Abstract
Mathematical modeling refers to any use of mathematics to do theoretical science. As such, it incorporates mathematical techniques into a larger structure that is seldom taught in mathematics courses. Models can be derived from mechanistic principles or based on empirical data; there are some commonalities between these types of modeling as well as important differences. The first two sections present the concepts of mathematical modeling. Three sections on empirical modeling develop the least squares method for fitting linear models, extend the method for use with a large class of nonlinear models, and present the Akaike information criterion (AIC) for model selection. Two sections on mechanistic modeling present basic methods for model derivation and nondimensionalization. Biological examples in this chapter include a discussion of the use and misuse of the Lotka–Volterra predator–prey model, the derivation of the Holling type II predation model, and a compartment model of pollution in a lake. The problems include several that use chemostat and SIR disease models and an exploration of the evidence for global warming provided by an extensive data set of grape harvest dates.
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Notes
- 1.
Ideally, a model should be both empirically and mechanistically based, but the methods for the two types of modeling are distinct.
- 2.
I am using the word “observation” to encompass both observation of the natural world and observation directed by experiments.
- 3.
Few students can resist the impulse to tap faster whenever they are having only minimal success.
- 4.
Given unit time for the experiment and unit area for the environment, we can interpret y as the consumption rate per predator, in prey animals per unit time, and x as the prey density, in prey animals per unit area.
- 5.
Demographic stochasticity of virus particles is not an issue in a model that tries to predict quantities of these particles in a person suffering from a communicable disease; however, demographic stochasticity in a population of people could be quite significant.
- 6.
An additional complication occurs if, as would usually be the case, the behavior of each individual is influenced by the behavior of the other individuals in the population.
- 7.
The connection between number of individuals and predictability is explored in Chapter 4.
- 8.
See Section 3.1.
- 9.
We might use this model for P. steadius, but we ought not use it for P. speedius.
- 10.
We return to this scenario later in this section. More appropriate models are presented in Chapter 7.
- 11.
See Section 7.1.
- 12.
Aphids are particularly convenient organisms for the study of population dynamics because many aphid species exist for long periods as asexual females that reproduce by cloning. Simple models are more likely to yield accurate results when applied to simple systems.
- 13.
One could not design a better organism for rapid population growth than the aphid. When reproduction is by cloning, individuals do not need to mature before they begin the reproduction process. Indeed, pea aphids are born pregnant and begin to give birth to live young within hours after becoming adults.
- 14.
See Problem 2.2.1.
- 15.
Most descriptions of predator–prey models interpret the variables as the numbers of individuals, but the models are more realistic if the variables are viewed as being the total biomass of the individuals.
- 16.
The point here is that using the Lotka–Volterra model to demonstrate that something can’t happen in the real world is a logical fallacy when the model itself contains the assumption that the thing can’t happen.
- 17.
The symbols in these models are generic; that is, they represent whatever actual variables are in a given model. For example, a model H = CL q is a power function model with independent variable L, dependent variable H, exponent parameter q, and coefficient parameter C. Most symbols in mathematics are not standard, so the reader must be able to identify models as equivalent when the only difference is the choice of symbols. This theme is extended much further in Section 2.6.
- 18.
- 19.
For ease of reading, I use a simplified form of summation notation. What I have as ∑ xy, for example, is more properly given as \(\sum _{i=1}^{n}x_{i}y_{i}\). In the given context, the extra notation decreases readability unnecessarily.
- 20.
Context is crucial. As noted earlier, the parameter m functions as a constant in the model y = mx (narrow view) but as a variable in the total discrepancy function F (broad view). Meanwhile, x and y are variables in the model, but the data points (x i , y i ) function as parameters in the total discrepancy calculation because we have a fixed set of data.
- 21.
The proof of Theorem 2.3.1 is given as Problem 2.3.10.
- 22.
See [10] for a much more complete discussion of this topic.
