Systems Theory

Tiago Garcia de Senna CarneiroPedro Ribeiro de AndradeGilberto Câmara

Münster, 2013

Geoinformatics enables crucial links between nature and society

Nature: Physical equations Describe processes

Society: Decisions on how to Use Earth´s resources

How to model Natural-Society systems?

If (... ? ) then ...

Desforestation?

Connect expertise from different fieldsMake the different conceptions explicit

“A hypothesis or theory [model] is clear, decisive, and positive, but it is believed by no one but the man who created it. Experimental findings [observations], on the other hand, are messy, inexact things, which are believed by everyone except the man who did that work”Harlow Shapley (1885-1972), American astronomer

“[The] advantage of a mathematical statement is that it is so definite that it might be definitely wrong…..Some verbal statements have not this merit; they are so vague that they could hardly be wrong, and are correspondingly useless.”Lewis Fry Richardson (1881-1953) – first to apply mathematical methods to numerical weather prediction

Models

How reality is conceived Any measurable part of reality can be modelled as a system

Systems are represented as stocks and flows Stocks represent storages of energy, matter, or information Flows connect and transport stocks

Real systems are opened only theoretical ones are closed

Environment System 2

System 3

System 1

System 4

What is a System?

Definition: A system is a group of components with different functions, which interact with each other

Example: The climate system includes the atmosphere, oceans, polar caps, clouds, vegetation…and lots of other things

How do we study systems?

• Identify the components

• Determine the nature of the interactions between components

Atmospheric Physics/Dynamics

Tropospheric Chemistry

Global Moisture

Ocean Dynamics

MarineBiogeochemistry

Terrestrial Ecosystems

Terrestrial Energy/Moisture

Climate Change

Pollutants

CO2

CO2

Soil

Land Use

Physical Climate System

Biogeochemical Cycles

Human Activities

(from Earth System Science: An Overview, NASA, 1988)

Earth as a system

Systems Theory

Provides a unified classification for scientific knowledge. Enunciated by biologist Ludwig Von Bertalanffy:

1920s: earliest developments 1937: Charles Morris Philosophy Seminar, University of Chicago 1950: “An Outline of General Systems Theory”, Journal for the Philosophy

of Science

Scientists that introduced Systems Theory in their fields: Parsons, sociologist (1951) J.G Miller, psychiatrist & psychologist (1955) Boulding, economist (1956) Rapoport, mathematician (1956) Ashby, bacteriologist (1958)

A system

Can you identify parts? and Do the parts affect each other? and Do the parts together produce an effect that is different from

the effect of each part on its own? and perhaps Does the effect, the behavior over time, persist in a variety of

circumstances?

Source: (Meadows, 2008)

A system

Can you identify parts? and Do the parts affect each other? and Do the parts together produce an effect that is different from

the effect of each part on its own? and perhaps Does the effect, the behavior over time, persist in a variety of

circumstances?

Source: (Meadows, 2008)

Systems can grow in different ways...População

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forever...

explode...

stabilize...

Run code #1 – Linear Growth

Feedbacks

Feedback is how the system affect itself

Essential to systems be able to reach their goal

Inflow OutflowSystem

Feedback

Population growth

Births Deaths

Fertility

Mortality

Population

Positive Coupling

AtmosphericCO2

Greenhouseeffect

• An increase in atmospheric CO2 causes a corresponding increase in the greenhouse effect, and thus in Earth’s surface temperature• Conversely, a decrease in atmospheric CO2

causes a decrease in the greenhouse effect

Negative Coupling

Earth’s albedo(reflectivity)

Earth’ssurface

temperature

• An increase in Earth’s albedo causes a corresponding decrease in the Earth’s surface temperature by reflecting more sunlight back to space• Or, a decrease in albedo causes an increase in surface temperature

The interesting thing to do is to putcouplings together in feedback loops…

person A’sbodytemperature

person A’sblankettemperature

Negative Feedback Loops:Electric Blankets

person B’sblankettemperature

person B’sbodytemperature

person A’sbodytemperature

person A’sblankettemperature

A Positive Feedback Loop:Mixed-up Electric Blankets

person B’sblankettemperature

person B’sbodytemperature

A Positive Feedback Loop:Mixed-up Electric Blankets

Any perturbation will cause both people to adjust their blanket controls, but with undesired consequences.

Ultimately, one person will freeze (become infinitely cold) and the other person to swelter (become infinitely hot).

Equilibrium State:

Conditions under which the system will remain indefinitely

--If left unperturbed

Reinforcing feedbacks

Also named: positive, self-reinforcing, discrepancy-enhancing, degenerative

Self-enhancing behavior that leads to growth or even collapses

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Run code #2 – Exponential Growth

Balancing feedback

Also named: negative, self-correcting, discrepancy-reducing, regenerative

Equilibrating or goal-seeking structures

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Homeostasis

It is a tendency that all systems have to maintain their equilibrium state through negative feedbacks

Initial condition = 3.2

Initial condition = 8

Run code #3 – Homeostasis

Equilibrium state (some times steady-state)

Equilibrium means a state of balance.

There are no net flows of matter or of energy

Input flow == Output flow

Inflow OutflowSystem

Equilibrium state (some times steady-state)

Equilibrium means a state of balance.

There are no net flows of matter or of energy

Input flow == Output flow

Inflow OutflowSystem

An Unstable Equilibrium State

low resilience

An Unstable Equilibrium State

Perturbation

When pushed by a perturbation, an unstable equilibrium state shifts to a new, stable state.

A Stable Equilibrium State

higher resilience

A Stable Equilibrium State

Perturbation

When pushed by a perturbation, a stable equilibrium state, returns to (or near) the original state.

