Lecture 6Sums of infinities

xxfxxf

yxfdxdy

x

)()(lim)(' 0

The antiderivative or indefinite integral

)(')(

xfdxxdf

dxxfxF )()(

Cxdxx

Cxdxx

Caa

dxa

Cxa

dxx

Cxdxx

Cea

dxe

Caxdxa

xx

aa

axax

)sin()cos(

)cos()sin(

)ln(111

ln1

1

1

dxxgdxxfdxxgxf

dxxfadxxaf

)()()()(

)()(

Integration has an unlimited number of solutions. These are described by the integration constant

dxxdf

dxdC

dxxdf

dxCxfd )()())((

0 1000N

Assume Escherichia coli divides every 20 min. What is the change per hour?

1

2

3

1000*2*2*2

1000*2*2*2*2*2*2

1000*2 tt

N

N

N

3 30 1 0 0

3 3 3 31 2 1 1

3 3( 1) 31 1 1

1000*2 1000 (2 1)

1000*2 *2 1000*2 (2 1)

1000*2 1000*2 (2 1)t tt t t t

N N N N

N N N N

N N N N

How does a population of bacteria change in time?

11 ttt rNNNNFirst order recursive function1

trNtN

Difference equation

rNdtdN

tN

t

0lim

Differential equations contain the function and some of it’s derivatives

rtrtC KeeeN

CrtCN

rdtNdN

rdtNdN

rNdtdN

21)ln(

rt

r

eNN

KKeNt

0

000

Any process where the change in time is proportional to the actual value can be described by an exponential function.

Examples: Radioactive decay, unbounded population growth,First order chemical reactions, mutations of genes,speciation processes, many biological chance processes

0

20

40

60

80

100

0 2 4 6 8 10

N

t

The unbounded bacterial growth process

2ln002

tt eNNN How much energy is necessary to produce a given number of bacteria? Energy use is proportional to the total amount of bacteria produced during the growth process

8

20

8

2

2t

t

tt

tt NN

What is if the time intervals get smaller and smaller?

Gottfried Wilhelm Leibniz (1646-1716)

Archimedes (c. 287 BC – 212 BC)

Sir Isaac Newton (1643-1727)

The Fields medal

0

20

40

60

80

100

0 2 4 6 8 10

N

t

2ln002

tt eNNN

Dtf(t)

bt

at

bt

att ttfN )(

The area under the function f(x)

ttftFttF

tft

tFttF

tfdtdF

t

t

)()()(lim

)()()(

lim

)(

0

0

bt

at

bt

att

bt

att

bt

att tFttFtFttFN )()(lim)]()([lim 00

0

20

40

60

80

100

0 2 4 6 8 10N

t

2ln002

tt eNNN

Dtf(x)

)1())1((

))(())1((...)3()4()2()3()1()2(

)()(lim 0

recFnrecFN

nrecFnrecFrecFrecFrecFrecFrecFrectFN

tFttFN

bt

att

bt

att

bt

at

bt

att

bt

att

b

a

bt

atit

bt

atit

dttfdtafdtbfN

aFbFN

)()()(lim

)()(lim

0

0

bt

at

bt

att ttfN )(

b

a

bt

att dttfttfArea )()(lim 0

Definite integral

)()()( aFbFFdttfAreab

a

b

a

0

20

40

60

80

100

0 2 4 6 8 10N

t

2ln002

tt eNNN

Dtf(x)

0000

8

2

0

8

2

0 559.3632ln

2522ln

4

2ln

256

2ln2

2 NNNN

NNNt

ttotal

What is the area under the sine curve from 0 to 2p?

011)0cos()2cos()cos()sin(2

0

2

0

xdxxA

4)0cos(4)2/cos(4)cos(4)sin(42/

0

2/

0

xdxxA

0

20

40

60

80

100

0 2 4 6 8 10N

t

a

b

What is the length of the curve from a to b?

dxdxdy

dcL

dxdxdy

dxdxdydx

dydxdc dydxdydxdydxdc

2

2

0,2

222

0,22

0,0

1

1lim)(

lim)(limlim

What is the length of the function y = sin(x) from x = 0 to x = 2p?

