Modeling C-3 (1).pdf
- 1. Chapter 3 Modeling Discrete Dynamical Systems with Difference Equations 1 4/11/2022 Math 3111/484
- 2. Modeling Discrete Dynamical Systems with Difference Equations Outlines Introduction 3.1 Discrete Dynamic Systems and Difference Equations 3.2 Modeling with Linear First order Difference Equations 3.3 Modeling with Systems of Difference Equations 2 4/11/2022 Math 3111/484
- 3. One of the main concerns of modeling is to predict the future development of a system. A system that changes over time is called dynamical system. A powerful paradigm to model change is future value = present value + change change = future value – present value Introduction 3 4/11/2022 Math 3111/484
- 4. If a variable of our interest changes in discrete time intervals, the above formula leads to a difference equation and a dynamical system we work with difference equations is called discrete dynamical system. 4 4/11/2022 Math 3111/484
- 5. • The mathematical assumption is that the time variable n is incremented discretely and corresponds to the integers {0, 1, 2, 3, 4, . . . }. The value of a variable x of interest is then a sequence {x0, x1, x2, x3, x4, . . . }. • Discrete models can be used in population growth, interests in accounts, drug dosages, genetics and others. 3.1 Discrete Dynamic Systems and Difference Equations 5 4/11/2022 Math 3111/484
- 6. For discrete models, the difference equation (discrete dynamical system) has the form, Future value = Function of {Present value, Previous values and possibly time}. xn+1 = f (xn , xn-1, …, n), where xn+1 = Future value, xn = Present value, xn-1 = Previous value, n = time. 6 4/11/2022 Math 3111/484
- 7. Definitions Definition 1: A difference equation is of first order if the value xn+1 depends only on xn, n and constants. It is of second order if xn+1depends on xn, xn-1, n and constants. And so on. Example 1: xn+1= 3xn-n2+7n+2 is first order difference equation. Example 2: nxn+1- (n3-0.5n+1)xn -6xn-1+8n=10 is second order difference equation. Definition 2: A difference equation is said to be autonomous if its calculation does not depend explicitly on n. Example 3: xn+1=7xn-12 is autonomous difference equation. 7 4/11/2022 Math 3111/484
- 8. …Continued Definition 3: If a difference equation involves no products of sequence variables, no powers of sequence variables, nor functions of sequence variables such as exponential, logarithmic or trigonometric functions, then we call the difference equation linear. For otherwise, the difference equation is nonlinear. Example 4: xn+1= 3xn+n2 is linear difference equation. Example 5: xn+1= (xn)2+xnxn-1 + 3n+5 is nonlinear difference equation. 8 4/11/2022 Math 3111/484
- 9. …Continued Definition 4: If each term of a difference equation contains sequence of variables, then it is called homogeneous difference equation. For otherwise, it is called non-homogeneous difference equation. Example 6: (xn+1)2 +3xn-10xn-1 = 0 is homogeneous and xn+1 +5xn- xn-2=n2+6n-10 is non-homogeneous. Definition 5: A solution of a difference equation is a sequence xn given by the formula in terms of n, xn = f(n), n=0, 1, 2, … The solution can be given analytically, graphically or numerically. 9 4/11/2022 Math 3111/484
- 10. 3.2 Modeling with Linear First Order Difference Equations The simplest possible difference (Linear, Autonomous) equation is of the form xn+1= rxn, n = 0,1,2,3,… (*) with some initial condition x0, where r is a constant. The solution can be found by implementing the iteration x1=rx0, x2=rx1=r(rx0)=r2x0, x3=rx2=r(r2x0)=r3x0, . . . xn=rnx0. Therefore, the solution of the Difference equation given in (*) is xn=rnx0. 3.2.1 Difference Equation of the form xn+1= rxn 10 4/11/2022 Math 3111/484
