Modeling Nature January 2009 1 Modeling Nature LECTURE 1: Models of Growth * and some general information
2 Ronald Westra Department of Mathematics Faculty of Humanities & Sciences Universiteit Maastricht 3 Overview Practical details
Introduction to modelling A simple model A mathematical description A computer science description An intuitive description
A complex model The nature of models and models of nature Models of Growth and Decay 4 Practical details Coordinator; Ronald Westra Department of Mathematics, Faculty of Humanities & Sciences, Maastricht University
Office: room 1.017, Bouillonstraat 8-10, Maastricht/Netherlands e-mail: [email protected] For requests: contact your tutor All tutors meet on a weekly basis to discuss the tasks and any problems that may arise 5 General Information The course Modelling Nature consists of
six 2-hour Lectures, six times two Tutor Groups, and a Student research project (in groups of 2 to 3 students). The course will end with a final exam based on the lectures and the reading material. 6 General structure Lecture Tutor Group meeting 1 (TG-a)
Preparing this weeks task e.g., subtasks 1a and 1b Tutor Group meeting 2 (TG-b) Discussing this weeks task by discussing Required Readings 7 Weekly roster (week 1,2,3,4,5,6)
Mondays: 8.30-10.30h Lectures Attendance is required in order to be able to pass the exams Two tutorial group meetings: Monday and Friday (whatever ) 8
Complete time table 02/2 - Lecture 1 : Models, Growth & Decay 09/2 - Lecture 2 : Predator-Prey Systems 16/2 - Lecture 3 : Network Models 02/3 - Lecture 4 : Chaos and Fractals;
first draft report and presentation feedback from tutors 09/3 - Lecture 5 : Percolation and Phase Transitions 16/3 - Lecture 6 : Self-Organization and Collective Phenomena 23/3 - Week 7 : Student presentations on chosen topic; report 30/3 - Week 8 : Final exam
9 Tutor Groups Corresponding to each of these six lectures there are two 2-hour Tutor Groups (TG) sessions; one with a pre-discussion on the tasks, and one with post-discussions. 10
Final exam A written exam of two hours on Monday 30 March, 8:30-10:30, consisting of a number of open questions based on the lectures and the reading material. 11 Student research project Students will prepare a research project related to the course.
Students (in groups of 2 to 3) will present a model system and write a report. A 2-3 page proposal is submitted ultimately on Monday 2 March to the tutor of the students TG. The projects are presented (10 minutes each presentation) in the week of 23 March in the TGs. Moreover, each student submits a short paper (2500 words) on the subject. The grade consists of 50% for the report and 50% for the presentation. 12
Final mark of the course The final mark of the course consists of 40% of the project and 60% of the written exam. Only students with sufficient attendance may attend the exam and present his/her project, when only 1 TG lacks for a valid pass, the student receives an additional task, otherwise the students fails the course. 13 Required and other readings
Descriptions of modelling issues related to the task at hand Additional readings and Web pointers are available (not mandatory, but very useful) The Required Readings (mandatory) are made available through Blackboard on a weekly basis Reading the Required Reading (and browsing through the notes of the lectures and TG meetings) is the best preparation for the mid-term and final exams 14
Task Descriptions Task descriptions of are made available through Blackboard on a weekly basis Note that the task descriptions in the syllabus become over-ruled by these new descriptions !!! 15 Required and additional reading,
and web pointers 16 17 18 19 Example exam question
Why should long-term predictions of climate models be interpreted with care? Because the climate system is a chaotic system and therefore highly sensitive to initial conditions. Small errors in measurements of climate variables may give rise to completely different weather patterns in the long run. 20 Wrong answer
