清大數學系 許世壁 Jan. 30, 2011 2011高中教師清華營 (1) Malthusin Model (The Exponential Law) Malthus (1798) proposed a mathematical model which assume the rate of growth is proportional.
Download ReportTranscript 清大數學系 許世壁 Jan. 30, 2011 2011高中教師清華營 (1) Malthusin Model (The Exponential Law) Malthus (1798) proposed a mathematical model which assume the rate of growth is proportional.
清大數學系 許世壁 Jan. 30, 2011 2011高中教師清華營 (1) Malthusin Model (The Exponential Law) Malthus (1798) proposed a mathematical model which assume the rate of growth is proportional to the size of the population. Let x(t ) be the population size, then dx rx, dt x(0) x0 where r is called per capita growth rate or intrinsic growth. Then x(t ) x0e rt 馬爾薩斯在其書 ”An Essay on the Principle of population” 提出馬爾薩斯人口論。其主張為 人口之成長呈幾何級數，糧食之成長呈算術級數。 The rule of 70 is useful rule of thumb. 1% growth rate results in a doubling every 70 years. At 2% doubling occurs every 35 years. (since ln 2 0.7 ) (2) Logistic Equation Pierre-Francois Verhult (1804-1849) in 1838 proposed that the rate of reproduction to proportional to both existing population and the amount of available resources. Let x(t ) be the population of a species at time t , Due to intraspecific competition dx x 2 rx ax rx1 , K dt x ( 0) x 0 r int rinsicgrowt h rat e K carrycapacit y Besides ecology, logistic equation is widely applied in Chemistry: autocatalytical reaction Physics: Fermi distribution Linguistics: language change Economics: Medicine: modeling of growth of tumors K x' ' 0 K 2 x' ' 0 t xk 1 rxk 1 xk f ( xk ), x0 0 given As 0 r 4, f : [0,1] [0,1] Period-doubling cascade: r0 1 r 3, xk convergesto a fixed point r1 3 r 3.449, period2 r2 3.449... r 3.54409, period 4 2 2 r3 3.54409... r 3.5644, period 8 2 3 r4 3.5644... r 3.568759, period 16 2 4 3.569946... r , period 2 Logistic map shows a route to chaos by period-doubling rn rn1 lim 4.669... n r n 1 rn is called the universal number discovered by Feigenbaum. The number is independent of the maps, for example xk 1 r 1 xk , 2 xk 1 r sin xk . If you zoom in on the value r=3.82 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram xk 1 f ( xk ), f : [0,1] [0,1] is chaotic if (i) Period three period k, k (ii) If f has a periodic point of least period not a power of 2, then 0, “Scramble” set S (uncountable) s.t. (a) x y in S lim sup | f n ( x) f n ( y ) | , n lim inf | f n ( x) f n ( y ) | 0 (b) x S , n period point p of f lim | f ( x) f ( p) | n n n 2 Sharkovsky ordering 3 5 7 9 23 25 27 22 3 22 5 22 7 2 n 2 n 1 2 2 2 1 If p q and f has periodic point of period p Then f has a periodic point of period q . f : V V is chaotic on V if f has sensitive dependence on initial conditions. f is topological transitive (i) (ii) (iii) Periodic points are dense in V f is topological transitive if for U ,W V there exists k 0 such that f k (U ) V Fashion Dress, designed and made by Eri Matsui, Keiko Kimoto, and Kazuyuki Aihara (Eri Matsui is a famous fashion designer in Japan) This dress is designed based on the bifurcation diagram of the logistic map This dress is designed based on the following two-dimensional chaotic map: In the mid 1930’s, the Italian biologist Umberto D’Ancona was studying the population variation of various species of fish that interact with each other. The selachisns (sharks) is the predator and the food fish are prey. The data shows periodic fluctuation of the population of prey and predator. The data of food fish for the port of Fiume, Italy, during the years 1914-1923: 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 11.9% 21.4% 22.1% 21.2% 36.4% 27.3% 16.0% 15.9% 14.8% 10.7% He was puzzled and turn the problem to his colleague, Vito Volterra, the famous Italian mathematician. Volterra constructed a mathematical model to explain this phenomenon. Let x(t ) be the population of prey at time t . We assume that in the absence of predation, grows exponentially. The predator consumes prey and the growth rate is proportional to the population of prey, d is the death rate of predator dx * , ax bxy bx y y dt dy * cxy dy cy x x , dt x (0) x0 0, y (0) y0 0 * d a dy cy x x * * x , y , . * c b dx bx y y x x* b y y* By separationof variables dx dy 0 x c y Integrating above yields x b y * * V(x,y) x x x ln * y y y ln * C x c y * * y * y * (x , y ) * x * Periodic orbits in phase plane V(x,y) Energylevel x Independently Chemist Lotka(1920) proposed a mathematical model of autocatalysis k1 k2 k3 A X AX , X Y 2Y , Y B Where A is maintained at a constant concentration a . The first two reactions are autocatalytic. The Law of Mass Action gives dx dt k1ax k 2 xy, dy k xy k y. 2 3 dt , are competition coefficients is small, is large is large, is small , are small Weak competitio n , are large Strongcompetition We assume: x1 , x2 , x3 has same intrinsic growth rate r In the absence of x3, x2win over x1. In the absence of x1, x3 win over x2. In the absence of x2 , x1 win over x3. x3 x2 x1 1.2 x1 x2 x3 1 0.8 0.6 0.4 0.2 0 -0.2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Thank you for your attention.