- 11. Discussions on Long Term Behavior! Long- term behavior (i.e. n→ ∞) Suppose x0 > 0, then 0 0 0 0 0 0 0 0 0 If 1, . If 1, . If 0< 1, 0 . If -1< 0, 0 . even If 1, ( 1) , odd and thus no convergence as . If 1, , = as n and thus n n n n n n n n n n n n n n r x r x as n r x x as n r x r x as n r x r x as n x n r x x x n n r x r x x r x o convergence as . n 11 4/11/2022 Math 3111/484
- 12. MatLab Plot For x0 > 0 12 4/11/2022 Math 3111/484
- 13. Definitions Definition 1: A number x* is called an equilibrium point or fixed point or steady state of a discrete dynamic system, xn+1 = f(xn) if f(x*)= x*. For xn+1=rxn, an equilibrium point satisfy the equation x*=rx*. Thus, if r ≠ 1, then x* = 0 the only equilibrium point. if r = 1, every number x* is an equilibrium point. 13 4/11/2022 Math 3111/484
- 14. Example 1 1. For each of the following models, a) xn+1=1.2 xn, x0=1000 b) xn+1=-0.5 xn, x0=4000 c) xn+1= xn, x0=20 (i) find x1, x2, x3, x4, x5 (ii) find the solution (iii) determine the equilibrium point (iv) plot xn versus n. 14 4/11/2022 Math 3111/484
- 15. Solutions: a) Xn+1=1.2xn, x0 = 1000 (i) x1=1.2x0 = 1.2(1000) = 1200 x2 =1.2x1=1.2(1200) = 1440 x3 = 1.2x2 =1.2(1440) = 1728 x4 =1.2x3=1.2(1728) = 2073.6 x5 = 1.2x4 =1.2(2073.6) = 2488.32 (ii) xn=rnx0 = (1.2)n(1000) Using this, x5=(1.2)5(1000) =2488.32 (iii) Since r ≠1, the only equilibrium point is x* = 0. 15 4/11/2022 Math 3111/484
- 16. 0 1 2 3 4 5 6 7 8 9 10 0 1000 2000 3000 4000 5000 6000 7000 n xn 16 4/11/2022 Math 3111/484
- 17. Example 2 Suppose a certain population of owls is growing at the rate of 2% per year. If initially we have a population of 100 owls, (i) Develop a model to predict the owls population. (ii) What will the population be after 10 years? (iii) Plot the population versus years. (iv) After what year will the population be doubled? 17 4/11/2022 Math 3111/484
- 18. Solutions: (i) Let pn denote the population of owls after n years. Then, pn+1= pn+0.02pn = 1.02pn Hence the model is pn+1 = 1.02pn , p0 = 100. The solution is pn =(1.02)n 100. (ii) The population after 10 years is p10 = =(1.02)10 100 = 122 18 4/11/2022 Math 3111/484
- 19. (iii) 0 10 20 30 40 50 60 70 80 90 0 100 200 300 400 500 600 700 n xn 19 4/11/2022 Math 3111/484
- 20. …continued Therefore the doubling time of the population is about 35 years. 2 1.02 ( ) 200 (1.02) 100 200 (1.02) 2 100 log 2 log 35 years log1.02 n n iv n 20 4/11/2022 Math 3111/484
- 21. Exercises: 1) If 5000 birr is invested at the rate of 8% compounded annually, (i) Develop a model to describe the sum of the money after n years. (ii) What is the sum of the money at the end of 10 years? (iii) How long does it take the sum of the money to double itself? 21 4/11/2022 Math 3111/484
- 22. …Continued 2) Radium is a radioactive element which decays at a rate of 1% every 25 years. If the initial amount of radium is 500 grams, then (i) develop a model to describe the amount of radium left. (ii) find the amount left after 100 years. (iii) plot of the amount of radium left versus years. (iv) what is the half-life time of the radium? 22 4/11/2022 Math 3111/484
- 23. 3) Suppose that a bacterial colony starts with 100 bacteria and the bacteria divide every 20 minutes. a) How will the population size change over time? b) Plot the bacteria population versus time. c) What will be the bacteria population after 120 minutes? 23 4/11/2022 Math 3111/484
- 24. 4) CIPRO is a drug for combating many infections including anthrax. Let us assume that during one- hour period our kidneys purify ¼ of this drug. If a patient takes 16 mg of this drug, then a) write a model to predict the amount of the drug in the patient’s blood. b) how long will the drug it take to be 6.75 mg in the patient’s blood? c) plot the amount of drug in the blood versus time. 24 4/11/2022 Math 3111/484