Why should long-term predictions of climate models be interpreted with care? Because the weatherman is not very certain of tomorrows weather and often makes mistakes. Also, it can suddenly start to snow and there is global warming and CO2 emission and the failure of countries worldwide to obey the Kyoto protocol. The weather is simply very complex and very difficult to predict. Personally, I think that mankind will never be able to predict the weather perfectly because the earth is changing all the time and there are cosmic
influences (sun spots, meteors). I hope that this answer satisfies you, I could have written more but there is no more room left on the paper 21 General Information Any questions? Ask after the Lecture or your personal tutor 22
Lecture 1: Models, Growth & Decay 23 1.1. Introduction to modeling 24
Introduction to modeling Reality Observations Experiments Your understanding of how the reality works = model = buildingstone for the Theory 25 TYPES OF MODELS Qualitative model
Quantitative model 26 GOOGLE for model 27 GOOGLE for model 28
Example 1: Gravity Natural phenomenon: on earth all things fall down when released ... 29 Example 1: Gravity Natural phenomenon: on earth all things fall down
when released ... Model 1: things fall down with constant velocity Careful experimental verification: model fails Model 2: things fall down with constant accelaration Careful experimental verification: model is correct 30 The Scientific Method Experiment / Observation
Model 31 Isaac Newton (1642-1727) 32 Example: Gravity Observations on planetary orbits
Experiments pendulum, falling objects (Mathematical) Model 33 Newtons Law of Universal Gravity All bodies exert a gravitational force on each other. The force is proportional to the product of
their masses and inversely proportional to the square of their separation. 34 Example 1: Gravity Problem: Each answer generates new questions Answer to Model-1 for Gravity: things fall down with constant accelaration New Question:
Why do things fall down with constant accelaration? 35 Example 1: Gravity Problem: Each answer generates new questions Therefore models increase in complexity ... * Newtons Law of universal gravity: 1678 1911 * Einsteins Law of General Relativity (GR): 1911 - present (gravitational force is but an illusion: spacetime is curved!!!)
* Quantum Gravity: (work in progress ...) : combination of GR and Quantum Mechanics (QM) 36 Introduction to modelling What is a model? A simplified and fully transparant description of a (natural) phenomenon 37
Issues with modelling What to include? What to exclude? How specific? How general?
What is the scope of the model Modelling is abstraction 38 Types of models Descriptions (theories) Mathematical models (formalized theories) Computer models May be an automated description May be a mathematical model (simulation)
Visualisation models Maps, Simulations, 39 A simple model Modelling the walking behaviour of the UCM Dean 40
A mathematical model 1 Reduce the Dean to a point (simplification) Designate the horizontal position of the point by x (formalisation) Time is represented by t (e.g., in seconds) The position of the Dean at time t is given by x(t) (e.g., in meters) 41 A mathematical model 2
We can now ask questions such as: Where is the Dean at time t=10 seconds? x(10) = ? At what time does the Dean arrive at position x=20 meters? x(t) = 20, t = ? 42 Another simple model
Modelling the walking behaviour of the drunken UCM Dean 43 The random walk model Can be described in mathematics Can be readily modelled using a computer simulation (e.g., StarLogo) Is a fundamental model (diffusion)
44 Example of a much more complex (simulation) model 45 Visualisation models Maps form a special type of model As all models they emphasize some features while ignoring others
The emphasized features allow for the discovery of patterns Like mathematical models, maps allow for the discovery of patterns or trends and for prediction 46 Some maps See www.worldmapper.org to see the world as youve never seen it before
