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Lecture 6 Sums of infinities. The antiderivative or indefinite integral. Integration has an unlimited number of solutions . These are described by the integration constant. How does a population of bacteria change in time?. - PowerPoint PPT Presentation
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Lecture 6Sums of infinities
xxfxxfyxf
dxdy
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*21000*2*2*2*2*2*2
1000*2 tt
NN
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
rdtNdNrN
dtdN
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
tf(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
tf(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
)()()( aFbFFdttfArea b
a
b
a
0
20
40
60
80
100
0 2 4 6 8 10N
t
2ln002
tt eNNN
tf(x)
0000
8
20
8
20 559.363
2ln252
2ln4
2ln256
2ln22 NNNNNNNt
ttotal
What is the area under the sine curve from 0 to 2p?
011)0cos()2cos()cos()sin( 2
0
2
0
ppp
xdxxA
4)0cos(4)2/cos(4)cos(4)sin(4 2/
0
2/
0
ppp
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?
dxdxdydcL
dxdxdy
dxdxdydxdydxdc 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?
p2
0
2)cos(1 dxxL
c
x
y
2)cos(1 xLNo simple analytical solution
22016011
248262)cos(1
7532 xxxxdxx
625526.7
]22580480
11215362482
2[4)cos(147532/
0
2
ppppp
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
adx dxxfdxxfV 22
0 )()(lim pp
)(
)(
22
1
)(bfy
afy
dyygV p
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 ypp p
What is the volume of sphere?
34
322)
3(22
33
0
32
0
222 rrxxrdxxrVrr pppp
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.
2000100020000001000000
10
10
PPNN
100020000001000000 1
PNN
PP
NN
PPz
NN
PdPz
NdN
1)ln()ln( cPzNPdPz
NdN
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
axeabA
aby )(
xaxa BeAey bx
bx
AaeAbey
1
ayBeAeaaBeaAey xaxaxaxa )(''
bayabyae
abyay ax )()(' 0
2
2
2
2
)1()1(1)1(' ayby
AaeAbea
AaeAbeb
AaeAabeAbeAaeeAby 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
BADCK
DnCnBnAn
The sum of n1 and n2 determines the order of the reaction.
First order reaction
...2211 CmCmA
3252 NONOON
][][52
52 ONdtONd
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] EvESkESv
EkEv
SESE
ESES
][][][][
21 ][]0[])[()0(][][][
kEkEkEkkvEEkEkESkEkvvv
SESEES
SEESSEESSEES
21 ][][ kEkdtEd
tkCekkE 1
1
2][
Equilibrium is at
1
221 ][][0][
kkEkEk
dtEd
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
dF g fFdt
( ) (0) ftg gF 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
00.5
11.5
22.5
33.5
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
2NKrrN
KNKrN
dtdN
Fish population growth can be described by a logistic model.
mNKrrN
KNKrN
dtdN
2
Every year a constant number of fish is harvested
Constant harvesting term
Logistic growth with harvesting
mNKrrN
dtdN 2
04 mKr
4rKm
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
-5000000
5000001000000150000020000002500000
0 5 10 15 20
N
Time
-5000000
5000001000000150000020000002500000
0 5 10 15 20
N
Time
Constant harvesting might stabilize populations
Logistic growth with harvesting
mrNNKrN 2mN
KrrN
dtdN 2
N
N
N
N
r/K 0.000001r 1.9m 70000
With harvesting Without harvestingTime N N N 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$2*E22+E22+D23
mNKrrN
dtdN 2
The critical harvesting rate
-5000000
5000001000000150000020000002500000
0 5 10 15 20
N
Time
N
N rrKKmrKN
mNKrrN
24
0
2
2
404 2 rKmrKKm
Harvesting below the critical rate is the condition for positive population size
r/K 0.000001 K 1900000r 1.9 m 902500
For a population to be stable dN/dt must be positive.
-2000000
200000400000600000800000
1000000
0 5 10 15 20
N
Time
Proportional harvesting
fNNKrrN
dtdN 2
N
N
Critical harvesting rate
Proportional harvesting stabilizes populations.
rKfKrN
fNNKrrN
dtdN
02
rfKrKf
K 1000000r 2.1f 0.5
With harvestingTime N N
0 1000001 139000 2390002 262445.9 501445.93 274272.6597 775718.564 -22502.80057 753215.765 13743.85704 766959.626 -8141.425068 758818.197 4918.506768 763736.78 -2938.145108 760798.559 1767.364875 762565.92
10 -1058.767124 761507.1511 635.8462419 76214312 -381.2949333 761761.713 228.8531682 761990.5514 -137.2843789 761853.2715 82.38051644 761935.65
=(-$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