- 25. Consider the dynamic system xn+1= rxn +b, with initial condition x0, where r and b are constants. 3.2.2 Difference Equation of the form xn+1= rxn + b 25 4/11/2022 Math 3111/484
- 26. The solution of this system is obtained as follows: 1 0 2 2 1 0 0 2 3 2 3 2 0 0 2 1 0 0 0 ( ) (1 ) ( (1 )) (1 ) (1 . . . ) If 1, ( 1) 1 If 1, n n n n n n n x rx b x rx b r rx b b r x b r x rx b r r x b r b r x b r r x r x b r r r r r x r x b r r x x nb 26 4/11/2022 Math 3111/484
- 27. Equilibrium Points of the System: 1 Given a dynamical sysem . If 1, * is the equilibriumpoint. 1 If 1 and 0, everynumberis the equilibriumpoint. If 1 and 0, no the equilibrium point exists. n n x rx b b r x r r b r b Classifying Equilibrium Points (Stability of the equilibrium point): If all solutions of 𝒙𝒏+𝟏=r𝒙𝒏 +b approach to the equilibrium point 𝒙∗ = 𝒃 𝟏−𝒓 as 𝒏 ⟶ ∞, then 𝒙∗ is called stable equilibrium point. For otherwise, it is called unstable. 27 4/11/2022 Math 3111/484
- 28. Thus, depending on the value of r, we obtain the following long-term behavior for the given system: Value of r Long–term behavior observed 𝑟 < 1 Stable equilibrium point 𝑟 > 1 Unstable equilibrium point 𝑟 = 1 No equilibrium point 28 4/11/2022 Math 3111/484
- 29. Example 1 Consider the following dynamical systems: a) xn+1 = 2xn-1, x0 =3 b) xn+1 =-0.5xn + 6, x0 =2 (i) Find the solutions. (ii) Find the equilibrium points and check their stability. (iii) Plot the solution to observe long-term behavior and stability, determine the limit of solutions. 29 4/11/2022 Math 3111/484
- 30. Solutions: 0 1 .( ) The solution of thedifference equation is ( 1) 1 1(2 1) 2 (3) 3(2 ) 2 1 2 1 2 1 ( ) The equilibriumpoint 1 * 1 1 1 2 Since 2 2 1, * 1 is unstable equilibriumpoint. n n n n n n n n n n a i b r x r x r x x ii b x r r x 30 4/11/2022 Math 3111/484
- 31. (iii) MatLab Plot of the System 0 1 2 3 4 5 6 7 0 20 40 60 80 100 120 140 n xn 31 4/11/2022 Math 3111/484
- 32. 0 .( ) The solution of thedifference equation is ( 1) 1 6(( 0.5) 1) ( 0.5) (2) 0.5 1 2( 0.5) 4 ( ) The equilibriumpoint 6 * 4 1 1 ( 0.5) Since 0.5 0.5 1, * 4 is stable equilibriumpoint. n n n n n n n n b i b r x r x r x x ii b x r r x 32 4/11/2022 Math 3111/484
- 33. (iii) MatLab Plot of the System 0 2 4 6 8 10 12 14 16 18 20 2 2.5 3 3.5 4 4.5 5 33 4/11/2022 Math 3111/484
- 34. Exercises 1) You currently have $5,000 in a saving account that pays 0.5% interest each month. If you add $400 each month, then a) develop a model to calculate the amount in the account. b) how much is in the account after 3 years? c) determine when the amount in the account reaches $10,000. 34 4/11/2022 Math 3111/484
- 35. … Continued 2) Every day a person consumes 5 micrograms of toxin which leaves the body at a rate of 2% per day. a) Develop a model to describe the accumulation of toxin in the body. b) How much toxin is accumulated in 30 days? c) How much toxin is accumulated in the body in the long run? d) Plot toxin accumulation versus time. 35 4/11/2022 Math 3111/484
- 36. Assignment I Question 1 1) Suppose that there are currently 25,000 unemployed workers in Bahir Dar city. Each month 8% of all those unemployed find jobs but another 1500 become unemployed. a) How many will be unemployed 6 months from now? b) At what level will the number of unemployed workers stabilize over time? 4/11/2022 Math 3111/484 36
- 37. Assignment I Question 2 2) A certain drug is effective in treating a disease if the concentration remains above 100 𝑚𝑔 𝑙 . The initial concentration is 640 𝑚𝑔 𝑙. It is known from laboratory experiments that the drug decays at the rate of 20% of the amount present each hour. a) Formulate a discrete model that describes the concentration after each hour. b) At what hour does the concentration reach 100 𝑚𝑔 𝑙? c) Determine the maintenance doses that will keep the concentration above the minimum effective level of 100 𝑚𝑔 𝑙 and below the maximum safe level of 800 𝑚𝑔 𝑙. 4/11/2022 Math 3111/484 37
