F.12 MODULE—FIRST ORDER NON-LINEAR SELF-REFERENCING—FLOWING IN

F.12.1 General Description

This goal setting module is self-referencing, the reference level being set by Maximum Possible Value of ‘S’, which is applied as an exogenous constraint. For example, this might be a consequence of finite capacity of an area to sustain a population. The actual state variable ‘S’ STATE OF THE SYSTEM is continually monitored as a measure of relative density, that is, Actual ‘S’ Divided by Maximum ‘S’. This indicates proximity of actual state to maximum possible. In self-referencing systems, whose behaviour over time is shown in Figure F12-2, the system is highly sensitive to the maximum possible value of S.

Figure F12-1. Stock-and-flow Diagram—First-order Non-linear Self-referencing

Figure F12-2. Reference Mode Behaviour—First-order Non-linear Self-referencing—Flowing In

As that value is approached the value of Multiplier Based on Actual ‘S’ Divided by Maximum ‘S’ drops off dramatically. The direct consequence of this is that the value of Flowing In (which is a product of Nominal Rate of Flowing In and Multiplier Based on Actual ‘S’ Divided by Maximum ‘S’ ) drops off just as dramatically. The value of Flowing In, the rate at which contributions to S are made is sensitive to its constrained maximum value.

F.12.2 Influence Diagram Representation

The influence diagram representation of this module is shown in Figure F12-3.

Figure F12-3. Influence Diagram—First-order Non-linear Self-referencing—Flowing In

F.12.3 Reference Sources

This module is defined after Goodman (1989: 112-114) and Hannon and Ruth (1994: 21-24). Further details, in relation to this module as part of the non-linear first order system producing S-shaped growth, are provided by Sterman (2000: 285-288).

F.12.4 Application

By itself this module serves to explain how feedback structure can regulate the state of a first-order system. This has significant implications for sustainability and resource-limited systems. The commercial or residential development in areas where land availability is limited would be amenable to analysis using a model based on this module. Similarly, the allocation of resources to achievement of project activities might be modelled using this module.

It is most frequently given as an example of behaviour of populations, early studies being based on rat populations. A population example is depicted in Figure F12-4, 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). The variable Birth Rate Multiplier has that characteristic that for small values of Population as Decimal of Carrying Capacity it takes on values close to 1.0, but as the value of Population as Decimal of Carrying Capacity rises Birth Rate Multiplier drops off sharply to be zero when Population as Decimal of Carrying Capacity reaches 1.0. Net Birthing remains near the Birth Rate Multiplier until POPULATION approaches Carrying Capacity when it reduces dramatically. As time passes the POPULATION approaches Carrying Capacity more slowly.

The equivalent influence diagram is shown in Figure F12-5.

Figure F12-4. Stock-and-flow Diagram—First-order Non-linear Self-referencing—Flowing in *POPULATION*

Figure F12-5. Influence Diagram—First-order Self-referencing System—Inflow to *POPULATION*

F.12.5 Functional Description

The functions of the variables contained in the module depicted in Figures F12-4 and F12-5 are described in Table F12-1.

The summary results of running a simulation of the POPULATION module over a 15-year period using a 1-month time-step appear at Figure F12-7.

Table F12-1. Functional Description—First-order Self-referencing System—Inflow to POPULATION

Variable	Function	Comment
Initial Population	4000	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
Carrying Capacity	3000	Units: <<animals>>.
Population As Decimal of 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, 0}))	See graph at Figure F12-6.

Figure F12-6. Graph—Birth Rate Multiplier as Function of Population Divided by Carrying Capacity

Figure F12-7. *POPULATION* Inflow Module Showing Auto Reports

References

Goodman, M.R., 1989, Study Notes in System Dynamics, Productivity Press, Portland, Oregon.
Hannon, B. and Ruth, M., 1994, Dynamic Modelling, Springer-Verlag, New York.
Sterman, J.D., 2000, Business Dynamics: Systems Thinking and Modelling for a Complex World, Irwin McGraw-Hill.