Early tests of the Lotka Volterra models

According to the Lotka-Volterra equations, the response of a predator population to an increase in a prey population is to increase its own numbers. This increase may be through an increase in the birth rate of the predator (perhaps combined with a decrease in death rate) or through immigration. Again, this is termed a numerical response. According to the Lotka-Volterra equations, the result of this increase in predation is a coupled numerical response in the prey population, which declines. Once the prey population has declined sufficiently, the negative consequences for the predator population results in its decline. Once predator numbers have decreased sufficiently, the prey population begins to recover, leading eventually to an increase in the predator population, and so on. A graph of the Lotka-Volterra results versus time look like a stable limit cycle (Fig. 10.1), though it is not. In a true limit cycle, if the populations are pushed out of the cycle by density-independent factors, the populations return to the original limit cycle. Due to the neutral stability of the Lotka-Volterra equations, they would have no return tendency.

The Lotka-Volterra equations led some ecologists to adopt the view that predator-prey interactions were "inherently oscillatory," and research was directed to test this proposition. As discussed previously, some parasite-host and various small mammal and bird populations in boreal and arctic regions display regular oscillations in number. Do predators cause these cycles? Already in 1924 Charles Elton had published "Periodic fluctuations in the numbers of animals: their causes and effects." In this and later publications (Elton and Nicholson 1942) he presented the now famous and infamous data on the populations of snowshoe hare (Lepus americanus) and the lynx (Lynx canadensis). As in the case of the Kaibab deer, these data were not based on systematic population surveys, but rather on the numbers of pelts brought into the Hudson's Bay Company by trappers. The data do show regular fluctuations of great magnitude, but we must ask, how reliable are these data? Are the fluctuations due to predator-prey interactions as envisioned by Lotka and Volterra? We will not try to answer these questions now, but note here that after 70 years of field experiments and time-series analysis, Krebs et al. (2001) concluded that the hare cycle can only be understood as an interaction involving the hare population, its food supply, and a community of predators (not just the lynx).

As discussed earlier, the competition equations of Lotka and Volterra were tested in the laboratory by Gause (1934). Gause also tried to test the predictions of the Lotka-Volterra predator-prey equations using microorganisms. He attempted to produce the predicted oscillations using as his prey populations of Paramecium caudatum grown in test tubes. To these tubes he introduced another ciliated protozoan, Didinium nasutum. Didinium is a voracious predator on Paramecium and it reproduces by binary fission, just as does its prey. In the simple test-tube environment Didinium was able to hunt down all of the Paramecium. Once its food supply was gone, Didinium starved. Thus, in any one tube, mutual extinction was assured. Gause next tried adding sediment to the bottom of the tubes as a refuge for Paramecium. This ensured the survival of the Paramecium, but the Didinium population eventually went extinct. With its predator eliminated, the Paramecium population rapidly grew to the expected carrying capacity. But Gause had more tricks. He now added one Paramecium and one Didinium every third day to each test tube. This finally resulted in coexistence of the prey and its predator for more than two weeks. Both the prey and the population went through two oscillations during this period.

Did Gause see his work as confirming the equations developed by Lotka and Volterra? Just the opposite. Gause stated that predator-prey interactions are not inherently oscillatory, and that coexistence was possible only through adding heterogeneity to the simple test-tube environment, or through constant interference of the system through the addition of immigrants.

Another early laboratory experiment illustrates the weaknesses in the simple Lotka-Volterra model. In Chapter 5 on metapopulations we described the work of Huffaker (1958), who was a California entomologist interested in biological control of pests in orange orchards. The prey species was the six-spotted mite (Eotetranychus sexmaculatus), which feeds on oranges. The predator was a carnivorous mite (Typhlodromus occidentalis) which preys on the six-spotted mite. Both species reproduce rapidly through parthenogenesis. In each experiment, Huffaker began with 20 prey females and introduced two predator females 11 days later.

In one experiment, Huffaker concentrated the food (oranges) in one area. The results mirrored those of Gause. The prey population rapidly increased, then was located by the predators, which also rapidly increased. Within a short time (25-30 days) both populations were extinct. Huffaker then began creating a heterogeneous environment. He set up a complex laboratory environment consisting of three 40-cell trays with a total of 120 feeding positions. Although each position contained one orange, he controlled the feeding surfaces by dipping the oranges in wax, leaving only 5% of the orange available for feeding. This forced the herbivorous mite to constantly seek out new feeding surfaces. He added small wooden pegs as launching pads for the six-spotted mites to speed their dispersal from one orange to another. And he added a maze of Vaselineā„¢ barriers across the trays to slow the dispersal of the predatory mites, which could travel only by foot. Once a predator arrived on an orange already colonized by the prey species, it quickly killed and consumed all of the herbivorous mites on that particular orange. But the rapid immigration and emigration of the herbivorous mite, along with the complex, heterogeneous environment created by Huffaker, allowed the two species to coexist in this laboratory environment for over 200 days.

In both of these laboratory studies, the coexistence of the prey with the predator depended upon environmental heterogeneity. Secondly, both systems required regular immigration of the prey and/or the predator to avoid extinction. Neither of these requirements was anticipated by the Lotka-Volterra approach.

Another instructive laboratory experiment was that of Utida (1957). In this case stable oscillations between the azuki bean weevil (Callosobruchus chinensis) and a parasitic wasp

(Heterospilus prosopidis) were maintained for more than 25 generations in a laboratory Petri dish (1.8 cm high by 8.5 cm diameter). The wasp is actually a specialized predator known as a parasitoid, which paralyzes its prey without killing it. The wasp lays an egg on the paralyzed host, usually an insect larva. The egg hatches and the wasp larva slowly consumes the host, leading to its eventual death. In this case, the wasp only lays eggs on certain instar larvae. The wasp does not parasitize adult or pupal beetles. Therefore, although these two populations undergo the regular oscillations of a limit cycle, both populations persist due to the nonrandom predation by the wasp. Again, the violation of the Lotka-Volterra assumption of random hunting is what allows the coexistence of prey and predator.

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