Quantifying the strengths of feedback loops can bring insight into the way the structure of a system dynamics model helps determine behaviour. This paper proposes the concept of loop impact, using the functional relationship between the second and first derivative from the Pathway Participation Metric method, and describes a numerical method to derive the impacts of feedback loops within a system dynamics model. An algorithm is presented that will identify which loop, or loop combination, explains stock behaviour. The method is applied to the yeast, epidemic and market growth models and compared with previous work. It is shown that the method can deal with loops that change polarity, hidden loops and those that self cancel. A procedure is set out to calculate impacts when loops have junctions and graphical converters. It is anticipated that the method will be adopted by system dynamicists and applied to a broader range of models. Copyright © 2014 System Dynamics Society

This paper proposes an interpretative framework for system dynamics models using concepts from Newtonian mechanics. By considering the second derivative form of a model, it is shown that Newton's three laws of motion have their equivalent in system dynamics, with forces between stocks being determined using the loop impact method. The concepts of mass, inertia, momentum and friction are explored as to their usefulness in understanding model behaviour. The Newtonian framework is applied to two standard system dynamics models-inventoryworkforce and economic long-wave-where their behaviour is analyzed using force dominance on the stocks. Results show improved intuitive understanding of system behaviour compared with existing dominance methods, particularly for models with exogenous effects, oscillations and many loops. The framework is commended for further exploration.address the limitations raised above. Firstly, the concept of force has a clear intuitive connection with variable behaviour, enabling informal explanations in terms of model structure. Secondly, the force concept is easily extended to exogenous inputs, enabling comparison with forces from stocks within feedback loops. Thirdly, using stock-to-stock pathways as the main structural element, implicit in the PPM and loop impact methods, enables the unique identification of system forces. Finally, concepts of mass and friction can help explain oscillations arising from higher-order feedback. The proposed Newtonian interpretative framework builds on Hayward and Boswell's (2014) loop impact method by recasting the concept of impact in terms of the classic definition of force. Sato (2016) notes that force has been used informally in SD in the past (e.g. Forrester, 1961;Goodman, 1974) and more recently (e.g. Richardson, 2011). The proposed interpretative framework will provide a formal definition of force and other Newtonian concepts for the SD methodology.The structure of the paper is as follows. Firstly, the concept of loop impact, as introduced by Hayward and Boswell (2014), is investigated further by discussing the notion of the "impact" of a cause on motion generally. The concept is reinterpreted as the impact of a force on a stock. Secondly, laws of SD are proposed by analogy with Newton's laws of motion, with the concepts of force, momentum, mass, inertia and friction explored as to their usefulness in understanding the behaviour of any SD model. Thirdly, a notation is introduced to enable the analytical computation of force impact. Finally, the ideas presented are applied to two existing SD models to evaluate their use. (Models are available in an online supplement as supporting information.)The concept of "impact"The interpretive framework described in this paper employs the effect that one stock has on another as its principal focus of system behaviour analysis. The proposal is to use Hayward and Boswell's (2014) definition of loop impact as a measure of the "force" of one stock on the "motion" of another stock. In order to aid understanding of the c...

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The possibility of using mathematics to model church growth is investigated using ideas from population modeling. It is proposed that a major mechanism of growth is through contact between religious enthusiasts and unbelievers, where the enthusiasts are only enthusiastic for a limited period. After that period they remain church members but less effective in recruitment. This leads to the general epidemic model which is applied to a variety of church growth situations. Results show that even a simple model like this can help understand the way in which churches grow, particularly in times of religious revival. This is a revised version of Hayward (1999) using System Dynamics and some small modifications to the SIR model 1 .

The Loops that Matter (LTM) approach to understanding behavior has proven easy to use and broadly applicable, but it has a shortcoming in its original formulation. This is because the original formulation treats the impact of a flow on a stock relative to the net flow, so that all scores tend to get very large in magnitude as a stock approaches equilibrium, but how big depends strongly on how the flows are specified. By reformulating the link scores from a flow to a stock, this topological dependency is removed. The mathematics behind this approach makes clear the relationship of LTM to the Pathway Participation and Loop Impact analysis methods. The result of this, when applying the analysis to a variety of models, is that the determination of the structure responsible for behavior is clearer, and more clearly tied to work already documented using other techniques.