2

0

2)cos(1 dxxL

Dc

Dx

Dy

2)cos(1 xL

No simple analytical solution

220160

11

248262)cos(1

7532 xxx

xdxx

625526.7

]22580480

11

2153624822[4)cos(14

7532/

0

2

dxx

We use Taylor expansions for numerical calculations of definite integrals.

Taylor approximations are generally better for smaller values of x.

1 2.2214412 -0.456773 0.1408784 0.000828

Sum 1.9063814 times 7.625526

What is the volume of a rotation body?

y y

x

x

b

a

b

a

dx dxxfdxxfV 220 )()(lim

)(

)(

22

1

)(bfy

afy

dyygV

What is the volume of the body generated by the rotation of y = x2 from x = 1 to x = 2

44 2

2

11

7.5 23.562

V y dy y

What is the volume of sphere?

34

32

2)3

(2233

0

32

0

222 rrx

xrdxxrVrr

y

x

Allometric growth

In many biological systems is growth proportional to actual values.

A population of Escherichia coli of size 1 000 000 growths twofold in 20 min. A population of size 1000 growths equally fast.

20001000

20000001000000

10

10

PP

NN

1000

20000001000000 1

P

NN

PP

NN

PP

zNN

PdP

zNdN

1)ln()ln( cPzNPdP

zNdN

zzc cPPeN 1

Proportional growth results in allometric (power function) relationships.

Relative growth rate

Differential equations

baydxdy

First order linear differential equation

aydx

yd 2

2

Second order linear differential equation

byaydxdy 2

First order quadratic differential equation

axeab

Aab

y )(

xaxa BeAey bx

bx

AaeAbe

y

1

ayBeAeaaBeaAey xaxaxaxa )(''

bayab

yaeab

yay ax )()(' 0

2

2

2

2

)1()1(1

)1(' ayby

AaeAbe

aAae

Abeb

Aae

AabeAbeAaeeAby bx

bx

bx

bx

bx

bxbxbxbx

Every differential equation of order n has n integration constants.

Chemical reactions and collision theory

21

43

4321

nn

nn

BADC

K

DnCnBnAn

The sum of n1 and n2 determines the order of the reaction.

First order reaction

...2211 CmCmA

3252 NONOON

][][

5252 ON

dt

ONd

The change in concentration is proportional to the number of available reactants, thus to the current concentration.

teONON 05252 ][][

teAA 0][][

The number of molecules decides about the number of colllisions and therefore about the number of reactions.

The speed of the reaction (the change in time in the number of reactants is proportional to the number of reactants).

K describes the reaction equilibrium.

)1(][][][][ 000tt eAeAAC

E + S ↔ ES [E] + [ES] = [E0]

[S] + [ES] = [S0]

[E] - [S] = [E0] - [S0] Ev

ESkESv

EkEv

SESE

ESES

][][

][][

21 ][]0[])[(

)0(][][][

kEkEkEkkv

EEkEkESkEkvvv

SESEES

SEESSEESSEES

21 ][][

kEkdtEd

tkCekk

E 1

1

2][

Equilibrium is at

1

221 ][][0

][kk

EkEkdtEd

First order chemical reactions result in equilibrium concentrations of enzyme and substrate

Enzyme Substrate Enzyme – substrate complex

First order reactions

Substrate concentration does not contribute to reaction speed

Autonomous first order differential equation

dFg fF

dt ( ) (0) ftg g

F t F ef f

What is the concentration of Insulin at a given time t?

Assume that Insulin is produced at a constant rate g. It is used proportional to its concentration at rate f

0

0.5

1

1.5

2

2.5

3

3.5

4

0 5 10 15 20 25 30

Time

F(t

) g / f

f = 0.2; g = 0.5

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3

F(t)

dF

(t)/

dt)

f = 0.2; g = 0.5F0 = 5

F0 = 1

g / f

A process where a substrate is produced at a constant rate and degraded proportional to it’s concentration is a self-regulating system.

