Mathematical Biology Aim : To understand exponential growth Objectives: 1) Understand derivation of the model 2) Introduce relative and absolute rates 3) Solve a simple differential equation (separable variables) 4) Obtain expressions for doubling time 5) Introduce radio-carbon dating 6) Summarise the properties of the exponential distribution Lecture 2: Growth without limit

Steps of model making 1)Collect data 2)Identify main processes 3)Write a “word” model for these main processes 4)Express the model as mathematical formulae 5)Solve the model 6)Interpret properties of solution in biological terms 7)Make testable predictions 8)Test, and see that model is not perfect 9)Back to step 1)

USA Population Population x10 6 Year

Collared Dove Population x10 3 Year

Turbidity Time (h) Exponential Growth of E. coli

Solving differential equations Procedure: 1) Classify the equation (for now ignore this) 2) Find general solution (includes arbitrary constant) 3) Find particular solution (constant fixed to a value) 4) Rearrange the solution if necessary (i.e. Y=…) 5) Check the solution - using the differential equation + init. cond. - using dimensional analysis

Dimensional Analysis Allows us to interpret parameters in our equations…and to check that our maths has all worked out correctly This just convention, but I shall use square brackets for dimensions, and introduce the following generic classes L to represent some sort of length (cm, feet, miles, etc.) T to represent a time (seconds, years, days, etc.) M to represent a mass (grams, kilos, etc.) Rules… 1) if you have A = B then must have [A] = [B] 2) if you have A + B then must have [A] = [B] (= [A+B]) 3) [AB] = [A][B] and [A/B] = [A]/[B] 4) if you have exp(A), sin(A) etc., then [A] = 1 5) [dY/dt] = [Y]/[t] = [Y] T -1

Concentration Time (min) Maths all works fine for exponential decay : e.g. drug concentration in blood Maths all works fine for exponential decay : e.g. drug concentration in blood Rate of metabolism is proportional to concentration …just take  < 0 to reflect decreasing concentration

Radioactive decay (again  < 0)

Radio Carbon Dating

Summary: the exponential function Cumulative Y t Y0Y0 Rate t Linearized t lnY lnY 0 t Y0Y0 Y 0 t t lnY lnY 0 0