Compel your students to analyze evolution by involution of these intriguing cases.*
*in-vo-lu-tion. 1. a: the act or an instance of enfolding or entangling: involvement. b: complexity, intricacy. 2. exponentiation: the mathematical operation of raising a quantity to a power. 3. a: an inward curvature or penetration. (http://www.merriam-webster.com/dictionary/ involution)
Thomas Henry Huxley's reaction to Darwin's argument for natural selection is reported to have been "How truthfully simple. I should have thought of it." Each premise is truthfully simple, and the conclusion follows directly from them:
* Each and every species has a potential for geometric increase, yet rarely will all individuals in a population produce offspring according to this potential.
* This difference is not random: Some fare better than others because of heritable attributes that they gained from their parents and will most likely pass on to their offspring.
* The result is a natural form of selection within a variable population, leading to greater representation of heritable traits that confer reproductive and/or survival advantage, ergo "fitness."
Do our students really appreciate the simple power of this argument? Can they understand the nuances that have fostered lively discussion, theoretical progress, and clever investigations over the past 150 years? Hopefully, students have gained a qualitative understanding by the end of a unit on evolution. We should help them develop a quantitative understanding as well.
To that end, I use the case studies described here to estimate the power of selection mathematically. I challenge college freshmen in a biology majors course to develop confidence in quantitative methods, ask questions about the evolutionary process that would not have occurred to them otherwise, and apply critical-thinking skills. These case studies develop skill by repeatedly using a powerful tool and concept: the algebra of exponential growth.
While these may not be "case studies" in the traditional narrative sense, they are based on real situations described in outside readings that I assign to supplement the textbook (Raven et al., 2008). I begin the semester with a unit on evolution, introducing the Hardy-Weinberg equilibrium, mutation, selection, genetic drift, and the founder effect. For these "challenges," the students are encouraged to work together in finding solutions, but each is responsible for written discussion in his or her own words. For this article, I've abridged the examples, included blanks and tables that the students fill in, and inserted solutions for ABT readers in square brackets. First, I introduce the method with an intuitive example that raises issues relevant to our first case study, the growth of the world's human population.
* The Method: Exponential Population Growth
Let [w.sub.0] = [1.3 x [10.sup.8]] be the world's population at the
beginning of year 1 AD and [w.sub.50] = [2.6 x [10.sup.8]] be the
world's population 50 generations later, at 1000 AD. You can find
[w.sub.0] and [w.sub.50] on page 1161 in your textbook. Use
scientific notation. To find w use [w.sub.1] = [w.sub.0](1 + r)
where r = the rate of population increase. To find w use [w.sub.2]
= [w.sub.1] [(1 + r).sup.2] and, by substituting the definition
above for [w.sub.1], find that [w.sub.2] = [w.sub.0](1 + r)(1 + r)
= [w.sub.0] [(1 + r).sup.2]. This means that generally, after n
generations, we can find any [w.sub.n] = [w.sub.0] [(1 + r).sup.n]
if we know r, which is powerful in itself, but first we must find
In this case, the world's population doubled in 1000 years (n = 50
generations, assuming 20 years/generation), so [w.sub.50]/[w.sub.0]
= 2 = [(1 + r).sup.50]. To solve for r, we follow the steps and
calculations in the first two columns of Table 1. …