Math Land

(This column was published on January 27, 1997.)

If any geographic locale in the world should be referred to as Math Land, it 
is Isle Royale, the island in Lake Superior I visited with Wally Neal last 
September.

I wrote then describing our forty mile backpacking expedition over that remote 
island. Now I will tell you what piqued this sometime math teacherıs original 
interest in that place.

Until about fifty years ago mathematics was applied almost exclusively to 
problems of engineering, physics and statistics. But then mathematicians began 
to explore new fields -- among them biology. Many of their investigations 
involved advanced mathematics, but a few are uncomplicated enough to display 
the power of mathematics to non-specialists.

Arguably the simplest of those applications is the two species predator-prey 
feedback loop. That title refers to what would happen if a single predator and 
the single species it fed upon were isolated from other factors and allowed to 
interact.

Here is what mathematicians predict would happen. Start with large numbers of 
both predators and prey. The predators would kill off many prey, reducing the 
prey population significantly. Clearly, if they killed all their prey, the 
predators themselves would then all die of starvation, but more likely 
predator numbers would decline while some prey individuals survived. Soon both 
populations would have been reduced to their lowest point. Then, with fewer 
predators to threaten them, the prey population would rise again. When those 
prey numbers had rebounded to their original level, the predator population, 
now with food again readily available, would increase as well, returning both 
species to the point at which they -- and we -- started.

A graph of that interaction with one axis representing prey numbers, the other 
predator numbers, produces an oval, thus the "loop" in that complex title.

Like so many theoretical concepts, however, it is not easy to find a real 
world application. In almost all settings predators feed on several prey 
species. Foxes, for example, feed on rabbits, mice and farmersı chickens; and 
mice are fed upon by foxes, crows and hawks. Wildlife biologists were, 
however, able to provide an example, and that is where Isle Royale came in.

On that island, separated from the northern Minnesota mainland by a broad 14 
mile channel, live moose and wolves, a single prey species and a single 
predator species that interact in isolation. Indeed, over a period of several 
decades the numbers of these two species went through at least one cycle in 
the manner predicted by mathematicians. From a low of 20 wolves and 500 moose 
in 1960, moose numbers climbed to 1500 in the early 1970s and wolf numbers 
followed to a high of 50 in 1980, both populations declining after that.

But by the time we explored the island late last summer, things had become 
more complicated. In 1994 there were less than 20 wolves despite a moose herd 
that peaked at 2500. Even with ready access to food, the wolves were not 
breeding; the alpha females who controlled reproduction were too old.

Then the moose population crashed; half the herd died during the 1994-1995 
winter. Their high numbers had exhausted their primary food source, mountain 
ash, and the moose were starving. We found stripped trees throughout the 
island.

And in 1995 some of the wolves separated into new packs and younger females 
began to bear young. That created a situation not predicted by the feedback 
loop: decreasing numbers of prey and increasing numbers of predators.

So things are not so simple after all. The mathematical feedback loop gives us 
interesting insights into species interactions but, as we should expect in 
this ever challenging world, not nearly all the answers.