1 / 21

ENGG2013 Unit 16 Tipping point

ENGG2013 Unit 16 Tipping point. Mar, 2011. Content . How to cut a pizza Population growth model Rabbit Fish Fish with harvesting. HOW TO CUT A PIZZA?. 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.

matha
Download Presentation

ENGG2013 Unit 16 Tipping point

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related