System Dynamics

Overview

System dynamics models systems as networks of stocks (accumulations) and flows (rates of change), connected by feedback loops. Developed by Jay Forrester at MIT in the 1950s, it excels at modeling systems where feedback, delays, and nonlinearity create counterintuitive behavior.

System dynamics operates at an aggregate level — it models populations, not individuals. This makes it complementary to agent-based modeling.

Stocks and Flows

Stocks are quantities that accumulate over time: inventory, population, money in an account, water in a reservoir. Mathematically, a stock S evolves as:

dS/dt = inflow(t) - outflow(t)
S(t) = S(0) + ∫₀ᵗ [inflow(τ) - outflow(τ)] dτ

Flows are rates: births per year, dollars per month, liters per second. Flows fill or drain stocks.

         inflow          outflow
  ──────►[████████]──────►
          Stock S

STRUCTURE Stock
    value: real

PROCEDURE UPDATE_STOCK(stock, inflow, outflow, dt)
    stock.value ← stock.value + (inflow - outflow) * dt
    IF stock.value < 0.0 THEN stock.value ← 0.0   // physical constraint

// SIR epidemic as stocks and flows
STRUCTURE SirModel
    susceptible: real
    infected: real
    recovered: real
    beta: real     // contact rate * transmission probability
    gamma: real    // recovery rate (1 / average duration)

PROCEDURE STEP(model, dt)
    n ← model.susceptible + model.infected + model.recovered
    infection_flow ← model.beta * model.susceptible * model.infected / n
    recovery_flow ← model.gamma * model.infected

    model.susceptible ← model.susceptible - infection_flow * dt
    model.infected ← model.infected + (infection_flow - recovery_flow) * dt
    model.recovered ← model.recovered + recovery_flow * dt

Causal Loop Diagrams (CLDs)

CLDs capture the qualitative feedback structure before building a quantitative model.

Positive link (+): A increase causes B to increase (or A decrease causes B to decrease)
Negative link (-): A increase causes B to decrease

    Population ──(+)──► Births
         ▲                  │
         └──────(+)─────────┘
    (Reinforcing loop R)

    Population ──(+)──► Deaths
         ▲                  │
         └──────(-)─────────┘
    (Balancing loop B)

Reading Loop Polarity

Count negative links in the loop:

Even number (or zero) of negative links → Reinforcing (R) loop → exponential growth or collapse
Odd number of negative links → Balancing (B) loop → goal-seeking, oscillation, or equilibrium

Feedback Loops

Positive (Reinforcing) Feedback

Amplifies change. Examples:

Compound interest: more money → more interest → more money
Word of mouth: more users → more recommendations → more users
Arms race: more weapons by A → more weapons by B → more weapons by A

Unchecked reinforcing loops produce exponential growth or collapse.

Negative (Balancing) Feedback

Counteracts change, driving toward a goal. Examples:

Thermostat: temperature gap → heating → reduced gap
Inventory management: shortage → increased orders → replenished stock
Predator-prey: more prey → more predators → fewer prey

Delays

Delays in feedback loops cause oscillation and overshoot. A supply chain with ordering delays will oscillate around the target inventory rather than smoothly converging.

  Material delay (pipeline):
    Outflow at time t = Inflow at time (t - delay)

  Information delay (exponential smoothing):
    dPerceived/dt = (Actual - Perceived) / delay_time

System Archetypes

Recurring feedback structures that produce common behavioral patterns:

Limits to Growth

A reinforcing loop drives growth, but eventually a balancing loop engages a constraint.

Growth slows, peaks, potentially declines
Example: product adoption hitting market saturation

Shifting the Burden

A symptomatic fix weakens the fundamental solution.

Short-term relief undermines long-term capacity
Example: overtime masking the need to hire

Tragedy of the Commons

Multiple agents exploit a shared resource, each benefiting individually while depleting the commons.

Individual rational behavior leads to collective disaster
Example: overfishing, overuse of shared infrastructure

Fixes that Fail

A fix produces unintended side effects that worsen the original problem after a delay.

Example: pesticide use → resistance → more pesticide needed

Growth and Underinvestment

Growth approaches a limit that could be raised by investment, but investment is not made until performance drops.

Example: infrastructure not expanded until congestion is severe

Forrester Diagrams (Stock-Flow Diagrams)

The quantitative counterpart of CLDs. Standardized notation:

  ┌─────┐
  │Stock│ ← Rectangle (accumulation)
  └──┬──┘
     │
  ◄──┤──►  ← Valve/flow regulator (hourglass shape)
     │
  ───┘      ← Pipe (flow channel)

  (cloud)   ← Source or sink (boundary of model)

  ────○──── ← Information link (dashed arrow with circle)

Simulation Mechanics

System dynamics models are systems of ODEs solved by numerical integration.

Euler Integration

Simplest method; adequate for many SD models:

S(t + dt) = S(t) + dS/dt · dt

Runge-Kutta (RK4)

More accurate; recommended when dynamics are fast or nonlinear:

PROCEDURE RK4_STEP(f, t, state, dt) → new_state
    n ← LENGTH(state)
    k1 ← f(t, state)
    s2 ← array: FOR i ← 0..n-1: state[i] + 0.5 * dt * k1[i]
    k2 ← f(t + 0.5 * dt, s2)
    s3 ← array: FOR i ← 0..n-1: state[i] + 0.5 * dt * k2[i]
    k3 ← f(t + 0.5 * dt, s3)
    s4 ← array: FOR i ← 0..n-1: state[i] + dt * k3[i]
    k4 ← f(t + dt, s4)

    RETURN array: FOR i ← 0..n-1:
        state[i] + dt/6 * (k1[i] + 2*k2[i] + 2*k3[i] + k4[i])

Time Step Selection

dt should be small relative to the shortest time constant in the model
Rule of thumb: dt ≤ (shortest delay or time constant) / 4
Halve dt and compare results to check numerical stability

Captures feedback and delay effects that spreadsheet models miss
Forces explicit articulation of causal assumptions
Accessible to stakeholders through visual stock-flow diagrams
Computes quickly (small ODE systems)

Limitations:

Aggregate — cannot represent individual heterogeneity
Requires continuous approximation of discrete processes
Structural assumptions baked into diagram topology can be hard to validate
Nonlinearity in table functions is opaque compared to explicit equations

Modeling Best Practices

Start with a reference mode — Sketch the expected behavior pattern (growth, oscillation, S-curve) before building the model
Build incrementally — Start with one feedback loop, verify behavior, then add complexity
Use dimensional analysis — Every equation must be dimensionally consistent (stocks in units, flows in units/time)
Test extreme conditions — Set stocks to zero or very large values; behavior should remain physically plausible
Document every equation — Record the reasoning behind each functional relationship