- 23.
Equivalent models are the subject of Section 2.6.
- 24.
This could be done with the exponential model \(z ={ \mathit{Ae}}^{-\mathit{kt}}\) as well, if the goal is to minimize the fitting error in the original data. However, for reasons beyond the scope of this discussion, it is usually better to fit exponential models in the linearized form \(\ln z =\ln A -\mathit{kt}\) rather than the original form.
- 25.
See Problem 2.4.8 for an illustration of how important this is. Other authors (see [10], for example) also state that one should use nonlinear regression rather than using linear regression on a linearized model, but they don’t always explain the reason carefully or present an illustrative example.
- 26.
In Example 2.7.3, we will use this method with a model derived in Section 2.5.
- 27.
It would be better to fit the data for individual lengths rather than averages; however, the raw data sets are quite large and not generally available.
- 28.
A 1975 paper presents compelling evidence that other methods in use in the mid-1970s were preferable to the Lineweaver–Burk method [2]. Unfortunately, Lineweaver–Burk was entrenched by then, and scientific progress has failed in this case to overcome the inertia of standard practice. The Lineweaver–Burk method remains in common usage today. The 1975 tests included an implementation due to Wilkinson of the nonlinear method [14], which of course produced the best results with simulated data under reasonable assumptions about the types of error in the data. There can be no question that the semilinear least squares method produces the best fit on a plot of the original data, nor is there any reason in the world of fast computing to settle for a method that is not as good simply because it is faster for hand computation. In general, a good understanding of the theories of various methods for solving problems helps us to identify cases, such as this one, where older methods should be replaced by newer computer-intensive methods.
- 29.
The simulated experiment in Example 2.4.1 was based on this general observation, with 8 % used as the percentage for the hypothetical substance.
- 30.
Formally, this result can be proved by mathematical induction.
- 31.
Note the distinction between dimensions, such as length, and the associated units of measurement, such as meters, feet, and light-years.
- 32.
Of course, this does not happen in a real feeding scenario; however, this is an assumption in the conceptual model. In practice, this discrepancy between the real experiment and the conceptual model causes difficulties in the measurement of the parameters.
- 33.
Note that x need not be very large. In our human version, we could have a restaurant the size of a shopping mall with an average of one serving of food in each of the stores.
- 34.
See Example 2.7.2.
- 35.
In this text, we will usually adopt the practice of using one case for all of the original variables in a model and the opposite case for the corresponding dimensionless variables. Other systems, such as those that add accent marks for either the dimensional or the dimensionless quantities, have greater flexibility but other disadvantages. Distinguishing by case has the advantage of easy identification of corresponding quantities without the clumsiness of accent marks.
- 36.
See Chapter 7.
- 37.
For models that have more than one nonlinear parameter, one can use a fully nonlinear method. These can be found in any mathematical software package, such as R or Maple. Other packages, such as spreadsheets, that fit exponential or power function models to data use the linearization method.
- 38.
See Example 2.4.4.
- 39.
This statement is a mathematical version of Occam’s razor, a well-known scientific principle attributed to the fourteenth-century philosopher William of Ockham, although not found in his extant writings and actually appearing in some form before Ockham. The most common form was written by John Punch in 1639 and translates literally from the Latin original as “Entities must not be multiplied beyond necessity.” My interpretation is much more in keeping with the actual statement than its more common renderings in English.
- 40.
The usual formula for AIC ends with 2K rather than 2(k + 1), where K is the number of parameters that have to be fit using statistics. This includes the statistical variance along with the k model parameters. For those readers not well versed in statistics, it is easier to count the number of model parameters than the number of statistical parameters, so our version builds the extra parameter into the formula.
- 41.
See Table 2.1.1.
- 42.
See Examples 2.4.4 and 2.7.3.
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Ledder, G. (2013). Mathematical Modeling. In: Mathematics for the Life Sciences. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7276-6_2
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