Lecture 6 Sums of infinities The antiderivative or indefinite integral Integration has an unlimited number of solutions. These are described by the integration constant

Assume Escherichia coli divides every 20 min. What is the change per hour? How does a population of bacteria change in time? First order recursive function Difference equation Differential equations contain the function and some of its derivatives 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

The unbounded bacterial growth process 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 What is if the time intervals get smaller and smaller? Gottfried Wilhelm Leibniz ( ) Archimedes (c. 287 BC – 212 BC) Sir Isaac Newton ( ) The Fields medal

t f(t) The area under the function f(x)

t f(x) Definite integral

t f(x) What is the area under the sine curve from 0 to 2 ?

a b What is the length of the curve from a to b? What is the length of the function y = sin(x) from x = 0 to x = 2 ? c x y

No simple analytical solution We use Taylor expansions for numerical calculations of definite integrals. Taylor approximations are generally better for smaller values of x.

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

Allometric growth In many biological systems is growth proportional to actual values. A population of Escherichia coli of size growths twofold in 20 min. A population of size 1000 growths equally fast. Proportional growth results in allometric (power function) relationships. Relative growth rate

Differential equations First order linear differential equation Second order linear differential equation First order quadratic differential equation Every differential equation of order n has n integration constants.

Chemical reactions and collision theory The sum of n 1 and n 2 determines the order of the reaction. First order reaction The change in concentration is proportional to the number of available reactants, thus to the current concentration. 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.

E + S ES [E] + [ES] = [E0] [S] + [ES] = [S0] [E] - [S] = [E0] - [S0] Equilibrium is at 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

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

Logistic growth with harvesting Fish population growth can be described by a logistic model. Every year a constant number of fish is harvested Constant harvesting term

Logistic growth with harvesting 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 Constant harvesting might stabilize populations Logistic growth with harvesting N N N N r/K r1.9 m70000 With harvestingWithout harvesting Time N N N N =- $C$1*C22^2+$ C$2*C22-$C$3 +C22+B23 =- $C$1*E22^2+ $C$2*E22 +E22+D23

The critical harvesting rate N N Harvesting below the critical rate is the condition for positive population size r/K K r1.9m For a population to be stable dN/dt must be positive.

Proportional harvesting N N Critical harvesting rate Proportional harvesting stabilizes populations. K r2.1 f0.5 With harvesting Time N N =(- $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 literature Refresh: Logistic growth Lotka Volterra model Sums of series Asymptotes Integral Prepare to the next lecture: Probability Binomial probability Combinations Variantions Permutations Literature: Mathe-online Logistic growth: function n/logistic.html