- 38. 3.3 Modeling with Systems of Difference Equations Motivational problems (Systems) Population interaction (Prey-Predator interaction) Commodity distribution Disease spread Consider system of first order linear homogeneous difference equation xn+1 = a11xn+a12yn +a13zn yn+1 = a21xn+a22yn +a23zn zn+1 = a31xn+a32yn +a33zn with initial conditions x0, y0, and z0. 3.3.1 Systems of Linear homogeneous Difference Equations 38 4/11/2022 Math 3111/484
- 39. In matrix form, we can write the above system as 39 1 11 12 13 1 21 22 23 1 31 32 33 0 0 0 with initial condition n n n n n n x a a a x y a a a y z a a a z x y z 4/11/2022 Math 3111/484
- 40. This can be written more concisely as 1 1 1 1 1 11 12 13 21 22 23 31 32 33 0 0 0 0 , , with initial condition n n n n n n n n n n X RX x where X y z a a a R a a a a a a x X y z x X y z 0 The solution for the system is n n X R X 40 4/11/2022 Math 3111/484
- 41. Remarks: 1) Although the above solution is correct, it presents a daunting practical problem. It is difficult (if not impossible ) to calculate Xn for large values of n. (As such calculation involves excessive computation of matrix multiplications to evaluate Rn.) 2) In the next section we use a systematic way of obtaining the general solution, using the eigenvalues and eigenvectors of the matrix R. 41 4/11/2022 Math 3111/484
- 42. Let λ1, λ2 , . . . ,λn be n distinct eigenvalues of R, and let v1, v2,. . ., vn be the corresponding eigenvectors. An eigenvalue λ of multiplicity k has k linearly independent eigenvectors, then the contribution to the general solution will be of the form: 42 Then the general solution is: 𝑿𝒏=𝑐1𝜆1 𝑛 𝒗𝟏+𝑐2𝜆2 𝑛 𝒗𝟐 + ⋯ + 𝑐𝑛𝜆𝑛 𝑛 𝒗𝒏 where 𝒄𝟏, 𝒄𝟐, … . 𝒄𝒏 are real constants such that 𝑿𝟎 = 𝑐1𝒗𝟏+𝑐2𝒗𝟐 + ⋯ + 𝑐𝑛𝒗𝒏. 𝒄𝟏𝝀𝒏𝒗𝟏+𝒄𝟐𝒏𝝀𝒏𝒗𝟐 + ⋯ + 𝒄𝒌𝒏𝒌−𝟏𝝀𝒏𝒗𝒌. 4/11/2022 Math 3111/484
- 43. Equilibria and stability analysis of System of Linear Homogeneous Difference Equations The equilibrium vector of the system is X* such that X*=RX* ⇒ (I-R)X*= 0, where 0 is the zero vector. NB: 1) The zero vector X* = 0 is always an equilibrium vector. 2) If I−R ≠0, then the system has no nonzero vector as equilibrium vector.(i.e. X* = 0 is the only equilibrium vector.) 3) If I−R = 0, then the system has nonzero vector as equilibrium vector. 43 4/11/2022 Math 3111/484
- 44. Definition 3.3.1 An equilibrium vector X* is stable if the general solution Xn tends to X* regardless of the initial conditions. Otherwise, it is called unstable. Criteria of stability in terms of nature of eigenvalues: Definition 3.3.2 If λ1, λ2 , . . . ,λn are eigenvalues of the matrix R, then 𝜌=max { 𝜆1 , 𝜆2 , . . . , 𝜆𝑛 } is called the spectral radius of R. If 𝜌 >1, then the solution will grow without bounds. Hence, the equilibrium vector is unstable. If 𝜌 <1, then the solution tends to the zero vector. Thus, the equilibrium vector is stable. If 𝜌 =1, then the solution converges to a multiple eigenvector associated to λ where 𝜌 = 𝜆. Hence, an equilibrium vector is unstable. 44 4/11/2022 Math 3111/484
- 45. Example 1 A battle is to start between 10,000 troops of army x and 5,000 troops of army y. Given that the destroying rate of x is 0.1 and that of y is 0.15. Develop a model to predict the outcome of the battle. 45 4/11/2022 Math 3111/484
- 46. Solution: • Let 𝑥𝑛 and 𝑦𝑛 be the number of troops of army x and y after n time interval respectively. Thus, 46 1 1 1 1 0 0 0.15 0.1 0.1 1 0.15 0.1 1 10,000, 5,000 n n n n n n n n n n n n n x x y y y x x y x x y y x y 1 1 1 0 1 0.15 , , 0.1 1 10000 , 5000 n n n n n n x Let X R y x X X y 4/11/2022 Math 3111/484
