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.