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 data2) Identify main processes3) Write a “word” model for these main processes4) Express the model as mathematical formulae5) Solve the model 6) Interpret properties of solution in biological terms7) Make testable predictions8) Test, and see that model is not perfect9) Back to step 1)

USA Population

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Collared Dove

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1956 1958 1960 1962 1964 1966

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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…andto check that our maths has all worked out correctlyThis 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

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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

0 expY Y t

dY Ydt

(again < 0)

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ln(2)T

Radio Carbon Dating

Summary: the exponential function

0tY Y e

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CumulativeY

tY0

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ddYt

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Linearized

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Y0

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ddYt

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t t

lnYlnY0

0