Run code #4 – Logistic Growth

Verify and analyse models with visualizations TerraME provides you different types of Observers However, it can only observes TerraME types: Cell, Agent, CellularSpace, Timer,

Environment, etc.

Ant agents eat sugar on a cellular space

Run codes #5, #6 – Logistic Growth

Discrete & Continuous Systems

Discrete systems jump from one state to other without intermadiate valuas, like the traffic light.

Continuous system change from a state to other going through all intermadiate states, like the speed of a car.

Depending on your point of view you can model a system as discrete or continuos, like a lift.

ht+1 =ht ± 1 = ± 0.1 hdt

dh

There are different types of equlibrium

Discrete systems: Fixed point - System converges to a one-dimension fixed

value.

N-dimensional attractors – System converges to attractors composed by several N fixed points

Deterministic CAOS – System is locked in a high dimensional attractor composed theorically by a infinite number of fixed points and will never repeat itself (this is the caos).

Run codes #7 – Discrete Logistic Growth

As the system is discrete we should use a difference equation istead of a differential equation:

)/1(1 KNrNN ttt

Logistic Map

From smooth behavior to deterministic caos through duplication of periods. Feigenbaum, M. (1983) – in Physics. May, R. (1976) – in Ecology.

Discrete Growth – It is no error propagation!

(a) r = 1,2, (b) r = 3,0, (c) r = 3,5 e (d) r = 4,0.

There are different types of equlibrium

Cotinuous systems: One single system

Static equilibrium - System converges to a one-dimension fixed value.

Coupled sytems (like prey-predator) Static equilibrium - System converges to a one-dimension fixed value.

Dynamic equilibrium – System converges to cyclical behavior and keep repeating itself

Erratic outcomes of deterministics rules should be treated as error propagation in the integration method

Run code #8 – Contiuous System

How CONTINUOUS systems grow?

Linear growth

Exponential growth

Logistic growth

rdt

dN

rNdt

dN

kNkrNdt

dN/)(

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N

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dN

How to implement stochastic models?

Create a random object that is able to generate numbers in a uniform distributionrandom = Random()

probability density function

Call function number(a, b) to generate real numbers within the interval [a, b]:

n = random:number(0,1)

Call function integer(a,b) to generate integers within the interval [a,b]: n =

random:integer(10,20)

How to implement stochastic models?

random = Random()

counter = 0for i = 1, 1000 do

local n = random:number(0, 1)if ( n < 0.3) then -- try 0.0 0.5 0.95 1.0

counter = counter + 1end

end

print( (counter/1000) * 100 ) -- 30%

Run codes #9 and #10 – Stochastic process

Coupled systems – Dynamic Equilibrium

Run code #11 – Prey-predator model

Short History of System Dynamics

The System Dynamics approach was developed in the 1960s at M.I.T. by Jay Forrester.

A system in Modelica

Conception of Reality

Any measurable part of reality can be modeled Systems are represented as stocks and flows

Stocks represent energy, matter, or information Flows connect and transport stocks

Systems are opened or closed

A system

Can you identify parts? and Do the parts affect each other? and Do the parts together produce an effect that is different from

the effect of each part on its own? and perhaps Does the effect, the behavior over time, persist in a variety of

circumstances?

Source: (Meadows, 2008)

slide 59

Systems Building Blocks

Stocks Flows Information Links Decision Points Converters Auxiliary Variables

slide 60

Stocks

“Things” that accumulate in a system Physical or non-physical things Value is a quantity or level Persistent (remain even if all flows stop) Conservation (stock units enter from environment

and return to environment)

slide 61

Flows

Movement of “things” in and out of stocks Not persistent (can be stopped and started) Value is a rate of change (will always have a time

dimension) Flow unit = stock unit / time The unit of measurement for a flow will always be

the unit of measurement of a stock divided by an element of time

slide 62

Stock and Flow Diagram

Stocks in boxes Flows as straight double arrows Information Links as thin curved arrows Decision Points as closed in X

Control Material Flaw

to Stock

Add New information

Send informationfrom the Stock

Control Material Flaw

from Stock

Stock

System Dynamics Modelling

Shrimp farming

Simple model for shrimp farm

Results?

Figure 7

An Unstable Equilibrium State

An Unstable Equilibrium State

Perturbation

When pushed by a perturbation, an unstable equilibrium state shifts to a new, stable state.

A Stable Equilibrium State

A Stable Equilibrium State

Perturbation

When pushed by a perturbation, a stable equilibrium state, returns to (or near) the original state.

Tools for system dynamics

Dinamo Vensim Simile STELLA

Water in the tub

Initial stock: water in tub = 40 gallons water in tub(t) = water in tub(t – dt) – outflow x dt t = minutes dt = 1 minute Runtime = 8 minutes Outflow = 5 gal/min

Cell

Not yet

(description extracted from “TerraME types and functions”)

Event

Not yet

Not yet

Temporal model

Source: (Carneiro et al., 2013)

1:32:10 ag1:execute( )

1:38:07 ag2:execute( )

1:42:00 cs:save()

. . .

(4) ACTIONreturn value

true

(1) Get first EVENT

1:32:00 cs:load( ) (2) Update current time

(3) Execute the ACTION

false

(5) Schedule EVENT again

Observer

Not yet

Water in the tub

Initial stock: water in tub = 40 gallons water in tub(t) = water in tub(t – dt) – outflow x dt t = minutes dt = 1 minute Runtime = 8 minutes Outflow = 5 gal/min

Water in the tub 2

Initial stock: water in tub = 40 gallons water in tub(t) = water in tub(t – dt) – outflow x dt t = minutes dt = 1 minute Runtime = 8 minutes Outflow = 5 gal/min Inflow = 40 gal every 10 min

Conclusions

Two ways to increase stocks Stocks act as delays or buffers Stocks allow inflows and outflows to be decoupled