Holly Peters Hirst

Models of one-species population growth are perhaps the easiest to build and explain, as is evidenced by their ubiquity in calculus, differential equations and numerical analysis texts. With some simple computational tools, these models are accessible at lower levels [1]. The material described here is designed to give students a first exposure to population models. It can be adapted to the algebra level by considering only the discrete versions of the models. I use this material to (re)introduce students to discrete and continuous rates of change and proportionality, and to develop students' scientific modeling skills.

I start with a careful development of proportionality, a notion fundamental to modeling [6]. Then I introduce the modeling process with the simple one-species Malthusian model. I examine two different ways of incorporating intra-species competition, both of which lead to the logistic model: Slowing the birth rate and counting individual interactions. Next, species interaction models are introduced starting with the simplest Lotka-Volterra predator-prey models. Limitations and improvements of the LV models are discussed, such as those proposed by Kolmogorov, Leslie, Gause, May, Thompson, etc. [3, 4]. Then harvesting, mutualism and competition are considered as alterations of the basic LV model [2, 4, 5].

I use this material every fall semester with great success; my students cite the review of rate of change and proportionality as helping them to understand "equations" in their science classes. The materials I have gathered thus far are not in hyper-textual form but are highly modular and non-linear in the sense that the more advanced models can be developed "in parallel," allowing class discussion to drive the order of development. I plan on having one of my students help me to pull an http version together, eventually including links to some of the software.