The book of nature is written in

the language of mathematicsGalileo Galilei

1. Introduction

2. Basic operations and functions

3. Matrix algebra I

4. Matrix algebra II

5. Handling a changing world

6. The sum of infinities

7. Probabilities and distributions

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

Our program

In this lecture we will apply basic mathematics and statistics to solve ecological problems.

The lecture is therefore application centred.

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!!!

www.uni.torun.pl/~ulrichw

Modelling Biology

Basic Applications of Mathematics and Statistics in the Biological Sciences

Part I: Mathematics

Script A

Introductory Course for Students of

Biology, Biotechnology and Environmental Protection

Werner Ulrich

UMK Toruń 2007

Modelling Biology

Basic Applications of Mathematics and Statistics in the Biological Sciences

Part I: Mathematics

Script B

Introductory Course for Students of

Biology, Biotechnology and Environmental Protection

Werner Ulrich

UMK Toruń 2007

Modelling Biology

Basic Applications of Mathematics and Statistics in the Biological Sciences

Part II: Data Analysis and Statistics

Script A

Introductory Course for Students of

Biology, Biotechnology and Environmental Protection

Werner Ulrich

UMK Torun 2007

Older scripts

Mathe onlinehttp://www.mathe-online.at/

http://tutorial.math.lamar.edu/

Additional sources

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.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

Asymptote

Root

The logarithmic function

The logarithmic function is not defined for negative values

Log 1 = 0

1)(log

0)1log(

aand

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ya

ya

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y

y

ax

a

a

1

log

log

log

John Napier(1550-1617)

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logloglog

Logarithms and logarithmic functions

dcbxay )ln(A general logarithmic function

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4

y

x

4)53ln(2 xyShift at x-axis

Shift at y-axis

Increase

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5

53

)53ln(24

2

2

ex

xe

xRoot

Curvature

What is the logarithm of base 2 of 59049 if the the logarithm of 59049 of

base 3 is 10?

590492

85.15585.1*103log10

85.15

2

xx

1631.0*585.12log*3log631.02log585.13log

32

3

2

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)/(log)(log)(log)/(log

43log415.0

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11.5

22.5

3

0 1 2 3 4

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x

The number e

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Leonhard Paul Euler (1707-1783)

01ieThe famous Euler equation

Logarithmic equations

bax )ln(

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ee

bax

bb

bax

)ln(

)ln(

)1ln(2 xx

1

)1ln(22

xe

xxx

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-2.5 -2 -1.5 -1 -0.5 0 0.5 1

x = 0x -0.8

cbax )ln()ln(

aex

eaxcb

cb

/ln

ln

Mixed equations often do not have analytical solutions.

1)1ln( xy

1....718282.211

1)1ln(01)1ln(

0

0

0

exex

xyxy

)ln( 1 xxey

111

1

0)ln(

0)ln(

0

10

10

10

1

0

0

0

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x

x

1)1ln( xexy

existssolution no0 always are powers

)1ln(

01)1ln(

00

x

x

ex

yexy

Roots

The commonly used bases

Logarithms to base 10Logarithms to base 2 Logarithms to base e

Log10 x ≡ lg xLog2 x ≡ lb x Loge x ≡ ln x

Digital logarithmBinary logarithm Natural logarithm

1 byte = 32 bit = 25 bit

232 = 4294967296

1 byte = lb( number of possible elements)

Classical metrics pHDeziBel

The scientific standardStandard of softwarePublicationsStatistics

Weber Fechner law

Sensorical perception of bright, loudness, taste, feeling, and others increase proportional to the

logarithm of the magnitude of the stimulus.

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loglog

0

Logarithmic function

05

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0 10 20 30

Effec

t E

Magnitude of c

kccE 0The power function law of

Stevens approaches the Weber-Fechner law at k = 0.33

Stevens’ power law

33.05.9 cE

ccE log201

log20 10

Power functions and logarithmic functions are sometimes very

similar.

Human brightless perception

0102

0

2

10 log20log10][PP

PPdBL

Loudness in dezibel

Dezibel is a ratio and therefore dimensionless

P: sound pressure

The rule of 20.

The magnitude of a sound is proportional to the square of sound pressure

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)?

PaxP

PP

3

1010510

102

52log2log10*2

log2040

0

50

100

150

200

0.00001 0.001 0.1 10 1000

dB

Magnitude of P [Pa]

x100

+40

40100log20][ 10 dBL

Logarithmic scale

Line

ar s

cale

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?

601000log20][

log201000log20][

10

010

010

PPdBL

PP

PPdBL

The machine appears to be 60 dB louder

To what level should the sound pressure increase to hear a sound 2 times louder?

510

510

10*2log20

10*2log20

2P

kP

55

2

5

510510

10*210*210*2

10*2log

10*2log2

PkkPP

kPP 010203040506070

0 0.0005 0.001 0.0015

k

PThe multiplication factor k is linearly (directly) proportional to the sound

pressure P.

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.

pH is the negative log10 of H+ concentration.

)27.(6)72.(71108*1005*10 pOHpHpH

)(log 310 OHpH

14 pOHpH

What is the pH of 0.5mol*l-1 NaOH?

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1][

10]*[5.0*]*][[10]][[

1014

3

14113

143

pHOH

lmollmolOHOHOH

143 10]][[ OHOH

The mass effect inphysics, chemistry, biochemistry,

and ecology

NaClClNa

KNaClClNa

][]][[

The Arrhenius model assumes that reaction speed is directly proportional to the number of contacts an therefore the number of reactive atomes.

14

2

3 10][

]][[

OHOHOH

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.

OHNaHCONaCOOH 2333

43.78.0*1.0.1

)8.0*1.01(10][10][

]][[ 5.7

35.7

3

33

pHOHNaHCO

NaCOOH

][][][][][ 232 OHAOHOHHA

][][log

][][log][loglog

][]][[

1010310103

HAApHpK

HAAOHK

HAAOHK

Henderson Hasselbalch equation

What is the pH of 0.2 mol l-1 C2H5COOH (pK = 4.75) and 0.1 mol l-1 NAOH?

75.4)1.01.0(log75.4 10 pH

OHCOOCHCOONaCHOHNaCOOCHOH 23333 1.01.01.01.02.0

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Time

Pre

dato

r ab

unda

nce

Magicicada septendecimPhoto by USA National Arboretum

A B C D E

1 Generation Predator A Predator B Predator CSum of predator densities

2 0 1 1.5 2 4.53 1 0.5 0.75 1 2.254 2 1 0.75 1 2.755 3 0.5 1.5 1 36 4 1 0.75 2 3.757 5 0.5 0.75 1 2.258 +A7+1 +B6 +C5 +D4 +SUMA(B8:D8)

A first model

0

1

2

3

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Time

Pre

dato

r abu

ndan

ce

A B C D E1 Generation Predator A Predator B Predator C Sum2 1 =2*LOS() =3*LOS() =4*LOS() =SUMA(M34:O34)3 =A2+1 =B2*LOS() =C2*LOS() =D2*LOS() =SUMA(M35:O35)4 =A3+1 =2*LOS() =C2*LOS() =D2*LOS() =SUMA(M36:O36)5 =A4+1 =B2*LOS() =3*LOS() =D2*LOS() =SUMA(M37:O37)6 =A5+1 =2*LOS() =C2*LOS() =4*LOS() =SUMA(M38:O38)

Magicicada septendecimPhoto by USA National Arboretum

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