# Lotka-Volterra rules

The Lotka-Volterra rules, also known as Lotka-Volterra laws or just Volterra rules , comprise three rules for the quantitative description of the population dynamics in predator-prey relationships .

The mathematical Lotka-Volterra equations on which the three rules are based were formulated independently of one another in 1925 and 1926 by the Austrian-American chemist Alfred J. Lotka and the Italian mathematician and physicist Vito Volterra . The three rules are found as "Laws" (loi) in Volterra's 1931 book; Volterra formulates the statements mathematically precisely and gives a strict derivation from the equations. In the biological textbook literature (see e.g. Müller 1991), under the (in this case incorrect) name Lotka-Volterra rules, there are occasionally deviating statements that cannot be justified from the equations.

## The rules

The Lotka-Volterra rules describe the numerical development of two populations over large periods of time. All three rules only apply provided that there is only a predator-prey relationship between the two species under consideration and that the other biotic and abiotic environmental factors are constant or negligible.

• First Lotka-Volterra rule (periodic population fluctuation): The population sizes of predator and prey fluctuate periodically. The fluctuations in the predator population follow those of the prey population with a phase delay. The length of the periods depends on the initial conditions and on the growth rates of the populations.
• Second Lotka-Volterra rule (constancy of mean values): The sizes ( mean value ) of the predator or prey population averaged over sufficiently long periods of time are constant. The size of the mean values ​​depends only on the growth and decline rates of the populations, not on the initial conditions.
• Third Lotka-Volterra rule (disturbance of the mean values): If the predator and prey population are decimated proportionally to their size, the mean value of the prey population increases in the short term, while the mean value of the predator population decreases in the short term.

Strictly speaking, the Lotka-Volterra rules can only be applied if they are rarely met. Nevertheless, they are of great importance in practical ecology because it has been shown that they still provide useful estimates even with more complex food relationships and fluctuating environmental factors.

### First Lotka-Volterra rule

The First Lotka-Volterra Rule states that the number of individuals of predator and prey fluctuate periodically and with a time offset under otherwise constant conditions.

Population fluctuations in predator and prey

The population curves thus form waves with temporally offset extremes, the curve of the predator population trailing. For example, a maximum in the prey population is followed by a maximum in the predator population. The reason for this is that with a high number of prey animals or plants, the predators have more food and thus increased chances of reproduction. Since the young of the predator need some time to grow, the maximum of the predator comes about much later. As the number of predators increases, so does the pressure on the prey population, it shrinks. However, with decreasing population density of the prey, the hunting success of the predators also decreases, so that their population also decreases due to lack of food. The reduced enemy pressure now increases the prey population again, etc.

As a textbook example of the First Volterra Rule, the Hudson's Bay Company catch records , which were kept for over 90 years. After that, the receipt of skins from lynxes (predators) and snowshoe hares (prey) fluctuated with a period of 9.6 years. However, this example is strictly influenced by a second predator, namely the hunters of the Hudson's Bay Company.

In mathematical terms, the result is coupled differential equations , which are also known as Lotka-Volterra equations . According to this, the increase in the number of predators depends on both the general birth rate and the probability with which the predators will eat a prey animal. The decrease in prey depends not only on the general death rate, but also on the frequency of contact.

### Second Lotka-Volterra rule

The Second Lotka-Volterra Rule states that the average size of the populations of predator and prey in a predator-prey relationship is constant over a longer period of time if the environmental conditions are otherwise stable. As with the first Lotka-Volterra rule, the statement results mathematically from the underlying differential equations.

The second rule was also empirically proven by the statistics of the Hudson's Bay Company from 1845 to 1935 by the number of lynx and snowshoe hares brought in. Although the number of pelts delivered annually for lynxes fluctuated between 1,000 and 70,000 and for snowshoe hares between 2,000 and 160,000, the mean values ​​when considering several periods are around 20,000 (lynxes) and 80,000 (snowshoe hares).

Basically, the Second Volterra Rule applies regardless of the initial size of the population. It can also be reproduced in experiments (e.g. with different protozoa ). In real systems, however, the initial sizes and the available area must be at least large enough that enough prey animals can be kept (e.g. by hiding places) even with high feeding pressure to ensure reproductive capabilities.

### Third Lotka-Volterra rule

The Lotka-Volterra Third Rule makes a statement about the effects of a disturbance in a predator-prey relationship . If the predator and prey population are decimated at the same time and by the same percentage, the mean value of the prey population increases for a short time, and the mean value of the predator population decreases for a short time.

In contrast to periodic fluctuations, the decrease in the predator population coincides with the decimation of the prey population. Not infrequently, the lack of food in this situation leads to a collapse of the predator population. Without predators, the remaining prey population then finds optimal conditions and grows faster than usual. By contrast, it takes longer than usual for the predator population to recover, because of the small number of individuals. In most predator-prey relationships, an additional factor is that the generation time of predators is longer than that of their prey due to their height.

This relationship must be observed in particular with pest control measures . For example, insecticides have the effect of decimating not only the pests, but also their predators to a much greater extent. In the end, this can mean that the damage is greater after such a measure than without control measures.

Synthetic toxins such as DDT , which are also accumulated in vertebrates , are particularly fatal because of their long-lasting effects.

But other insecticides are also problematic. B. consists in disturbing the molting of insects. Thus, both Juvenil - hormones and ecdysteroid -hormones, which often seems as ecologically more acceptable means have been proposed for. B. ground beetles , predatory bugs and other predatory insects, in addition to the lack of food, damage their development as well as their herbivorous prey. Due to the third Volterra rule, they lead to long-term damage to the biological balance and natural pest control by predators is prevented.

## Further statements

As Volterra notes in his book, even further statements follow from the mathematical formulations of the Lotka-Volterra equation:

• If only the predator population is decimated, the mean value of the prey population increases for a short time. The long-term mean of the predator and prey population remains unchanged despite the decimation.
• If only the prey population is decimated (and the decimation rate remains below the reproduction rate), the mean value of the predator population is reduced for a short time. The long-term mean of the predator and prey population remains unchanged despite the decimation.

## literature

• Vito Volterra: Leçons sur la théorie Mathématique de la lutte pour la vie. Éditions Jaques Gabay, Paris 1990, ISBN 2-87647-066-7 (Authorized reprint of the original edition published by Gauthier-Villars in 1931. The three laws are dealt with on pages 15-27.)
• Hans Joachim Müller (Ed.): Ecology (=  UTB . Volume 1318 ). 2nd revised edition. Gustav Fischer, Jena 1991, ISBN 3-334-00398-1 , p. 224 .