So I sat down the other day (as I often do), and thought, “What about exponential and logarithmic functions might a high-school student find interesting?” Yes yes, exponential functions are essential in a number of financial math applications, but figuring out mortgage calculations is not the most engaging activity for the average teenager. Surely I could find something more worthy of student attention.

Soon enough, I found my rabbit hole: “Malthusian catastrophe.” Since stumbling on this half-familiar phrase nearly 24 hours ago, I have devoted an unusual proportion of my time to flesh it out through internet research. This post is an attempt to organize this information and maybe even get some suggestions for new resources. Well anyway, behold! :

“The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction, and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world”.

—Malthus T.R. 1798.

An essay on the principle of population. Chapter VII, p61 [via Wikipedia]

Famine, pestilence, and plague … I’m in!

[side note: Check out the Math Pickle and their values for student learning: “Mathematics and Beauty are Inseparable” ; “War, Death, and Nastiness Engage Students” ; “Hard Fun! That’s How We Learn Best” ]

Right away we have debatable questions: are humans doomed to catastrophe? are there too many of us? what is the maximum possible human population, and what will happen when we get close to that level?

**Resources**

**1. Introduction.** The best resource I’ve found so far is this site: http://www.math.montana.edu/frankw/ccp/modeling/discrete/snooping/learn.htm

It’s a sequence of modules that have taught me a good deal about different models for population growth. This particular section focuses on discrete methods (most appropriate for my class), but there are sections invoking continuous data and more Calculus-based approaches as well. Anyway, the discrete model module naturally leads to the study of recursive sequences, arithmetic and geometric scenarios, and other goodies. The algebra requirement is robust but not insurmountable. And there’s the sense that the advancing models show different levels for differentiation: the earliest models definitely hit at my core curriculum expectations for the semester, while later models become as complex as you want them to be. For example, I would expect all student to be able to use the exponential model, while more advanced students can use logistic models and beyond.

Model 1: . Simple exponential growth.

Model 1b: . Allows for “immigration/emigration”

Model 2: . C represents “carrying capacity,” so as P approaches C, the possibility for growth is diminished, with interesting behavior based on the value of R.

[I’ve had some difficulty with disparities in quoted discrete logistic models. This site was immensely helpful in straightening things out.]

Now that I feel like the math is appropriate for my class, I can look for my next requirement: there should be the possibility to play — play with calculations to look for patterns, play with a simulation to make conjectures, and play with an equation to see how a model works out.

**2. M&M’s in their natural environment. ** First, place 20 M&M’s (representing fish) on their fishbowl (plate) [sidenote: I’ve also seen this done with coins, but M&M’s are much more inherently interesting (and edible).]. Each time, put the fish in the cup, shake ‘em up, and pour them on the plate. The ones showing “M” get eaten. Second scenario, start with two, and for each one showing an “M”, add one to the bowl. The PBS activity sheet clearly tells you when the fish should be eaten. The follow-up activity leads the students to develop NOW-NEXT models to match the fishbowl data.

Supplementing this activity, there is a simulation of the fish population for different growth coefficients. The page also gives a model including logistic growth, so this could be explored later.

**3. Dice simulation.** As an extension to the M&M’s simulation, I want to use dice to let students invent different scenarios that would vary the growth rate. For example, start with 2 dice. Each turn, roll the dice, and for each 6 you see, add one dice to the population. What is the long-term behavior? Can you write a model?

I’ve spent an inordinate amount of time trying to make up a variation that would lead to logistic-like behavior. Here’s my best attempt at such a game:

Start with any number of dice. Each time, roll the dice onto a sheet of paper (or other region). If the dice land on the paper, they survive, then they may reproduce according to whatever rules you like. If they do not land on the paper, they die. What is the maximum population you can achieve? What is the long-term (and short-term) behavior? Can you write a model? (the model is not nearly as clear in this case)

This game is not perfect in terms of reproducing a lot of the quirks of the logistic model, but I’m awaiting my bulk shipment of dice to try it out.

**4. The model should be intuitive.** The Wikipedia article on “Matrix Population Models” explains that “all populations can be modeled by one simple equation:

Which is often called a BIDE model (Birth, Immigration, Death, Emigration). This is a rather intuitive equation that can let you build some good models. Also of note is that when considering the world population, there is (currently) no immigration and emigration from our planet! This is rather convenient.

**5. Vast selection of computer simulations.** By no means a complete or optimized list.

- Rabbits and Wolves
- Same idea, sustainability game
- Fertility dynamics – nice applet simulation
- Demographics/population simulation
- Animation / interactive showing births deaths CO2 use worldwide

**6. For further reading.**

- Readable explanation of exponential and logistic models
- Periodicity and Chaos — includes TI – 83 fun (part of a nice set of “vignettes”)
- Policy issues, math and teaching notes too
- Incidentally, the Montana website was produced as part of the “Connected Curriculum Project,” and I found the Duke site here.

**7. For further thought / processing.** Thanks for your hard work, intrepid reader!

Have students mark their data in a Google Doc spreadsheet, which can allow for easier aggregation of data and class comparisons. This could also aid discussions about random variation and deviation from the NOW-NEXT model.

BINARY FAMILY TREE: Create tree with you at the trunk, and biological parents above. How many great-great-great-great-…-great-grandparents do you have? Realization: This number is greater than the human population. You must be related!

RABBIT PROBLEM: Note that Fibonacci is proposing his problem at the dawn of population growth! Fibonacci – c. 1200. World pop. in 1200: about 400 million.

Thanks so much for sharing this. I won’t get to exp & logs for a while, but it’s in the file and I look forward to playing with it!

Thanks for the “thanks” ! Hopefully I’ll be able to share any practical materials or guidance after I run it through my classroom.

Here’s my suggestion for the logistic dice: the number of dice in play determines the survival rules. One possible way this could go:

Less than 10 dice: all dice survive to the next turn

10-19 dice: any “1” roll = dead

20-29: “1” and “2” = dead

30-39, 40-49, 50-59: same thing for 3, 4, 5

(This can be remembered by students quickly: the tens digit determines the death rule)

Your “growth” rule can be whatever you want: for example, I like “every 6 makes three babies”. This lets you talk about population equilibrium: in this case it is anywhere in the 30s since expected value says births and deaths are equally likely. But the upper limit of population is still quite high.

I’d try that and see how it goes. Neat stuff.

One last thing while I’m here: where did this NOW-NEXT thing come from, and can we change it to LAST-NOW or something? NOW-NEXT leads to students writing rules in the form f(n+1) = f(n) + … and I’d rather they come up with rules in the form f(n) = f(n-1) + … Picky, I know, but most functions I know give f(n), not f(n+1), and when you’re typing in the formula for cell B2 in a spreadsheet, you’re calculating NOW based on LAST, not NEXT based on NOW. It gets confusing, even more confusing than the dumb explanation I just gave.

Thanks, I like the logistic dice suggestion. I’ll try it out with the bulk set o’ dice, but I can see that it has one feature that I was looking for… the possibility for catastrophic population collapse! How exciting! (This could happen if the birth rate is very high and you end up with over 60 dice in play.)

And double thanks for the LAST-NOW suggestion. I’m sure you saved me a big headache down the line.

Researched it a bit more: it appears to come from either the NCTM Standards 2000 or the Core Plus curriculum series. I don’t know why they used the NEXT-NOW notation, but it doesn’t transition well. Good luck and let me know how it goes with the dice! The rules I suggested fit the kx(1-x) model pretty well; if k is small enough (low growth rules) it stabilizes, and if k is too big all hell breaks loose. Sweet!