Math 121 - Calculus for Biology I Spring Semester, 2009
	© 2001, All Rights Reserved, SDSU & Joseph M. Mahaffy San Diego State University -- This page last updated 19-Jan-09

 Outline of Chapter

1. Chirping Crickets and Temperature
2. Equations of Lines
3. Metric System Conversion
4. Worked Examples
5. Juvenile Height
6. References

 Chirping Crickets and Temperature

• Folk method for finding temperature (Fahrenheit) - Count the number of chirps in a minute and divide by 4, then add 40
• In 1898, A. E. Dolbear [3] noted that "crickets in a field [chirp] synchronously, keeping time as if led by the wand of a conductor."
• He wrote down a formula in a scientific publication (first?)

• Does this formula of Dolbear match the folk method described above?

• Mathematical models for chirping of snowy tree crickets, Oecanthulus fultoni, are similar
• Data from C. A. Bessey and E. A. Bessey (8 crickets) from Lincoln, Nebraska during August and September, 1897
• The least squares best fit line to the data is T = 60 + (N - 92)/4.7
• Java applet below to try to fit data

source for linear_ckt.java -Alternate link

Dolbear's Cricket Equation as a Linear Model

The line creates a mathematical model - the temperature as a function of the rate snowy tree crickets chirp

1. How well does the line fitting the Bessey & Bessey data agree with the Dolbear model given above?
2. When can this model be applied from a practical perspective?
3. Over what range of temperatures is this model valid?
4. How accurate is the model and how might the accuracy be improved?

These questions show the complex relationship between the biology of the problem and the mathematical model. The first two are biological with mathematics playing a very limited role.

1. Clear discrepancies: Different species, Approximations, Still similar

2. This biological thermometer has limited use: Snowy tree crickets only chirp for a couple months of the year and mostly at night with temperatures above 50^oF.

Last 2 questions are more mathematical.

3. The range of validity in temperatures for the model gives the domain, which in this case is mostly between 50^oF and 85^oF.

4. The folk formula is less accurate than the model formed by data, but it is much more easily applied.

The Bessey model used a linear least squares, but would a quadratic fit the data better? Would it give better insight into the problem.

Equations of Lines

The slope-intercept form of the line.

Some definitions:

1. The variable x is the independent variable.

2. The variable y is the dependent variable.

3. The slope is m.

4.The y-intercept is b.

Note that the variables x and y are only used for convenience. When describing a mathematical model using a linear model, one often chooses variables that more closely match the objects being observed. The cricket equation given above can be written

The independent variable is N, the rate that the crickets are chirping (number of chirps per minute). The temperature, T, is the dependent variable. The slope is 1/4, and the T-intercept is 40. A graph of this equation is seen in the applet above by pressing the button for Dolbear.

The point-slope form of a line.

A line passing through the point (x₀, y₀ ) with slope of m has the form

Given two points (x₀ , y₀ ) and (x₁ , y₁), then the slope m is given by:

Examples are found in the hyperlinked Worked Examples section.

Metric System Conversion

All of the conversions for measurements, weights, temperatures, etc. are linear relationships.

Converting Fahrenheit to Celsius

• The freezing point of water is 32^oF and 0^oC, so take (f₀ , c₀) = (32, 0).
• The boiling point of water is 212^oF and 100^oC (at sea level), so take (f₁, c₁ ) = (212,100).

The slope is

The point-slope form of the line gives

or

The temperature f in Fahrenheit is the independent variable. The equation of the line gives the dependent variable c in Celsius.

Below are JavaScript programs for a variety of linear conversions or linear relationships.

Example: Find a linear relationship that determines a weight in pounds given a weight in kilograms.

Solution: From the JavaScript program, we find 1 kilogram is 2.2046 pounds. If we let y be the weight in pounds and x be the weight in kilograms, the linear relationship is given by

Juvenile Height

Average juvenile height as a function of age [4]

The height, h, is graphed as a function of the age, a, with the data from the table. The data almost lie on a line, which suggests a linear model.

source for linear_hta.java ---Alternative Figure -Alternate link

This model is good for finding the average height of a child for any age between 1 and 13. The linear least square best fit to the data is

The next section will explain finding the linear least squares best fit or linear regression to the data.

For modeling, it is valuable to place units on each part of the equation.

• The height, h, from our data has units of cm.
• Thus, ma and the intercept b must have units cm.
• Since the age, a, has units of years, the slope, m, has units of cm/year.
• The slope is the rate of growth.

What questions can you answer with this mathematical model?

1. What is the average height of an eight year old?
2. What height does the model predict for a newborn baby?
3. If a six year old child is 110cm, then estimate how tall she'll be at age 7.

Answers:

1. The model predicts that the average eight year old will be 124 cm, which is found by setting a = 8 in the model.

2. The height intercept represents the height of a newborn., so this model predicts that a newborn would be 72.3 cm. However, this is outside the range of the data, which makes its value more suspect.

3. The model indicates that the growth rate is about 6.5 cm/year, so the six year old should grow about 6.5 cm and be 116.5 cm at age 7 though the average 7 year old as predicted by the model would be 117.5 cm.

What are some of the limitations and how might the model be improved?

• The domain of this function is restricted to some interval around 1 < a < 13. (The model predicts the average 20 year old to be 201.5 cm.)
• A better prediction of average 8 year olds would be to average the heights of 7 and 9 year olds (125.5 cm). (Local analysis)
• Similarly, a newborn is better estimated using only the data given for one and three year olds (66.5 cm).
• Growth rates for girls and boys differ, so you might want to split the data according to sex.
• The data shows faster growth rates between 0 and 5 and again between 9 and 13, which agrees with the common idea that growth occurs in spurts. A more complicated model?

References

 [1] H. A. Allard, The chirping rates of the snowy tree cricket (Oecanthus niveus) as affected by external conditions, Canadian Entomologist (1930) 52, 131-142.

[2] C. A. Bessey and E. A. Bessey, Further notes on thermometer crickets, American Naturalist (1898) 32, 263-264.

[3] A. E. Dolbear, The cricket as a thermometer, American Naturalist (1897) 31, 970-971.

[4] David N. Holvey, editor, The Merck Manual of Diagnosis and Therapy (1987) 15^th ed., Merck Sharp & Dohme Research Laboratories, Rahway, NJ.