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Dynamical Systems Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks 6 1 Chapter 6 Dynamical System[.]
Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen Chapter Dynamical Systems Discrete Mathematics II/Mathematical Modelling Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks Nguyen An Khuong, Huynh Tuong Nguyen Faculty of Computer Science and Engineering University of Technology, VNU-HCM 6.1 Contents Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen Introduction Contents Malthusian Growth Model Introduction Malthusian Growth Model Properties of the systems Properties of the systems System of ODEs Homeworks System of ODEs Homeworks 6.2 Dynamical Systems Change?! Nguyen An Khuong, Huynh Tuong Nguyen • Dynamical systems: Tools for constructing and manipulating models • So we often have to model dynamic systems • Discrete −→ difference equations (“linear" vs “nonlinear", “single variable" vs “multivariate") • Continuous −→ differential equations (“ordinary" vs “partial"; “linear" vs “nonlinear") • We will formulate the equations, analyze their properties and learn how to solve them • To start there are many good references on this subject, including: Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks • F.R Giordano, W.P Fox & S.B Horton, A First Course in Mathematical Modeling, 5th ed., Cengage, 2014 • A Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd Cambridge University Press, 2008 6.3 Single Species Equations: Growth Dynamical Systems Nguyen An Khuong, Huynh Tuong Nguyen • Basic concept that individuals divide to increase a population can be modeled mathematically using a differential equation • Can loosely be applied to populations that don’t divide to populate • Attributed to Malthus, who in 1798 found small group of organisms obeyed growth law Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks • The solution to the equation concerned him greatly 6.4 Dynamical Systems Exponential Growth Nguyen An Khuong, Huynh Tuong Nguyen • As an example, consider the classic example of bacteria on a petri dish • Let’s say the number on the plate grows by 10% each hour and the initial population is x0 = 1000 Contents • After the first hour: x1 = 1000 + (0.1) ∗ 1000 = 1000 ∗ (1.1) Introduction Malthusian Growth Model • After the second hour: x2 = 1000 ∗ (1.1) + (0.1) ∗ (1000 ∗ 1.1) = 1000 ∗ (1.1) ∗ (1.1) = 1000 ∗ (1.1)2 Properties of the systems System of ODEs • After the third hour: x3 = 1000 ∗ (1.1)3 etc t Homeworks t • In general, xt = xo (1 + r) which can be written as xt = x0 a where a = (1 + r) The solution for the Bacteria is thus an exponential function of base + r • The (discrete) dynamical system: xt = axt−1 (recurrent relation/difference equation) 6.5 Dynamical Systems Instantaneous Exponential Growth Nguyen An Khuong, Huynh Tuong Nguyen • In the last slide we examined growth at some kind of finite increment using an average growth rate r over that increment • What happens if the growth process is continuous? • Divide the growth process over the time increment into nt stages with the growth rate for each stage being the average growth over the increment r n where r is Contents Introduction r nt ) (1) n • Look at the limit as we divide our interval into ∞ pieces x(t) = xo (1 + Malthusian Growth Model Properties of the systems System of ODEs Homeworks r lim x0 [(1 + )n/r ]rt n→∞ n • We can pull x0 out of the limit In brackets we have the irrational number e: x(t) = x0 ert (2) (3) 6.6 Dynamical Systems Instantaneous Exponential Growth (cont’d) Nguyen An Khuong, Huynh Tuong Nguyen • Our time-dependent population equation satisfies the fundamental differential equation: dx = x0 = rx (4) dt where r is the (in this situation) constant growth rate (See how to derive the equation above in page 462, [Giordano et al.], eqn(11.5).) • The solution to these equation can be found by separating variables and integrating x˙ = Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs ln|x(t)| = rt + c (5) Homeworks and applying the initial condition that x(0) = xo at t = to get ln |x(t)| = rt + ln |xo | (6) • Take the exponential of both sides to give the exact solution to the instantaneous growth equation x(t) = xo ert (7) 6.7 Dynamical Systems Exponential Growth: Solution properties Nguyen An Khuong, Huynh Tuong Nguyen • We can calculate the time to double the population (known as the "rule of 70" in financial circles) 70 ln(2) ∼ r r • What does the solution look like? On a log plot, it is a straight line of slope r tdouble = (8) Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks • Is this realistic? 6.8 Dynamical Systems What if our growth rate was negative?: Mortality Nguyen An Khuong, Huynh Tuong Nguyen x˙ = −mx (9) • We have exponential decay Solution will tend toward zero, no matter what the initial condition was Contents Introduction Malthusian Growth Model Properties of the systems System of ODEs Homeworks • Most systems have both growth and decay terms • Example, our phytoplankton equation will have terms for growth as a function of light, temperature, and nutrients, and decay (mortality) 6.9 Dynamical Systems More realistic model: Finite Resources Nguyen An Khuong, Huynh Tuong Nguyen • We now know that growth cannot continue forever because of finite resources and in fact in simplified scenarios will reach a given constant level K know as the carrying capacity of the system • Verhulst noticed that simple populations appear to be capped x˙ = rx(1 − x) Introduction Malthusian Growth Model and added an additional term to remove excess capacity x x˙ = rx(1 − ) K • We can always nondimensionalize this by the carrying capacity to give Contents Properties of the systems (10) System of ODEs Homeworks (11) • This is commonly known as logistic growth 6.10