Systems Theory Tiago Garcia de Senna Carneiro Pedro Ribeiro de Andrade Gilberto 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 fields Make 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 ( ), 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 ( ) – 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

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... 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 Outflow System Feedback

Population growth Births Deaths Fertility Mortality Population

Positive Coupling Atmospheric CO 2 Greenhouse effect An increase in atmospheric CO 2 causes a corresponding increase in the greenhouse effect, and thus in Earth’s surface temperature Conversely, a decrease in atmospheric CO 2 causes a decrease in the greenhouse effect

Negative Coupling Earth’s albedo (reflectivity) Earth’s surface 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 put couplings together in feedback loops…

person A’s body temperature person A’s blanket temperature Negative Feedback Loops: Electric Blankets person B’s blanket temperature person B’s body temperature

person A’s body temperature person A’s blanket temperature A Positive Feedback Loop: Mixed-up Electric Blankets person B’s blanket temperature person B’s body temperature

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

Run code #2 – Exponential Growth

Balancing feedback  Also named: negative, self-correcting, discrepancy- reducing, regenerative  Equilibrating or goal-seeking structures

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 Outflow System

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 Outflow System

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. h t+1 =h t ± 1 = ± 0.1 h

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:

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 N r k

How to implement stochastic models?  Create a random object that is able to generate numbers in a uniform distribution random = 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 = 0 for i = 1, 1000 do local n = random:number(0, 1) if ( n < 0.3) then -- try counter = counter + 1 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)

slide59 Systems Building Blocks  Stocks  Flows  Information Links  Decision Points  Converters  Auxiliary Variables

slide60 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)

slide61 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

slide62 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 information from 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

Perturbation

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

Temporal model Source: (Carneiro et al., 2013) 1:32:10ag1:execute( ) 1:38:07ag2:execute( ) 1:42:00cs:save()... (4) ACTION return value true (1) Get first EVENT 1:32:00cs: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