# The Lotka-Volterra Equations: Nature's Ode to Boom and Bust

Why should investors pay attention to cycles? For the same reason farmers pay attention to seasons. It is pointless for a farmer to try to harvest in the spring (or plant in the fall).

Investors, too, have actions that apply to given cycle points that are shaped by human behavior, which make them harder to pin down.

Investment cycles don't follow a fixed calendar pattern. Some are multiple years or even decades long. This adds to the challenge of cycle awareness, and adjusting behavior accordingly.

A classic example of cycles thinking is found in the old saying: "The best cure for high prices is high prices." That saying also works in reverse: "The best cure for low prices is low prices."

When applied to most commodities, high or low-price extremes have a direct impact on human behavior. The behavior change then shapes the cycle in a self-perpetuating feedback loop.

In a free market system, price discovery functions as a signal, with producers and consumers adjusting supply and demand in response, causing the price to fluctuate between extremes.

And yet, cycles would still exist, even in the absence of market price and of human behavior. The back-and-forth nature of cycles is embedded in nature itself.

We can see this in the Lotka-Volterra equations, which drive the predator-prey model. The Lotka-Volterra equations — named after Alfred James Lotka and Vito Volterra, two mathematicians from the late 1800s and early 1900s — can help us understand why cycles show up everywhere. Mother Nature, it turns out, is a fan of boom and bust.

Without getting into the math, we can observe that the predator-prey model revolves around three basic inputs: Predators, Prey, and Grass.

The predators eat the prey; the prey eat the grass; and the grass grows back quickly, slowly, or not at all, depending on how fast it gets eaten. These three inputs are all you need for a self-sustaining cycle.

Let’s say the predator is the lynx, a form of wildcat, and the prey is the Snowshoe Hare. Here is a simplified version of the model:

• The hare population booms thanks to a rich diet of grass.
• The lynx population also booms via plenty of hares to eat.
• The grass supply grows scarce, due to a glut of hares.
• At the same time, there are lynxes eating hares left and right.
• These pressures cause a "bust" — a hare population collapse.
• The lynx population, now starved for food, collapses, too.
• The reduction of the hare population is a respite for the grass, which again becomes plentiful.
• At this point, the cycle repeats from step one.

We can see this boom-bust pattern via recorded data from the Hudson's Bay Company, which sold lynx and hare fur pelts in large quantities between 1845 and 1935. You can see the peaks and troughs in the cycle pattern of total lynx and hare furs sold in the Wikipedia chart below.

In the absence of some disturbing outside factor, the Lotka-Volterra equations describe a predator-prey relationship that can cycle indefinitely (as long as the sun keeps nourishing the grass). The cycle simply repeats, with highs and lows in the lynx and hare population self-correcting through natural means.

So while the Lotka-Volterra equations describe a steady long-term relationship, there is a natural boom-bust rhythm within that pattern. On a regular basis, the status of the hare population depends on the particular cycle point and various interrelated factors.

This is how markets behave, too. Investors have the potential to greatly improve their long-term results by understanding this dynamic.

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The views and opinions expressed herein are the views and opinions of the author and do not necessarily reflect those of Nasdaq, Inc.

## Dr. Richard Smith

Dr. Smith is a regular speaker and lecturer and particularly enjoys opportunities to share his knowledge and help others gain an edge in the market.