47 THE WORLD 48 Mark Newman (http://www-personal.umich.edu/~mejn/) POPULATION 49 Mark Newman (http://www-personal.umich.edu/~mejn/) ENERGY CONSUMPTION (INCLUDING OIL)
50 Mark Newman (http://www-personal.umich.edu/~mejn/) GREENHOUSE GAS EMISSIONS 51 Mark Newman (http://www-personal.umich.edu/~mejn/) TOTAL SPENDING ON HEALTHCARE 52 Mark Newman (http://www-personal.umich.edu/~mejn/)
PEOPLE LIVING WITH HIV/AIDS 53 Mark Newman (http://www-personal.umich.edu/~mejn/) CHILD MORTALITY 54 Mark Newman (http://www-personal.umich.edu/~mejn/) Global distribution of population
55 The year 1 56 The year 1500 57 The year 1900 58
The year 1960 59 The year 2050 60 The year 2300 61 Models and theories Models are intimately related to theories
A good model Is transparent Is based on a good theory Makes predictions that are testable And therefore falsifiable Is general Provides insight 62 An odd model A UCM sorting hat model
Can you recognize the concentration from a students appearance? Model takes (an image of) a face as input and generates a concentration as output 63 1.2 Growth and Decay
64 Growth and Decay Examples of Growth and Decay Unlimited growth Limited growth Modelling growth and decay in nature Exponential decay of foraging paths Growth of knowledge Relation to future tasks
65 Growth and decay Growth and decay: two sides of the same coin Growth At each step: replace each element by n elements Decay At each step: replace n elements by one element 66
Mathematical description Mathematicians like to make short statements Instead of saying: At time t seconds the quantity is n times the quantity at t-1 seconds They say: P(t) = n P(t-1)
67 plot for P(t) = nP(t-1) 6000000 5000000 4000000 n=2 3000000 n=3 n=4
2000000 1000000 20 18 16 14
12 10 8 6 4
2 0 0 P(t) t 68
Logarithms The rapid growth makes it hard to draw Trick: express quantities in terms of their number of zeros LOG(x) is the number of zeros of x LOG(10) = 1 LOG(1000) = 3 LOG(1000000) = 6 A logarithmic plot of P(t) = n P(t-1) makes the curves straight
69 Logarithmic plot for P(t) = nP(t-1) 10000000 1000000 100000 n=2 10000 n=3
1000 n=4 100 10 1 0 1 2 3
4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19 20 Log(P(t)) t
70 From growth to decay We can perform the same trick with decay Instead of saying: At time t seconds the quantity is 1/n times the quantity at t-1 seconds Mathematicians say: P(t) = (1/n) P(t-1)
71 plot for P(t) = (1/n)P(t-1) 1 0.9 0.8 0.7 n=2 0.5
n=3 0.4 n=4 0.3 0.2 0.1 20
18 16 14 12 10
8 6 4 2 0 0
P(t) 0.6 t 72 Logarithmic plot for P(t) = (1/n)P(t-1) 0
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20 1
0.1 0.01 Log(P(t)) n=2 0.001 n=3 0.0001
n=4 0.00001 0.000001 0.0000001 t 73 Unlimited growth
P(t) = nP(t-1) In most cases, there is a limit to the growth Although this is obvious, it is often forgotten, e.g., World population growth
Spreading of disease (AIDS) Internet hype Success 74 75 76 77
78 Bounded growth Apparently, growth is generally bounded An S-shaped curve is characteristic for bounded growth The logistic curve 79
Bounded growth 80 Bounded growth 81 Bounded growth (Verhulst) P(t+1) = n P(t) (1-P(t)) Logistic model a.k.a. the Verhulst model
How do you state this model in a linguistic form? Pn is the fraction of the maximum population size 1 n is a parameter 82 Balancing growth and decay The Verhulst model balances growth: P(t+1) = n P(t) With decay P(t+1) = n (1-P(t))
83 P(t+1) = 1.5 P(t) (1-P(t)) 0.35 0.3 0.25 Pn 0.2
0.15 0.1 0.05 n 29 27 25
23 21 19 17 15
13 11 9 7 5 3
1 0 84 Another example of a bounded growth model: y = size x = time
the generalised logistic curve y A C 1 Te B x M
1 T Parameters (knobs) of the model: A, the lower asymptote C, the upper asymptote M, the time of maximum growth B, the growth rate T, near which asymptote maximum growth occurs
85 Y A C 1 Te B x M
1 T T = 0.25 T=1 T=2 86 Fitting cow weight data
243 968 1 0.15e 0.1955( X 20.1) 0.15
87 END of LECTURE 1 88
