The Lotka-Volterra model
We let x denote the population of rabbits, which reproduce at a rate proportional to their population, and let y denote the population of lynxes (bobcats), which die unless their main food source (rabbits) is present. The equations we use are:dx/dt = ax - bxy
dy/dt = bxy - cy where a is a rate constant reflecting how fast rabbits reproduce, b specifies how fast lynxes reproduce given a number, x, of rabbits to eat, and c indicates the mortality rate of lynxes. For any set of these constants, the numbers of rabbits and lynxes will oscillate with a period that depends upon a, b, and c.The next figure shows typical oscillations in the rabbit and lynx populations, for the special case where all constants are a=b=c=1.
The rabbits reproduce because their main food source is plentyful.
The lynx population will also increase, but only after the rabbit population has grown. Once the lynx population gets too high, rabbits will be eaten more rapidly than new rabbits are born, and their population will begin to decrease, which in turn will lead to a decrease in the lynx population. The rabbit population can then begin to rise again. Thus there will be a time lag between changes in the two populations.
Observational evidence
The next figure shows the number of lynx furs turned in to the Hudson Bay Company from 1820 to 1920. Distinct oscillations are seen with a period of about nine years. No data were available on the rabbit population, so one cannot be sure that the oscillations are due to a predator-prey interaction.
Experimental evidence
To test the model, experiments have been performed in the laboratory with paramecia (paramecium aurelia) that eat the yeast saccharomyces exiguns; as shown in the next figure. Notice how the predator population lags behind the population changes in the prey.
Oscillations in the populations of paramecia and yeast, from D'Ancona, 1954