ForceOfInterest Description: Discrete compounding rate being converted to CCR
	identicalRate	Type: PassThrough Description: Use indicated rate if it is continuously compounding

Used in Components (1)

LogisticGrowth

BusinessSimulation.CausalLoop

S-shaped logistic growth process

Revisions

Added type choice for K and defaulted optional constant parameters to unspecified.