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Lecture 6 Sums of infinities

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?.

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Lecture 6 Sums of infinities

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  1. Lecture6 Sums of infinities Theantiderivativeorindefiniteintegral Integrationhas an unlimitednumber of solutions. Thesearedescribed by theintegrationconstant

  2. Howdoes a population of bacteriachangein time? AssumeEscherichia coli dividesevery 20 min. Whatisthechange per hour? First order recursivefunction Differentialequationscontainthefunction and some of it’sderivatives Differenceequation Any processwherethechangein time isproportional to theactualvaluecan be described by an exponentialfunction. Examples: Radioactivedecay, unboundedpopulation growth, First order chemical reactions , mutations of genes, speciationprocesses, many biologicalchanceprocesses

  3. Theunboundedbacterial growth process How much energy isnecessary to produce a givennumber of bacteria? Energy useisproportional to thetotalamount of bacteriaproducedduringthe growth process Whatisifthe time intervalsgetsmaller and smaller? Gottfried Wilhelm Leibniz (1646-1716) Sir Isaac Newton (1643-1727) Archimedes (c. 287 BC – 212 BC) The Fields medal

  4. f(t) Dt Thearea under thefunction f(x)

  5. f(x) Dt Definiteintegral

  6. f(x) Dt Whatisthearea under the sine curvefrom 0 to 2p?

  7. b Dc Dy a Dx Whatisthelength of thecurvefrom a to b? What is the length of the function y = sin(x) from x = 0to x = 2p?

  8. No simpleanalyticalsolution We use Taylor expansions for numericalcalculations of definiteintegrals. Taylor approximationsaregenerallybetter for smallervalues of x.

  9. y y Whatisthevolume of a rotation body? x x What is the volume of the body generated by the rotation of y = x2 from x = 1 to x = 2 y What is the volume of sphere? x

  10. Allometric growth In many biological systems is growth proportional to actualvalues. A population of Escherichia coli of size 1 000 000growthstwofoldin 20 min. A population of size 1000 growthsequallyfast. Relative growth rate Proportional growth resultsinallometric(powerfunction) relationships.

  11. Differentialequations Second order lineardifferentialequation First order lineardifferentialequation First order quadraticdifferentialequation Everydifferentialequation of order n has n integrationconstants.

  12. Chemical reactions and collisiontheory Thenumber of moleculesdecidesaboutthenumber of colllisions and thereforeaboutthenumber of reactions. Thespeed of thereaction (thechangein time inthenumber of reactantsisproportional to thenumber of reactants). The sum of n1 and n2 determines the order of the reaction. K describesthereactionequilibrium. First order reaction Thechangeinconcentrationisproportional to thenumber of availablereactants, thus to thecurrentconcentration.

  13. First order reactions Substrateconcentrationdoes not contribute to reactionspeed Enzyme – substratecomplex Substrate Enzyme E + S ↔ ES [E] + [ES] = [E0] [S] + [ES] = [S0] [E] - [S] = [E0] - [S0] Autonomous first order differentialequation Equilibriumisat First order chemical reactionsresultinequilibriumconcentrations of enzyme and substrate

  14. Whatistheconcentration of Insulin at a given time t? Assumethat Insulin isproducedat a constantrateg. Itisusedproportional to itsconcentrationatrate f A processwhere a substrateisproducedat a constantrate and degradedproportional to it’sconcentrationis a self-regulating system.

  15. Logistic growth withharvesting Everyyear a constantnumber of fishisharvested Fish population growth can be described by a logistic model. Constantharvesting term

  16. Logistic growth withharvesting First order quadraticdifferentialequationwithconstant term Test of logic The model predictsthattheharvestingrate m must be smallerthanrk. Otherwisethepopulationgoesextinct.

  17. Logistic growth withharvesting First order quadraticdifferentialequation N Constantharvestingmightstabilizepopulations DN N DN

  18. Thecriticalharvestingrate For a population to be stabledN/dtmust be positive. N DN Harvestingbelowthecriticalrateisthecondition for positivepopulationsize

  19. Proportionalharvesting N Proportionalharvestingstabilizespopulations. DN Criticalharvestingrate Theharvestingratemust be smallerthantherate of populationincrease.

  20. Home work and literature • Refresh: • Logistic growth • Lotka Volterra model • Sums of series • Asymptotes • Integral • Prepare to thenextlecture: • Probability • Binomialprobability • Combinations • Variantions • Permutations Literature: Mathe-online Logistic growth: http://en.wikipedia.org/wiki/Logistic_function http://www.otherwise.com/population/logistic.html

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