ENGG2013 Unit 16 Tipping point Mar, 2011.

Content How to cut a pizza Population growth model – Rabbit – Fish – Fish with harvesting kshumENGG20132

HOW TO CUT A PIZZA? kshumENGG20133

Pizza Cutting What is the maximal number of pieces you can get by cutting a pizza n times? kshumENGG20134 One cut: 2 pieces two cuts: 4 pieces

How many pieces for 3 cuts? kshumENGG20135 Three cuts: 7 pieces

How many pieces for 4 cuts? kshumENGG20136 Four cuts: 11 pieces

How many pieces for 5 cuts? kshumENGG20137 Five cuts: 16 pieces

Is there any pattern? Let P(n) be the maximal number of pieces you can get by n cuts. P(1) = 2, P(2) = 4, P(3) = 7, P(4) = 11, P(5) = 16 kshumENGG20138 A recursive formula: P(n) = P(n–1)+n

COUNTING RABBITS kshumENGG20139

A Population Growth Model for Rabbits Model – One pair of baby rabbits, one male and one female at the beginning. – Baby rabbit becomes adult rabbit after one month. – Each pair of adult rabbits produces one male and one female baby rabbit in each month. Q: How many pairs of rabbits after n months? kshumENGG

A Recursion Formula Let F(n) be the number of pairs of rabbits in the n-th period of time F(1) = 1, F(2) = 1, F(3) = 2, F(3) = 3, F(4) = 5 kshumENGG F(n) = F(n-1) + F(n-2) This sequence of integers are called the Fibonacci numbersFibonacci numbers

A Puzzle kshumENGG x8=64 5x13 = 65

LOGISTIC POPULATION GROWTH MODEL kshumENGG201313

Logistic population growth – n(t) = population of fish in a lake in the t-th year – t = 0,1,2,3,4 kshumENGG Increase in population Proportionality constant An example for K=1 Graph of n(1-n) Slow growth fast growth Slow growth negative growth

Example Take a=0.5, K = 1 for example kshumENGG Matlab program: T = 20; n = zeros(1,T); % Initialize n to the zero vector n(1) = 0.7; % Initial population for t = 2:T n(t) = n(t-1)+0.5*n(t-1)*(1-n(t-1)); end plot(1:T,n,'ro-') xlabel('time') ylabel('Population')

Fish Population with harvesting Population growth with harvesting kshumENGG Matlab program: T = 20; n = zeros(1,T); % Initialize n to the zero vector n(1) = 0.7; % Initial population for t = 2:T n(t) = max(n(t-1)+0.5*n(t-1)*(1-n(t-1)) - 0.1,0); end plot(1:T,n,'ro-') xlabel('time') ylabel('Population')

Tipping point ( 臨界點 ) If h < 0.125, stable population If h > 0.125, fish will disappear eventually kshumENGG h=0.12 h=0.13 Below the tipping pointAbove the tipping point

Tipping point Compare the different effects on the fish population – if we harvest h=0.124 per year – If we harvest h=0.126 per year Plots of the fish population for 150 years: kshumENGG h=0.124 h=0.126 Slightly below the tipping pointSlightly above the tipping point

Tipping point kshumENGG h=0.1249h= Marginally below the tipping pointMarginally above the tipping point Compare the different effects on the fish population – if we harvest h= per year – If we harvest h= per year Plots of the fish population for 500 years:

Discussion A system may depend on some external parameters which we may choose. – The amount of harvest per year in the previous example. There may be a sudden change in system behavior by modifying a parameter slightly. Cannot cross the tipping point if we want a sustainable environment. kshumENGG201320

Discrete-time dynamical system How to solve? – Given some initial condition, find how the system evolve. How to solve linear difference equation in general? – Pizza cut, and Fibonacci numbers satisfies some linear difference equations – How to compute F(1000)? The logistic growth model (with or without harvesting) is not linear. kshumENGG201321