System dynamics

System Dynamics

System Dynamics is a methodology used to understand, model, and analyze the behavior of complex systems over time using:

• Feedback loops
• Stocks and flows
• Differential equations

It was developed by Jay W. Forrester (MIT).

Key Features

• Focuses on long-term behavior
• Uses feedback control
• Deals with non-linear systems
• Useful for policy and decision-making

Examples

• Population growth
• Economic development
• Environmental systems
• Industrial production systems

Elements of System Dynamics

Element	Meaning	Example
Stock	Accumulation	Population
Flow	Rate of change	Birth rate
Feedback	Cause–effect loop	Supply–demand
Delay	Time lag	Policy impact

Exponential Growth Model

An exponential growth model describes a situation where the rate of growth is proportional to the current size of the system.

Mathematical Model

$$\frac{dX}{dt}=rX$$

Where:

• X = system variable
• r = growth rate
• t = time

Solution

$$X(t)=X_0{e}^{rt}$$

Graph Shape

X
│       /
│     /
│   /
│ /
└───────── t

Examples

• Population growth (unlimited resources)
• Bank compound interest
• Spread of virus (initial phase)

Exponential Decay Model

An exponential decay model represents a system where the quantity decreases at a rate proportional to its current value.

Mathematical Model

$$\frac{dX}{dt}=-rX$$

Solution

$$X(t)=X_0{e}^{-rt}$$

Graph Shape

X
│\
│ \
│  \
│   \
└───────── t

Examples

• Radioactive decay
• Discharging capacitor
• Cooling of hot object

Comparison: Growth vs Decay

Basis	Exponential Growth	Exponential Decay
Change	Increasing	Decreasing
Sign of r	Positive	Negative
Curve	Rising	Falling
Example	Population	Radioactivity

Logistic Curve Model

A logistic model represents growth that:

• Starts exponentially
• Slows down
• Finally stabilizes due to limited resources

Mathematical Model

$$\frac{dX}{dt}=rX(1-\frac{X}{K})$$

Where:

• K = carrying capacity

Logistic Curve Shape (S-Curve)

X
│        ───────
│      ─
│   ─
│ ─
└──────────── t

Phases

• Initial slow growth
• Rapid growth
• Saturation

Examples

• Population with limited resources
• Market product life cycle
• Technology adoption

System Dynamics Diagrams

A. Stock and Flow Diagram

      Inflow
        ↓
   ┌─────────┐
   │  Stock  │
   └─────────┘
        ↓
      Outflow

B. Feedback Loop Diagram

Positive Feedback

A → B → C
↑         ↓
└─────────┘

(Amplifies change)

Negative Feedback

A → B → C
↑         ↓
└─(–)─────┘

(Stabilizes system)

Feedback Comparison Table

Feedback	Effect	Example
Positive	Growth	Population
Negative	Stability	Thermostat

World Model (Forrester World Model)

The World Model is a large-scale system dynamics model developed to study global problems.

Main Variables

• Population
• Industrial output
• Pollution
• Natural resources
• Food production

World Model Diagram

Population → Resources → Pollution
     ↑           ↓           ↓
     └──── Food Production ──┘

Objectives of World Model

• Study long-term global trends
• Analyze sustainability
• Support policy decisions

Applications

• Climate change studies
• Economic planning
• Environmental management

Summary Table

Topic	Key Point
System dynamics	Time-based modeling
Exponential growth	Unlimited increase
Exponential decay	Continuous decrease
Logistic curve	Limited growth
Diagrams	Stocks, flows, feedback
World model	Global system analysis

Exam Writing Strategy

• Start with definition
• Write mathematical equation
• Draw clear diagram
• Add real-life example
• Use tables for comparison