STOCKS AND FLOWS IN SYSTEM DYNAMICSSTOCKS AND FLOWS IN SYSTEM DYNAMICS

Esmaeil Khedmati MorasaeCommunity Based Participatory Research Center

Tehran University of Medical SciencesSpring, 2013

Causal Loop Diagrams (CLDs) are well suited for showinginterdependencies and feedback processes (mentalmodel).

However, there are some limitations to CLDs namelytheir inability to represent stock and flow structure ofsystems.

Causal Loop Diagrams (CLDs) are well suited for showinginterdependencies and feedback processes (mentalmodel).

However, there are some limitations to CLDs namelytheir inability to represent stock and flow structure ofsystems.

Stocks are accumulations. They characterize the state ofthe system and generate the information upon whichdecisions and actions are based.

Stocks give systems inertia and provide them withmemory.

Stocks create delays by accumulating the differencebetween the inflow to a process and its outflow.

Stocks are accumulations. They characterize the state ofthe system and generate the information upon whichdecisions and actions are based.

Stocks give systems inertia and provide them withmemory.

Stocks create delays by accumulating the differencebetween the inflow to a process and its outflow.

By decoupling rates of flow, stocks are the source ofdisequilibrium dynamics in systems.

Example, the balance of your bank account.

By decoupling rates of flow, stocks are the source ofdisequilibrium dynamics in systems.

Example, the balance of your bank account.

DIAGRAMMING NOTATIONS OF STOCKS AND FLOWS.

- Stocks are represented by rectangles.- Inflows are represented by a pipe (arrow) pointing into

(adding to) the stock.- 0utflows are represented by pipes pointing out of

(subtracting from) stock.- Valves control the flows.- Clouds represent the sources and sinks for the flow.

- Stocks are represented by rectangles.- Inflows are represented by a pipe (arrow) pointing into

(adding to) the stock.- 0utflows are represented by pipes pointing out of

(subtracting from) stock.- Valves control the flows.- Clouds represent the sources and sinks for the flow.

- A source represent the stock from which a floworiginating outside the boundary of the model arises.

- A sink represent the stocks into which flows leavingthe model boundary drain.

- A source represent the stock from which a floworiginating outside the boundary of the model arises.

- A sink represent the stocks into which flows leavingthe model boundary drain.

MATHEMATICAL REPRESENTATION OF STOCKS AND FLOWS

The stock and flow structure can be represented in anunambiguous mathematical formalism.

- Integral equation,

- Differential equation,

The stock and flow structure can be represented in anunambiguous mathematical formalism.

- Integral equation,

- Differential equation,

THE CONTRIBUTION OF STOCKS TO DYNAMICS

Stocks are critical in generating the dynamics of systemsfor the following reasons:

1- stocks characterize the state of the system andprovide the basis for actions.

The stocks in a system tell decision makers where theyare, providing them with the information needed toact.

Stocks are critical in generating the dynamics of systemsfor the following reasons:

1- stocks characterize the state of the system andprovide the basis for actions.

The stocks in a system tell decision makers where theyare, providing them with the information needed toact.

2- stocks provide systems with inertia and memory.Stocks accumulate past events. The content of a stock

can only change through an inflow or outflow. Withoutchanges in these flows, the past accumulation into thestock persists.

- Memories and beliefs are stocks. Your beliefs persistover time, generating inertia and continuity in yourattitudes and behavior.

2- stocks provide systems with inertia and memory.Stocks accumulate past events. The content of a stock

can only change through an inflow or outflow. Withoutchanges in these flows, the past accumulation into thestock persists.

- Memories and beliefs are stocks. Your beliefs persistover time, generating inertia and continuity in yourattitudes and behavior.

3- stocks are the source of delays.All delays involve stocks. A delay is a process whose

output lags behind its input. The difference betweeninput and output accumulates in a stock of material inprocess.

By definition, when the input to a delay changes, theoutput lags behind and continues at the older rate forsometime. During such adjustment, the stockaccumulating the difference between input and outputchanges.

3- stocks are the source of delays.All delays involve stocks. A delay is a process whose

output lags behind its input. The difference betweeninput and output accumulates in a stock of material inprocess.

By definition, when the input to a delay changes, theoutput lags behind and continues at the older rate forsometime. During such adjustment, the stockaccumulating the difference between input and outputchanges.

4- stocks decouple rates of flow and createdisequilibrium dynamics.

Stocks absorb the differences between inflows andoutflows, thus permitting the inflows and outflows to aprocess to differ.

