8 - Curve Fitting and Biological Modeling

Summary

Most of the models introduced in this text have been developed by making reasonable theoretical assumptions, which are then incorporated into a mathematical framework. However, the ultimate test of the validity of any model is that its behavior is in accord with real data. Because of the simplifications introduced in any mathematical model of a biological system, we must expect some divergence between even the most carefully collected data and wellconstructed model. How can we determine if a model describes data well? How can we determine the parameter values in a model that are appropriate for describing real data? These questions are much too broad to have a single answer. There are, however, mathematical tools that can be used in addressing them.

Imagine having collected data on a population size at successive time intervals. Plotting the population values as a function of time might give a plot that appears to grow roughly exponentially. We might, therefore, think the simple Malthusian model P_t+1 = λP_t, introduced in Chapter 1, is adequate for describing the population growth. Then, the data points should lie approximately on a curve P_t = λ^tP₀ = P₀e^{(In λ)t}. But what should λ be? Is there a “best” estimate of this parameter that locates the curve “closest” to the data points? How can we be confident the population is really growing exponentially, and not more slowly, with the data points actually lying on a parabola, for instance?