The book of nature is written in the language of mathematics Galileo Galilei Our program In this lecture we will apply basic mathematics and statistics to solve ecological problems. 1. Introduction 2. Basic operations and functions 3. Matrix algebra I 4. Matrix algebra II 5. Handling a changing world The lecture is therefore application centred. 6. The sum of infinities 7. Probabilities and distributions Students have to prepare the theoretical background by their own!!! For each lecture I’ll give the concepts and key phrases to get acquainted with together with the appropriate literature!!! This literature will be part of the final exam!!! 8. First steps in statistics 9. Moments and descriptive statistics 10. Important statistical distributions 11. Parametric hypothesis testing 12. Correlation and linear regression 13. Analysis of variance 14. Non-parametric testing 15. Cluster analysis Older scripts Modelling Biology Modelling Biology Basic Applications of Mathematics and Statistics in the Biological Sciences Basic Applications of Mathematics and Statistics in the Biological Sciences Part I: Mathematics Script A Introductory Course for Students of Modelling Biology Basic Applications of Mathematics and Statistics in the Biological Sciences Biology, Biotechnology and Environmental Protection Part I: Mathematics Werner Ulrich Script B Part II: Data Analysis and Statistics Script A Introductory Course for Students of Biology, Biotechnology and Environmental Protection Werner Ulrich Introductory Course for Students of Biology, Biotechnology and Environmental Protection Werner Ulrich UMK Toruń 2007 UMK Toruń 2007 www.uni.torun.pl/~ulrichw UMK Torun 2007 Mathe online http://www.mathe-online.at/ http://tutorial.math.lamar.edu/ Additional sources John Napier (1550-1617) Logarithms and logarithmic functions A logarithm is that number with which we have to take another number (the base) to the power to get a third number. The logarithmic function y a x x loga y 2 1.5 1 0.5 Root 0 -0.5 -1 0 1 2 3 4 5 Log 1 = 0 -1.5 -2 -2.5 Asymptote The logarithmic function is not defined for negative values 6 a loga y y 1 log a y a y log(1) 0 and loga (a) 1 Logarithms and logarithmic functions xy z y a ln(bx c) d a log x a log y a log z a loga x loga y a loga z log x log y log z 1.5 y 1 x/ y z a a log a x /a 0.5 0 log a y log a x log a y a a log a x a log a y log a z a log a z -0.5 0 Curvature 2 3 x y 2 ln(3x 5) 4 Shift at y-axis Increase Shift at x-axis xy z a 1 -1 log x log y log z log a x y A general logarithmic function 2 Root a y loga x a loga z y log x log z 4 2 ln(3x 5) e 2 3x 5 e2 5 x 0.7964 3 4 ax z by loga z x loga b y y loga b logb z y logb a x x logb a loga z y loga b logb z loga b x y loga b x x loga b logb a y x logb a 1 loga b logb a loga b What is the logarithm of base 2 of 59049 if the the logarithm of 59049 of base 3 is 10? x 10 log2 3 x 10*1.585 15.85 215.85 59049 1 logb a loga ( x / y) loga ( x) loga ( y) loga ( y / x) loga ( y) c loga ( y) c loga ( x) y x c loga ( x) log2 3 1.585 log3 2 0.631 log2 3 * log3 2 1.585* 0.631 1 log 2 4 3 0.415 log 2 3 4 Leonhard Paul Euler (1707-1783) The number e 40 35 30 25 20 15 10 5 0 The famous Euler equation y ei 1 0 2 3 i x x x x ex 1 ... 1! 2! 3! i 0 i! -6 -4 -2 y=ex 0 2 4 x 1 1 1 1 e 1 ... 1! 2! 3! i 0 i! e 3 2.5 1 e limn 1 n n e = 2.71828183…. y 2 1.5 1 e limn 1 n 1 0.5 n 0 0 1 2 x 3 4 3 Logarithmic equations ln(x a) b 2 x ln(x 1) ln(x a) b 2 x ln(x 1) e ln( x a ) e e2 x x 1 b xa e x e a b b ln(ax) ln(b) c 2 1.5 1 x -0.8 x=0 0.5 0 -2.5 -2 -1.5 -1 -0.5 -0.5 0 -1 Mixed equations often do not have analytical solutions. ax elnb c x elnbc / a 2.5 -1.5 Roots y ln(x 1) 1 y ln(xe x 1 ) y 0 y ln(x 1) 1 y 0 ln(x0 1) 1 ln(x0 e x0 1 ) 0 x0 e x0 1 1 ln( x 1) 1 x e ln(x 1) y 1 y 0 x e ln(x0 1) e x0 x0 1 e x0 1 e x0 1 1 powers are always 0 x0 1 no solut ionexist s x0 e 1 2.718282.... 