Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

Another example comes from a simplified model of predator-prey interactions in biology. Imagine a population of sharks preying on a population of fish. When the fish are at their maximum population level, the shark population grows at its maximum rate because there are so many fish to eat. The shark population continues to climb and reaches its own maximum level a quarter cycle later, by which time the fish population has dropped, having been preyed on so severely a quarter cycle earlier. An analysis of this model shows that the two populations oscillate out of phase by 90 degrees. Similar predator-prey oscillations are seen elsewhere in nature, for example, in annual fluctuations of Canadian hare and lynx populations as recorded by fur-trapping companies in the 1800s (though the real explanation for those oscillations is undoubtedly more complicated, as is often the case in biology).