Transcript NikMBLECT2

Mathematical Biology
Lecture 2: Growth without limit
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
Steps of model making
1)
2)
3)
4)
5)
6)
7)
8)
9)
Collect data
Identify main processes
Write a “word” model for these main processes
Express the model as mathematical formulae
Solve the model
Interpret properties of solution in biological terms
Make testable predictions
Test, and see that model is not perfect
Back to step 1)
USA Population
150
100
50
0
1750
1800
1850
Year
1900
1950
Collared Dove
20
15
10
5
0
1956
1958
1960
1962
Year
1964
1966
Exponential Growth of E. coli
400
300
200
100
0
0
1
2
Time (h)
3
4
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
Maths all works fine for exponential decay :
e.g. drug concentration in blood
Rate of metabolism is proportional to concentration
…just take b < 0 to reflect decreasing concentration
15
10
5
0
0
2
4
6
Time (min)
8
10
Radioactive decay
(again b < 0)
dY
 bY
dt
Y  Y0 exp  b t 
T1  
2
ln(2)
b
Radio Carbon Dating
Summary: the exponential function
Y  Y0e b t
b 0
Cumulative
Rate
Y
Linearized
lnY
dY
dt
Y0
b 0
t
b Y0
lnY0
t
lnY
lnY0
dY
dt
Y
Y0
0
t
b Y0
t
t
0
t