1 y ln(xex1 ) y 0.5 1 The commonly used bases Logarithms to base 2 Logarithms to base 10 Logarithms to base e Log2 x ≡ lb x Log10 x ≡ lg x Loge x ≡ ln x Binary logarithm Digital logarithm Natural logarithm 1 byte = 32 bit = 25 bit Classical metrics pH DeziBel The scientific standard Standard of software Publications Statistics 232 = 4294967296 1 byte = lb( number of possible elements) Weber Fechner law Human brightless perception Sensorical perception of bright, loudness, taste, feeling, and others increase proportional to the logarithm of the magnitude of the stimulus. c E k log k log c C c0 Logarithmic function Effect E Stevens’ power law 35 30 25 20 15 10 5 0 E c0c k E 9.5c 0.33 The power function law of Stevens approaches the WeberFechner law at k = 0.33 c E 20 log10 20 log c 1 0 10 20 Magnitude of c 30 Power functions and logarithmic functions are sometimes very similar. Loudness in dezibel The magnitude of a sound is proportional to the square of sound pressure Dezibel is a ratio and therefore dimensionless P: sound pressure 200 L[dB] 20log10 100 40 150 dB The rule of 20. Linear scale P2 P L[dB] 10 log10 2 20 log10 P0 P0 100 +40 50 x100 0 0.00001 0.001 0.1 10 1000 Magnitude of P [Pa] Logarithmic scale The threshold of hearing is at 2x10-5 Pascal. This is by definition 0 dB. What is the sound pressure at normal talking (40 dB)? P 40 20 log10 log10 P 2 log10 2 5 5 2 *10 P 2 x103 Pa The sound pressure is 100 times the threshold pressure. How much louder do we hear a machine that increases its sound pressure by a factor of 1000? 1000 P P 20 log10 L[dB] 20 log10 P0 P0 1000 P L[dB] 20 log10 60 The machine appears to be 60 dB louder P kP 20 log10 5 2 * 10 2 P 20 log10 5 2 * 10 2 log10 P kP log 10 2 *105 2 *105 2 kP P P k 5 2 *105 2 *105 2 *10 k To what level should the sound pressure increase to hear a sound 2 times louder? 70 60 50 40 30 20 10 0 0 0.0005 0.001 0.0015 P The multiplication factor k is linearly (directly) proportional to the sound pressure P. The mass effect in physics, chemistry, biochemistry, and ecology Na Cl NaCl [ Na ][Cl ] K [ NaCl] pH is the negative log10 of H+ concentration. The Arrhenius model assumes that reaction speed is directly proportional to the number of contacts an therefore the number of reactive atomes. [ H 3O ][OH ] 1014 [ H 2O] [H3O ][OH ] 1014 pH log10 ( H3O ) pH pOH 14 10 ml of a solution of H2S has a pH of 5. What is the concentration of OH- after adding 100 ml HCN of pH 8. 10* 5 100* 8 110pH pH 7.(72) pOH 6.(27) What is the pH of 0.5mol*l-1 NaOH? [ H 3O ][OH ] 1014 [ H 3O ][m ol* l 1 ] * 0.5[m ol* l 1 ] 1014 [ H 3O ] 1 *1014 pH (log10 2 14) 13.7 0.5 [HA] [H2O] [H3O ] [ A ] [H2O] [ H 3O ][ A ] [ A ] [ A ] K log10 K log10 [ H 3O ] log10 pK pH log10 [ HA] [ HA] [ HA] Henderson Hasselbalch equation What is the pH of 0.2 mol l-1 C2H5COOH (pK = 4.75) and 0.1 mol l-1 NAOH? H3O 0.2CH3COO 0.1Na 0.1OH 0.1CH3COONa 0.1CH3COO H2O 0.1 pH 4.75 log10 ( ) 4.75 0.1 Living organisms are buffered systems Blood is a CO2 – NaHCO3 buffer at pH 7.5 What is the pH after injection of 100 ml 0.8mol*l-1 CH3COOH. H3O NaCO3 NaHCO3 H2O [ H 3O ][NaCO3 ] 107.5 (1 0.1* 0.8) 7.5 10 [ H 3O ] pH 7.43 [ NaHCO3 ] 1. 0.1* 0.8 A first model Magicicada septendecim A B C D 1 Generation Predator A Predator B Predator C 2 3 4 5 6 7 8 0 1 2 3 4 5 +A7+1 1 0.5 1 0.5 1 0.5 +B6 1.5 0.75 0.75 1.5 0.75 0.75 +C5 2 1 1 1 2 1 +D4 E Sum of predator densities 4.5 2.25 2.75 3 3.75 2.25 +SUMA(B8:D8) Photo by USA National Arboretum Predator abundance 6 5 4 3 2 1 0 0 5 10 15 20 25 30 35 Time 40 45 50 55 60 65 A 1 Generation 2 1 3 =A2+1 4 =A3+1 5 =A4+1 6 =A5+1 B Predator A =2*LOS() =B2*LOS() =2*LOS() =B2*LOS() =2*LOS() 25 35 C Predator B =3*LOS() =C2*LOS() =C2*LOS() =3*LOS() =C2*LOS() D Predator C =4*LOS() =D2*LOS() =D2*LOS() =D2*LOS() =4*LOS() E Sum =SUMA(M34:O34) =SUMA(M35:O35) =SUMA(M36:O36) =SUMA(M37:O37) =SUMA(M38:O38) Magicicada septendecim Predator abundance Photo by USA National Arboretum 3 2 1 0 0 5 10 15 20 30 Time 40 45 50 55 60 65 Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Jota Kappa Lambda My Ny Xi Omikron Pi Rho Sigma Tau Ypsilon Phi Chi Psi Omega Home work and literature Refresh: • • • • • Greek alphabet Logarithms, powers and roots: http://en.wikipedia.org/wiki/Logarithm Logarithmic transformations and scales Euler number (value, series and limes expression) Radioactive decay Prepare to the next lecture: • • • • • • Logarithmic functions Power functions Linear and algebraic functions Exponential functions Monod functions Hyperbola