F.14 MODULE—FIRST ORDER NON-LINEAR SELF-REFERENCING

F.14.1 General Description

This module combines the functionality of the First-Order Positive-Feedback Self-referencing and First-Order Negative-Feedback Self-referencing modules. The descriptions of each of those modules apply. When First-Order Positive-Feedback Self-referencing and First-Order Negative-Feedback Self-referencing modules are combined we see that regardless of the initial population, the integrated module produces behaviour which results in ‘S’ STATE OF THE SYSTEM tending towards the Maximum Possible Value of ‘S’. As that value is approached, either from below or from above (depending on the initial value of ‘S’) the value of Multiplier Based on Actual ‘S’ Divided by Maximum ‘S’ returns a value which controls the values of Flowing In and Flowing Out. The result is a strong tendency towards the Maximum Possible Value of ‘S’. Feedback loop dominance depends on the initial conditions.

Figure F14-1. Stock-and-flow Diagram—First-order Non-Linear Self-referencing

F.14.2 Influence Diagram Representation

The influence diagram representation of this module is shown in Figure F14-2.

Figure F14-2. Influence Diagram—First-order Non-Linear Self-referencing

F.14.3 Reference Source

Further details of the non-linear first order system exhibiting shifting loop dominance are provided by Sterman (2000: 285-288).

F.14.4 Application

By itself this module serves to explain how feedback structure can regulate the state of a first-order system. In the process, the feedback loops having greatest influence change. This is an example of shifting feedback loop dominance. This has significant implications for sustainability and resource-limited systems. It is most frequently given as an example of behaviour of populations. A population example is depicted in Figure F14-3, where POPULATION directly influences population density which is measured as a decimal fraction of Carrying Capacity using the variable Population as Decimal of Carrying Capacity. Carrying Capacity is limited by space or area. Hence, the capacity to sustain the population is limited by an externally applied constraint (or exogenous factor).

Figure F14-3. First-order Non-Linear Self-referencing—*POPULATION* Example

F.14.5 Functional Description

The functions of the variables contained in the module depicted in Figures F14-4 are described in Table F13-1.

Table F14-1. Functional Description—First-order Non-Linear Self-referencing—POPULATION

Variable	Function	Comment
Initial Population	User defined	Units: <<animals>>.
POPULATION	Initial Population	Units: <<animals>>. Initial Population – dt * ( Net Birthing)
Net Birthing	'Nominal Birth Rate Fraction' * 'Fraction of Nominal Birth Rate' * POPULATION/1<<yr>>	Units: <<animals/yr>>.
Nominal Birth Rate Fraction	0.35	Dimensionless
Nominal Death Rate Fraction	0.20	Dimensionless
Carrying Capacity	5000	Units: <<animals>>.
Population Divided by Carrying Capacity	POPULATION/'Carrying Capacity'	Dimensionless.
Birth Rate Multiplier	(GRAPH(('Population Divided by Carrying Capacity') ,0.70,0.05,{1.0, 0.95, 0.9, 0.50, 0.1, 0.05, 0,00}))	Dimensionless. See graph at Figure F12-6.
Death Rate Multiplier	(GRAPH(('Population Divided by Carrying Capacity') ,1.0,0.1,{0.0,0.1,0.2,0.4,0.8,1.6,3.2,6.4}))	Dimensionless. See graph at Figure F13-6.

References

Sterman, J.D., 2000, Business Dynamics: Systems Thinking and Modelling for a Complex World, Irwin McGraw-Hill.