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ENGG2013 Unit 16 Tipping point

Discover fascinating insights on pizza cutting, rabbit population growth, and fish populations under harvesting conditions. Learn about tipping points and their effects on sustainability.

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ENGG2013 Unit 16 Tipping point

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  1. ENGG2013 Unit 16Tipping point Mar, 2011.

  2. Content • How to cut a pizza • Population growth model • Rabbit • Fish • Fish with harvesting ENGG2013

  3. HOW TO CUT A PIZZA? ENGG2013

  4. Pizza Cutting • What is the maximal number of pieces you can get by cutting a pizza n times? One cut: 2 pieces two cuts: 4 pieces ENGG2013

  5. How many pieces for 3 cuts? Three cuts: 7 pieces ENGG2013

  6. How many pieces for 4 cuts? Four cuts: 11 pieces ENGG2013

  7. How many pieces for 5 cuts? Five cuts: 16 pieces ENGG2013

  8. 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 A recursive formula: P(n) = P(n–1)+n ENGG2013

  9. COUNTING RABBITS ENGG2013

  10. 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? http://www.bbc.co.uk/radio4/science/5numbers3.shtml ENGG2013

  11. 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 F(n) = F(n-1) + F(n-2) This sequence of integers are called the Fibonacci numbers ENGG2013

  12. A Puzzle 8x8=64 5x13 = 65 ENGG2013

  13. LOGISTIC POPULATION GROWTH MODEL ENGG2013

  14. Logistic population growth • n(t) = population of fish in a lake in the t-th year • t = 0,1,2,3,4 An example for K=1 Graph of n(1-n) Increase in population fast growth Slow growth Proportionality constant Slow growth negative growth ENGG2013

  15. Example • Take a=0.5, K = 1 for example 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') ENGG2013

  16. Fish Population with harvesting • Population growth with harvesting 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') ENGG2013

  17. Tipping point (臨界點) • If h < 0.125, stable population • If h > 0.125, fish will disappear eventually Below the tipping point Above the tipping point h=0.12 h=0.13 ENGG2013

  18. 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: Slightly below the tipping point Slightly above the tipping point h=0.124 h=0.126 ENGG2013

  19. Tipping point • Compare the different effects on the fish population • if we harvest h=0.1249 per year • If we harvest h=0.1251 per year • Plots of the fish population for 500 years: Marginally below the tipping point Marginally above the tipping point h=0.1249 h=0.1251 ENGG2013

  20. 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. ENGG2013

  21. 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. ENGG2013

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