- 47. 1 0 Then, the model can be written as , with intial condition n n X RX X Let us use MATLAB to compute eigenvalues and eigenvectors of R. 47 4/11/2022 Math 3111/484
- 48. 1 2 1 2 Eigenvalues of R are 0.8775, 1.1225 The correspondingeigenvectorsare 1.2247 1.2247 , 1 1 v v 1 1 1 2 2 2 1 1 2 2 0 Thesolutionis where n n n X c v c v c v c v X 48 4/11/2022 Math 3111/484
- 49. 1 1 2 2 0 1 2 1 2 1 1 0 2 implies 1.2247 -1.2247 10000 c c 1 1 50000 1.2247 1.2247 10000 1 1 50000 1.2247 1.2247 , then 1 1 6582.6 1582.6 c v c v X c c If M c M X c 49 4/11/2022 Math 3111/484
- 50. Hence , the solution is 1 1 1 2 2 2 1.2247 1.2247 6582.6(0.8775) ( 1582.6)(1.1225) 1 1 6582.6(0.8775) (1.2247) ( 1582.6)(1.1225) ( 1.2247) 6582.6(0.8775) (1) ( 1582.6)(1.1225) (1) 8061.71(0. n n n n n n n n n n n n n n X c v c v x y x y x 8775) 1938.21(1.1225) 6582.6(0.8775) 1582.6(1.1225) n n n n n y Therefore, we observe that x army will win the battle. Moreover, 𝐼 − 𝑅 ≠0, then 𝑥∗ = 0 0 , the only equilibrium vector for the given system, which is unstable. (WHY?) 50 4/11/2022 Math 3111/484
- 51. MatLab plots of the Armies: 51 0 2 4 6 8 10 12 14 16 18 20 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x 10 4 n x y 4/11/2022 Math 3111/484
- 52. Example 2 Suppose in a small town, on any given day 50% ill people become healthy and 10% healthy people become ill. If we start with 5000 healthy and 500 ill people, a) What will be the situation after 5 days? b) What will happen in the long run? Solution: 52 Let Hn = Number of healthy people after n days In = Number of ill people after n days 4/11/2022 Math 3111/484
- 53. …continued 53 1 0 1 0 1 0 1 0 0.9 0.5 , 5000 0.1 0.5 , 500 0.9 0.5 5000 , 0.1 0.5 500 n n n n n n n n n n H H I H I H I I H H H I I I Therefore, the system will be modeled as : How do we obtain this model? Justify ! Now, let us find the general solution of the system using eigenvalue and eigenvector method: 4/11/2022 Math 3111/484
- 55. 1 1 1 2 2 2 5 5 1 5 416.6667(0.4) 916.6667(1) 1 1 416.6667(0.4) 4583.3 416.6667(0.4) 916.6667 4588 912 45883 917 n n n n n n n n n n n X c v c v H I H I H I H I a. b. 55 4/11/2022 Math 3111/484
- 56. 0 5 10 15 20 25 30 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 time(day) population healthy ill 56 4/11/2022 Math 3111/484
- 57. Assignment I Question 3 1) Let 𝑈𝑛 and 𝑉 𝑛 be the total amount of pollutant in lakes 𝐴 and 𝐵 respectively, in year 𝑛, and 38% of the pollutant from lake 𝐴 and 13% of the pollutant from lake 𝐵 are removed every year. Also, the pollutant that is removed from lake 𝐴 is added to lake 𝐵 due to the flow of water from lake 𝐴 to lake 𝐵. It is also assumed that 3 𝑡𝑜𝑛𝑠 of pollutant are directly added to lake 𝐴 and 9 𝑡𝑜𝑛𝑠 of pollutant are added to lake 𝐵. a) Develop a discrete dynamical system model to describe this system. Find the equilibrium points and state whether they are stable or not. 4/11/2022 Math 3111/484 57
- 58. … Assignment I Question 3 Contd. b) Suppose it is determined that an equilibrium level of a total of 10 𝑡𝑜𝑛𝑠 of pollutant in lake 𝐴 and a total of 30 𝑡𝑜𝑛𝑠 in lake 𝐵 would be acceptable. What restrictions should be placed upon the total amounts of pollutants that are added directly, so that these equilibria can be achieved? c) Plot 𝑈𝑛vs 𝑛 and 𝑉 𝑛 vs 𝑛 using MatLab.(MatLab code is part of the solution) 58 4/11/2022 Math 3111/484
- 59. Consider a linear systems of non-homogeneous difference equations Xn+1 = RXn + B, with initial condition X0 where B is a vector whose components are constants. Therefore, the solution is of the form: ( ) ( ) ( ) n ( ) 1 where is the solution of the associated homogeneous system: and is the particular solution. h p n n n h n p n n n X X X X X R X X 59 3.3.2 Systems of Linear non-homogeneous Difference Equations (Reading Assignment!) 4/11/2022 Math 3111/484
- 60. 4/11/2022 Math 3111/484 60 End of Chapter 3! Thank You!