Experimental Demonstration of Volterra’s Periodic Oscillations in the Numbers of Animals

In these equations b₁ is the birthrate of the prey, and d₂ is the deathrate of the predator. The rate of increase of the population of the prey is decreased by the rate at which the prey is consumed by the predator (k₁N₁N₂) and if the rate of reproduction of the predator depends on the number of prey available for consumption the net rate of increase of the predator will be given by the difference of two factors one of which depends on the number of prey available (k₂N₁N₂) and another which represents the death rate of the predator population (d₂N₂). The actual rate of increase of the population of either species can be positive or negative and a generalised solution of the equations (1) shows that, under the conditions laid down, a series of periodic fluctuations may be expected in the values of N₁ and N₂.

It is, however, necessary to consider a second type of fluctuation which is aperiodic. For example, an epidemic cannot start in a population containing a large number of immune individuals but if the concentration of the latter diminishes owing to a loss of immunity or the infiltration of non-immune individuals, then an epidemic is liable to start when a critical threshold of non-immune individuals has been reached. Once the epidemic has started it will continue until the original state of the immune population has been again reached—and the cycle will be repeated. This type of sudden fluctuation is obviously different from that discussed by Lotka and Volterra¹. It has been shown experimentally (Gause, 1934) that in a population of two infusoria (Didinium nasutum as predator, and Paramecium caudatum as prey) the interaction of the two species is aperiodic and leads to a complete disappearance of both species. If immigration is allowed from external sources, fluctuations of the second theoretical type occur analogous to those described by Topley and Green-wood (Greenwood, 1932). The present paper on the other hand deals with populations which exhibit fluctuations of the type considered by Lotka and Volterra.

The first set of experiments were performed with Paramecium bursaria and the yeast Schizosaccharomyses pombe. The culture medium had the following composition:

This stock solution was diluted 225 times with doubly distilled water before use. The yeast was cultivated on solid beer wort in 1·5 per cent, agar in a Petrie dish. A fixed amount of yeast was removed by a platinum loop and placed in the salt solution containing the Paramecium. As Paramecium (1928) has pointed out, S. pombe is a perfectly satisfactory food for Paramecium bursaria. In order to avoid an accumulation of waste products the yeast and Paramecium were transferred to fresh salt solution every other day by means of a centrifuge. During the course of an experiment the yeast was prevented from settling on the bottom of the container by passing air bubbles through the salt solution. In order that the system should exhibit periodic fluctuations of the Lotka-Volterra type it is necessary that the yeast should be capable of reproduction. For experimental purposes it was convenient to control the density of the yeast population by doubling (or increasing 1·5 times) the unconsumed population of yeast cells at fixed intervals of time (one day). Alternatively the density of Paramecium could be varied.

Fig. 1 represents the results of one of these experiments carried out in an Ehrlen-meyer flask containing 30 c.c. of the salt solution (at a temperature of 25° C.). The curve for P. bursaria shows the change in the number of individuals in 0·5 c.c. and that of S. pombe, the change in the number of cells in a large square of a Buerker-Reichert camera ⁠. The coefficient of multiplication of yeast was such that b₁= 0·65 (with 24 hours as a unit of time), and the coefficient of mortality (d₂) of P. bursaria was 0·32. Quite clearly periodic fluctuations of the Lotka-Volterra type occurred.

It will be noted that the first cycle in these experiments was larger than the subsequent cycles. This seems to be associated with the fact that a relatively large number of Paramecium (24 individuals per 0-5 c.c.) were introduced.

The relative values of the density of population of Paramecium (N₂) and of the yeast N₁ are shown in Fig. 2. A similarity to the theoretical curves of Lotka and Volterra is obvious.

From data of the type here presented it is possible to calculate the values of the constants k₁ and k₂ in equation (1). This has been done in Table I.

It will be noticed that the values of the constants do not change markedly over wide ranges of variation of population density; at the same time some variation undoubtedly occurs and requires further investigation.

Similar periodic oscillations have been observed in mixed populations of Paramecium aurelia and Saccharomyces exiguus. In this case the Osterhout salt solution was buffered (3 c.c. M/20 KH₂PO₄ + 100 c.c. of water + M/20 KOH, giving pH = 7-5). The coefficient of multiplication of the yeast (b1) had an average value of 1·0, and the coefficient of daily dilution of Paramecium (d₂) was 0·45. The system was incubated at 25° C. in 25 c.c. of liquid, and the culture was stirred by gentle rotation of the flask. There appeared to be a slight deficiency of oxygen in these experiments and the rate of mortality of the Paramecium was somewhat high. As is shown in Figs. 3 and 4, however, typical fluctuations were observed whose period was approximately constant. Fig. 4 shows that the magnitude of the fluctuations was also approximately the same in all cases.

It now seems fairly clearly established that periodic fluctuations of the Lotka-Volterra type actually occur under controlled experimental conditions and it is of interest to consider the fluctuations observed by Cutler (1923) and others (Russell, 1927). In these cases it looks as though essentially the same factors were at work (Nicholson, 1933). The bacteria in the soil did not fluctuate in numbers when grown in sterilised soil, but rose to high population densities and remained at those levels. As soon as amoebae were introduced the number of bacteria fell but no constant level was reached, the numbers fluctuating markedly—with an inverse proportionality between the number of bacteria and amoebae. It is difficult to avoid the conelusion that the system is definable as a periodic fluctuation of the Lotka-Volterra type.

It is of interest to note, on the other hand, that recent observations on the vertebrates does not reveal similar fluctuations (Sewertzoff, 1933 ; Jensen, 1933). Further investigation of these cases would be of interest.

In conclusion I wish to express my sincere thanks to Dr W. W. Alpatov for interest in the present investigation.