Logistic growth with harvesting

2NKr

rNKNK

rNdtdN

Fish population growth can be described by a logistic model.

mNKr

rNKNK

rNdtdN

2

Every year a constant number of fish is harvested

Constant harvesting term

Logistic growth with harvesting

mNKr

rNdtdN 2

04 mKr

4rK

m

First order quadratic differential equation with constant term

Test of logic

The model predicts that the harvesting rate m must be smaller than rk. Otherwise the population goes extinct.

First order quadratic differential equation

-500000

0

500000

1000000

1500000

2000000

2500000

0 5 10 15 20

N

Time

-500000

0

500000

1000000

1500000

2000000

2500000

0 5 10 15 20

N

Time

Constant harvesting might stabilize populations

Logistic growth with harvesting

mrNNKr

N 2mNKr

rNdtdN 2

N

N

DN

DN

r/K 0.000001r 1.9m 70000

With harvesting Without harvestingTime DN N DN N

0 100000 1000001 110000 210000 180000 2800002 284900 494900 453600 7336003 625383.99 1120284 855671.04 15892714 803503.3627 1923787.4 493832.5374 20831045 -115761.8084 1808025.5 -381423.717 17016806 96292.1652 1904317.7 337477.3877 20391577 -78222.29081 1826095.4 -283763.511 17553948 64956.81722 1891052.2 253840.9281 20092359 -53079.31085 1837972.9 -219478.076 1789757

10 44004.08423 1881977 197308.871 198706511 -36081.14598 1845895.9 -173004.769 181406112 29870.60197 1875766.5 155899.1213 196996013 -24543.54838 1851222.9 -137818.02 183214214 20297.25391 1871520.2 124325.8571 195646815 -16699.42536 1854820.7 -110477.131 1845991

=-$C$1*C22^2+

$C$2*C22-$C$3+C22+B23

=-$C$1*E22^2+

$C$2*E22+E22+D23

mNKr

rNdtdN 2

The critical harvesting rate

-500000

0

500000

1000000

1500000

2000000

2500000

0 5 10 15 20

N

Time

N

DN r

rKKmrKN

mNKr

rN

2

4

0

2

2

404 2 rK

mrKKm

Harvesting below the critical rate is the condition for positive population size

r/K 0.000001 K 1900000r 1.9 m 902500

For a population to be stable dN/dt must be positive.

-200000

0

200000

400000

600000

800000

1000000

0 5 10 15 20

N

Time

Proportional harvesting

fNNKr

rNdtdN 2

N

DN

Critical harvesting rate

Proportional harvesting stabilizes populations.

rKfKr

N

fNNKr

rNdtdN

02

rfKrKf

K 1000000r 2.1f 0.5

With harvestingTime DN N

0 1000001 139000 2390002 262445.9 501445.93 274272.6597 775718.564 -22502.80057 753215.765 13743.85704 766959.626 -8141.425068 758818.197 4918.506768 763736.78 -2938.145108 760798.559 1767.364875 762565.92

10 -1058.767124 761507.1511 635.8462419 76214312 -381.2949333 761761.713 228.8531682 761990.5514 -137.2843789 761853.2715 82.38051644 761935.65

=(-$C$2/$C$1)*(C22

^2)+$C$2*C22-$C$3*C22

+C22+B23

The harvesting rate must be smaller than the rate of population increase.

Home work and literatureRefresh:

• Logistic growth• Lotka Volterra model• Sums of series• Asymptotes• Integral

Prepare to the next lecture:

• Probability• Binomial probability• Combinations• Variantions• Permutations

Literature:

Mathe-onlineLogistic growth: http://en.wikipedia.org/wiki/Logistic_functionhttp://www.otherwise.com/population/logistic.html