In the 1920s, two mathematicians working on separate continents proposed the same set of mathematical equations for describing population growth and decline in biological systems.
Alfred Lotka, a Polish-American mathematician, chemist, and statistician, was the first to come up with the equations, though in a seemingly unrelated field. In 1910, he developed a model to describe autocatalytic chemical reactions, later noticing an analogy between biological and chemical systems, and extending them to ecological systems in the 1920s.
“In both systems, all processes could be reduced to two kinds of changes: those involving exchanges of matter between the components of the system, and those involving exchanges of energy,” a review of the topic by science historian Dr Sharon Kingsland explains of Lotka’s thinking. “In the chemical system the components were molecules. In the biological system the components were organisms plus the raw materials in their environment, and the exchanges of matter and energy took place through the web of food relationships, growth, and reproduction.”
Lotka, and later Italian mathematician Vito Volterra, derived equations to describe the populations of predator and prey in systems where the two groups interact, assuming that food for the prey population is ample and the environment is not significantly changing to the benefit of one of the groups. Though any mathematical equation used to describe the animal world is a simplification in an attempt to understand the complex dynamics going on, and makes huge assumptions within it, the Lotka-Volterra equations describe population growth with eerie levels of accuracy.
Lotka himself expressed surprise at how well the model, initially made for chemical reactions, translated to predator-prey relationships. As Volterra also later found, the equations tend to show oscillations between population sizes of both groups, as prey expand and then are reduced by the predator species. The predator population meanwhile grows as the prey grows, but then face more competition for their food as they reduce the abundant prey, and the result is a reduced predator population.
“Periodic phenomena play an important role in nature, both organic and inorganic. In chemical reactions rhythmic effects have been observed experimentally, and have also been shown, by the writer and others, to follow, under certain conditions, from the laws of chemical dynamics,” Lotka wrote in a 1920 paper. “However, in the cases hitherto considered on the basis of chemical dynamics, the oscillations were found to be of the damped kind, and therefore, only transitory (unlike certain experimentally observed periodic reactions).”
“It seemed that the occurrence of […] permanent oscillations, the occurrence of purely imaginary exponents in the exponential series solution presented, would demand peculiar and very specific relations between the characteristic constants of the systems undergoing transformation; whereas in nature these constants would, presumably, stand in random relation.”
“It was, therefore, with considerable surprise that the writer, on applying his method to certain special cases, found these to lead to undamped, and hence indefinitely continued, oscillations.”
The equations, though of course a simplification of the real-world, can help ecologists to study predator-prey dynamics, and model what will happen if, for example, an invasive species is introduced to an environment. Like Zipf’s law of languages, it shows how mathematical equations can appear to govern (or, really, adequately describe) extremely complex and variable systems.
Source Link: Population Growth Appears To Closely Follow The Lotka-Volterra Mathematical Equations