In equilibrium, the total inflow to a stock equals its totaloutflow so the level of stock is unchanging.

However, inflows and outflows usually differ becausethey are often governed by different decisionprocesses. Disequilibrium is the rule rather than theexception.

4- stocks decouple rates of flow and createdisequilibrium dynamics.

Stocks absorb the differences between inflows andoutflows, thus permitting the inflows and outflows to aprocess to differ.

In equilibrium, the total inflow to a stock equals its totaloutflow so the level of stock is unchanging.

However, inflows and outflows usually differ becausethey are often governed by different decisionprocesses. Disequilibrium is the rule rather than theexception.

Production and consumption.Whenever two coupled activities are controlled by

different decision makers, involve different resources,and are subject to different random shocks, a bufferor stock between them must exist, accumulating thedifference.

As these stocks vary, information about the size of thebuffer will feedback in various ways to influence theinflows and outflows.

Production and consumption.Whenever two coupled activities are controlled by

different decision makers, involve different resources,and are subject to different random shocks, a bufferor stock between them must exist, accumulating thedifference.

As these stocks vary, information about the size of thebuffer will feedback in various ways to influence theinflows and outflows.

Often, but not always, these feedbacks will operate tobring the stock into balance.

Whether and how equilibrium is achieved can not beassumed but is an emerging property of whole systemas its many feedback loops interact simultaneously.

Often, but not always, these feedbacks will operate tobring the stock into balance.

Whether and how equilibrium is achieved can not beassumed but is an emerging property of whole systemas its many feedback loops interact simultaneously.

IDENTIFYING STOCKS AND FLOWS

Different disciplines use different terms for stocks andflows;

Economics, levels (stock) and rate (flow)Public health, prevalence and incidence.

Stock: state of the systemFlow: the rate at which the state of the system changes.

Different disciplines use different terms for stocks andflows;

Economics, levels (stock) and rate (flow)Public health, prevalence and incidence.

Stock: state of the systemFlow: the rate at which the state of the system changes.

- Interest rate- Unemployment rate- Expected rate of inflation

- Interest rate- Unemployment rate- Expected rate of inflation

Any flow into or out of a stock can be either positive ornegative. The direction of the arrow defines the signconvention for the flow. An inflow adds to stock whenthe flow is positive; if the flow is negative it subtractsfrom the stock. When the outflow is positive, the flowsubtracts from the stock.

Any flow into or out of a stock can be either positive ornegative. The direction of the arrow defines the signconvention for the flow. An inflow adds to stock whenthe flow is positive; if the flow is negative it subtractsfrom the stock. When the outflow is positive, the flowsubtracts from the stock.

STATE-DETERMINED SYSTEMS

The theory of dynamic systems takes a state-determinedsystem or state variable approach.

The only way a stock can change is via its inflows andoutflows. In turn, the stocks determine the flows.

Systems evolve by feedback of information from the sateof the system to the flows that alter the system.

The theory of dynamic systems takes a state-determinedsystem or state variable approach.

The only way a stock can change is via its inflows andoutflows. In turn, the stocks determine the flows.

Systems evolve by feedback of information from the sateof the system to the flows that alter the system.

Systems therefore consist of networks of stocks andflows linked by information feedbacks from the stocksto the rates.

Systems therefore consist of networks of stocks andflows linked by information feedbacks from the stocksto the rates.

Networks of stocks and flows are coupled by informationfeedback.

Equation representation: the derivatives of the stocks indynamic systems are, in general, nonlinear functionsof the stocks, the exogenous variables, and anyconstant.

In matrix notation, the rates of change dS/dt are afunction f() of state vector S, the exogenous variablesU and the constants C:

dS/dt=f (S, U, C)

Networks of stocks and flows are coupled by informationfeedback.

Equation representation: the derivatives of the stocks indynamic systems are, in general, nonlinear functionsof the stocks, the exogenous variables, and anyconstant.

In matrix notation, the rates of change dS/dt are afunction f() of state vector S, the exogenous variablesU and the constants C:

dS/dt=f (S, U, C)

AUXILLARY VARIABLES

As mentioned, mathematical description of a systemrequires only the stocks and their rates of change.

For ease of communication and clarity, however, it isoften helpful to define intermediate or auxiliaryvariables.

Auxiliaries consist of functions of stocks.

As mentioned, mathematical description of a systemrequires only the stocks and their rates of change.

For ease of communication and clarity, however, it isoften helpful to define intermediate or auxiliaryvariables.

Auxiliaries consist of functions of stocks.