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Ronald Westra, Eric Postma Department of Mathematics Universiteit Maastricht

Ronald Westra, Eric Postma Department of Mathematics Universiteit Maastricht

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## Ronald Westra, Eric Postma Department of Mathematics Universiteit Maastricht

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**Introduction Knowledge Engineering**Ronald Westra, Eric Postma Department of Mathematics Universiteit Maastricht**Introduction Knowledge Engineering**Lecture 3 Modelling of Dynamical Systems**Introduction Knowledge Engineering**How to survive this course…**Introduction Knowledge Engineering**How to find this lecture …http://www.math.unimaas.nl/personal/ronaldw/Education/IKT/IKT_page.htm**Introduction Knowledge Engineering**3.1 On Growth and Decay**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 • …**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**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)**P(t)**t plot for P(t) = nP(t-1)**Logarithms**• The rapid growth makes it hard to draw • Trick: express quantities in terms of their number of zeros • A logarithmic plot of P(t) = n P(t-1) makes the curves straight…**Log(P(t))**t Logarithmic plot for P(t) = nP(t-1)**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)**P(t)**t plot for P(t) = (1/n)P(t-1)**Log(P(t))**t Logarithmic plot for P(t) = (1/n)P(t-1)**Introduction Knowledge Engineering**EXAMPLE:Growth of Bacterial Populations**Growth of a population of bacteria**• Consider a controlled laboratory environment: • Bacteria are single cell micro organisms which reproduce by cell division. • They live from nutrition provided in the laboratory setting • There is plenty of space to multiply**Modelling population growth**• Each cell divides after a constant amount of time. • Initially cells are of different, unrelated, ages • During a period of time Δ, the amount of cells which split is proportional to the size of the population.**Modelling population growth**• The rate of change is proportional to the population size: • Let N(t) be a function specifying the number of cells at time t, t ≥ 0. • Then it must hold that • N’ = β N. • (where N’ is the first order derivative of N ). Differential equation:**Which function N satisfies the equation?**• Could N be a polynomial? • N(t)= ao + a1 t + a2 t2 + …. + an tn. • N’(t) = a1 + a2 t + a3 t2 + …. + an tn-1. • N(t’) ≠ N(t) • N cannot be a polynomial!**Which function N satisfies the equation?**• Can N be an exponential function? • N(t) = αeγt • N’(t) = γαeγt • N’(t) = γ N(t) as required.**What is the value of α?**• Let no be the initial population size, that is, N(0) = no. • Then no = N(0) = αeγ0 = α.**Definitions**• N’ = β N is called a linear homogenous first order differential equation, because • It is a linear function, • it involves only the first order derivative, • it only considers the function and its derivative.**Conclusion**• The linear first order homogenous difference equation • xn+1 = a xn • has solution xn = an xo. • This problem can be solved without ‘algorithm’, the analytical solution is a formula. • Notice that xn converges, reaches an equilibrium if and only if |a| < 1.**Introduction Knowledge Engineering**3.2 Bounded Growth**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 • …**Bounded growth**• Apparently, growth is generally bounded • An S-shaped curve is characteristic for bounded growth • The logistic curve**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 indicating time**Balancing growth and decay**• The Verhulst model balances growth:P(t+1) = n P(t) • With decayP(t+1) = n (1-P(t))**Problems with the logistic function as a model for**population growth : Verhulst attempted to fit a logistic curve to 3 separate censuses of the population of the United States of America in order to predict future growth. All 3 sets of predictions failed. In 1924, Professor Ray Pearl and Lowell J. Reed used Verhulst's model to predict an upper limit of 2 billion for the world population. This was passed in 1930. A later attempt by Pearl and an associate Sophia Gould in 1936 then estimated an upper limit of 2.6 billion. This was passed in 1955.**Bounded growth**• Apparently, growth is generally bounded • An S-shaped curve is characteristic for bounded growth • The logistic curve (e.g., the Verhulst equation)**Introduction Knowledge Engineering**3.3 Predator-Prey Models**Recall the Logistic Model**• Pn is the fraction of the maximum population size 1 • is a parameter Large P slows down P Logistic model a.k.a. the Verhulst model**Interacting quantities**• The logistic model describes the dynamics (change) of a single quantity interacting with itself • We now move to models describing two (or more) interacting quantities**Fish statistics**• Vito Volterra (1860-1940): a famous Italian mathematician • Father of Humberto D'Ancona, a biologist studying the populations of various species of fish in the Adriatic Sea • The numbers of species sold on the fish markets of three ports: Fiume, Trieste, and Venice.**predator**prey Volterra’s model • Two (simplifying) assumptions • The predator species is totally dependent on the prey species as its only food supply • The prey species has an unlimited food supply and no threat to its growth other than the specific predator**Two Populations P and Q**xtis the prey-populationytis the predator-populationa,b,c,d are parameters**Behaviour of the Volterra’s model**Oscillatory behaviour Limit cycle**Effect of changing the parameters (1)**Behaviour is qualitatively the same. Only the amplitude changes.**Effect of changing the parameters (2)**Behaviour is qualitatively different. A fixed point instead of a limit cycle.**Exponential growth**Limited growth Exponential decay Oscillation Why are PP models useful? • They model the simplest interaction among two systems and describe natural patterns • Repetitive growth-decay patterns, e.g., • World population growth • Diseases • … time**Introduction Knowledge Engineering**3.4 Fibonacci**Fibonacci’s rabbits**• Around the year 1200, the italian mathematician Fibonacci asked himself the following question. • I start with a single newborn rabbit-pair. Mature rabbit pairs create offspring every month. Rabbit pairs are mature from the second month. How many rabbits do I have after t months? (assuming rabbits live forever)**A second order differential equation**• Let Kn be the number of rabbits after n month. Then it must hold that • Kn = Kn-1 + Kn-2. • Because all pairs of rabbits that lived in Kn-1are still alive in Kn, and all pairs that were alive in Kn-2 produced a pair of offspring. Linear homegenous second order difference equation**Solving the 2nd order difference equation**• Kn - Kn-1 - Kn-2 = 0. • Let’s guess a solution once again… • Kn = k λn. • Then it must hold that • k λn+2 -k λn+1 -k λn = 0. • k λn (λ2 -λ- 1)= 0.