Wolves Change Rivers — a Note on Mathematical Ecology.


This post was inspired by a wonderful video about the effects on Yellowstone National Park of returning wolves to the ecosystem after an absence of seventy years. Wolves are keystone predators and kill and eat other animals. The effect of introducing them was to … sharply increase both the diversity and number of animals in the park and the number of green plants in the park; the wolves even moved where the rivers flowed.

As you will find if you watch the video, this is a wonderful example of a trophic cascade. The wolves showed up and the ecosystem’s health increased a whole bunch. This is also an example of a mathematical phenomenon called a nonlinear system. Systems we understand easily tend to be linear — the response of the system is proportional to the input. If you jump on the bed harder, you bounce higher, unless the ceiling introduces a painful non-linearity by impacting your head. Non-linear systems are far more common than linear ones, but harder to fathom.

The mathematics of ecology in highly non-linear. This is why ecology is more like whack-a-mole than science, sometimes.

If you haven’t yet watched the linked video, Occupy Math urges you to do so. Occupy Math has done a post about non-linearity where cicadas evolve to discover prime numbers to avoid predators. Positive feedback, which Occupy Math used to explain how fake news escapes from reality into social media is another example of a non-linear effect. The wolves in Yellowstone are a wonderful example of non-linearity. If you want an example simple enough to do yourself (with a spread sheet) look at Chaos on Purpose which goes over the logistic map from population dynamics.

The key point is this: things you know are going on and can measure can affect lots of other things that are hard to measure. This sort of non-linear interaction matters a lot. If the wolves made the Yellowstone ecosystem much healthier and more robust, what happens to the north when the polar bears disappear from habitat loss? If climate change removes moose, the biggest herbivore from the northern forests, those forests will change radically.

How is this math? These systems are complex enough that we cannot measure them with sufficient precision to make tight, accurate mathematical models. In addition, if thermians from the klaatu nebula handed us a good model of earth’s ecosystem, solving it would require major advances in our mathematical ability to understand the implications. Hopeless? Not quite. We can make pretty good short-range local predictions and we can make good guesses — supported by mathematical reasoning — about where help and intervention will do the most good. Of course this will be much harder if the current US budget guts the Environmental Protection Agency which gathers the type of data needed for these models.

Let’s look at some brief examples of non-linearities in ecology.

At one point, when condo builders were using so much of Florida’s fresh water that the Everglades were getting short on water, well-intentioned environmental activists got a law passed to stabilize water flow into the Everglades. Normally there is a wet season and a dry season; using water management involving reservoirs, the law got the flow back up and more constant than it had been. The effect was to severely damage wading bird populations — the dry season concentrated the fish population into small pools during the bird’s nesting season, making feeding the young birds far easier. A simple, linear fix to the Everglades damaged the ecosystem because a relatively simple non-linearity was missed.

When Occupy Math was young, there was an effort to wipe out mountain lions in the part of California where his grandparents lived. The effort succeeded and the exploding deer population started starving because there was not enough accessible greenery. They started eating tree bark (which will kill the trees). A massive slaughter of deer was necessary to arrest the ecological death-spiral. A similar problem (with a different cause) is under way now in the eastern US.

The film Cane Toads, an Unnatural History is both hysterically funny and a good documentary about the nonlinear effect of introducing cane toads to Australia to control a sugar cane pest. They didn’t eat that pest, but they did eat everything else from mice to cat food left outdoors. They also look like excellent prey, but have poison glands on their skin and so killed a remarkable number of Australia’s marsupial predators. The cane toads were in addition to several other Australian ecological disasters caused by introducing everything from rabbits to prickly pear cactus. In general, invasive species are a wonderful example of an ecological change with the potential for violent non-linear response.

Let’s take a quick look at some simplified math about this!

Occupy Math’s student Meghan Timmins worked on a project that modeled the process of building stable ecological communities from a model of how much each species affected the others. The point of the project was that if the new species show up one at a time, you find smaller communities than if your community-building model permits several species to show up together — which can happen during a migration after a flood or forest fire. Let’s look at two different tracks of the number of animals of several types over time. Each colored line tracks the abundance of one species in the model.


The difference between the nice stable ecology and the one that’s jumping all over the place is just which creatures showed up. The two simulations are in the same overall ecosystem model. This illustrates the degree to which any firm prediction about an ecosystem, even a simple one, is guess-work.

Occupy Math notes that this week’s post is mostly informative. Treating the ecology cavalierly is a bad idea because we are very bad at predicting what consequences follow from our actions. Not all the surprises are bad — sometimes we get good news — but Occupy Math notes that he skipped about fifty examples of adverse ecological impacts from human activity when putting this post together. Do you have examples of non-linear effects, ecological or otherwise? Then please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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