Stock-and-Flow-Diagrams

Stock-and-flow diagrams (SFDs) are the formal modelling notation of system dynamics — a visual language that converts qualitative intuitions about feedback into quantifiable, simulatable models. Developed by Jay Forrester at MIT during the late 1950s, SFDs extend Causal-Loop-Diagrams by forcing modellers to distinguish what accumulates from what changes it.

Notation Elements

SFDs use a small set of standardised symbols:

• Stocks (rectangles / “bathtubs”): Accumulations — inventory, population, debt, trust, water in a reservoir. Stocks are the “state” of the system at any moment; they correspond directly to System-Stock concepts.
• Flows (double-line arrows with a valve): Rates of change that fill or drain stocks — hiring rate, birth rate, spending rate. These correspond to System-Flow concepts. Flows can change instantaneously; stocks cannot.
• Clouds: Sources or sinks outside the model boundary (e.g., an “infinite supply” of potential customers). Clouds acknowledge that every model has boundaries.
• Auxiliary variables (circles): Intermediate calculations that determine flow rates. For example, a “desired inventory” auxiliary drives an ordering flow.
• Connectors (thin arrows): Information links showing which variables influence which. These are the same causal relationships captured in CLDs, but now differentiated from material flows.

SFDs vs. Causal Loop Diagrams

Causal-Loop-Diagrams are fast and communication-friendly but have three limitations that SFDs address:

• No stock-flow distinction: CLDs conflate accumulations with rates, producing misleading diagrams. SFDs make the distinction explicit.
• No simulation: CLDs cannot be run on a computer. SFDs map directly to differential equations, enabling numerical simulation and sensitivity testing.
• No model boundary: CLDs imply unlimited scope. Clouds in SFDs force explicit acknowledgement of what is inside and outside the model.

When to use CLDs: Early hypothesis generation, stakeholder communication, exploring feedback structure without committing to equations.

When to use SFDs: When precision matters, when System-Delays and accumulations drive behaviour, when you need to test policies quantitatively.

Origins and Tools

Forrester’s Industrial Dynamics (1961) introduced SFDs to model industrial supply chains — the same supply-chain dynamics later popularised as The-Beer-Game. The notation was partly inspired by electrical circuit diagrams and the hydraulic metaphor of water flowing between reservoirs.

Modern software tools for SFD modelling:

• Vensim (Ventana Systems) — most widely used in academic and policy research
• STELLA / iThink (isee Systems) — popular in education
• AnyLogic — supports SFDs alongside agent-based and discrete-event modelling
• Powersim — common in Scandinavian business applications

Related Concepts

• System-Stock
• System-Flow
• Causal-Loop-Diagrams
• Balancing-Feedback-Loops
• Reinforcing-Feedback-Loops
• System-Delays
• Thinking in Systems - Meadows - 2008

Future Connections

Once created, link also to: System-Zoo (task 008 — the canonical SFD archetypes), Oscillation-in-Systems (task 009 — delay-driven oscillation best shown in SFDs).

Sources

• Meadows, Donella H. (2008). Thinking in Systems: A Primer. Chelsea Green Publishing. ISBN: 978-1-60358-055-7.

  • Chapters 1–2 introduce SFD notation and contrast it with verbal descriptions; uses SFDs throughout to illustrate feedback and delay dynamics.

• Forrester, Jay W. (1961). Industrial Dynamics. MIT Press. (Reprinted by Productivity Press, 1999). ISBN: 978-0-915299-88-7.

  • Original source for SFD notation; introduces stocks, flows, and auxiliary variables as a formal modelling language for industrial supply chains.

• Sterman, John D. (2000). Business Dynamics: Systems Thinking and Modeling for a Complex World. Irwin/McGraw-Hill. ISBN: 978-0-07-231135-8.

  • Chapters 6–7 provide the most comprehensive academic treatment of SFD notation, differential equation mapping, and simulation methodology.

• Richardson, George P. and Alexander L. Pugh III (1981). Introduction to System Dynamics Modelling with Dynamo. MIT Press. ISBN: 978-0-262-18091-2.

  • Early academic textbook establishing SFD as a modelling discipline; includes formal notation conventions still used in Vensim and STELLA.

• Kirkwood, Craig W. (1998). “System Dynamics Methods: A Quick Introduction.” Arizona State University working paper. Retrieved from: https://nutritionmodels.com/papers/Kirkwood1998.pdf

  • Accessible primer on SFD elements (stocks, flows, auxiliaries, connectors) and their mathematical interpretation as differential equations.

Note

This content was drafted with assistance from AI tools for research, organization, and initial content generation. All final content has been reviewed, fact-checked, and edited by the author to ensure accuracy and alignment with the author’s intentions and perspective.