predator–prey equations
Predator-prey relationships are one aspect of food chains or food webs, which are analyzed in field ecology. The technical term predator includes not only the real robbers and parasites, parasitoids and grazers. The model developments described below in ecology can be applied in principle to all four cases. Excluded are the theories, however, all types, which differ from dead organic matter to feed (eg, scavengers, detritus, decomposers ), because in this case there can be no biological response of the prey population.
In nature, many complex response patterns exist in the relationship between predator and prey, their explanation is an essential, central field of ecological theory. Due to the variety of different relationships, the transfer from one system to another is difficult. In some cases, a robber decimated a prey animal population to a fraction of their unaffected density, in other cases the effect of a predator on a prey population is barely detectable. Essentially this involves, first, whether a robber specializes in a certain prey species, or whether it is a generalist with numerous equivalent prey species, between these two extremes is a wide range of cases of different preference. Secondly, effects of other types and interactions with environmental factors are always significant.
Particularly interesting for the ecological analysis are systems in which the robber is the density of its prey regulated, or in which you obtained the density of both cyclical fluctuations. As a rule, many other factors such as food supply, climate, habitat competition, pathogens, parasites, stress and other predators also affect the population sizes (see population dynamics ).
With the goal of providing general dynamic properties of predator-prey relationships and to investigate various mathematical models have been created in theoretical biology. The easiest and most well known is the Lotka - Volterra model. Basis are the works of the Austrian mathematician Alfred J. Lotka and the Italian mathematician and physicist Vito Volterra, independently formulated in 1925 and 1926, the now named after them Lotka -Volterra equations. Is mathematical equations, where the quantitative aspect of the population dynamics for the first time as a function of time. They are based on the logistic equation. The biological applications of these equations are known today under the name of the first, second and third Lotka -Volterra rule.
A computer simulation, which makes the predator-prey relationship clearly, is the simulation Wator by Alexander K. Dewdney and David Wiseman.
The Lotka - Volterra model
In the Lotka - Volterra model show predator and prey species coupled frequency fluctuations. Simplified: there is much booty, takes the population of the predator to, then the prey is rare, the robbers can not find a more adequate food and to be rare, the prey population to recover, etc., etc. In the model, however, there are so-called neutral stable cycles. This means that the cycles occur without external influences, the cycle length is determined by the choice of variables ( without timer ), without interference from outside these cycles would continue without any deviation forever. But: In natural systems actually observable cycles can normally due to this mechanism does not occur, due to the inevitable and always acting fluctuations of environmental variables had populations that are subject to the model dynamics, fluctuate acyclic and erratic in reality. Populations whose variations could be explained solely by the model, there is not likely. Nevertheless, the model is useful as a first approximation to the explanation of coupled oscillations.
The most famous case in which coupled to the population of a robber and his prey, delayed cycles have been actually observed in nature, the cycles of the snowshoe hare ( lepus americanus ) and its predator, the Canada lynx (Lynx canadensis). The species are show over a vast area (a large part of northern North America, from Alaska to Newfoundland ) a cycle of about ten years length ( actually observed: 9-11 years). This example was even included in school books. Originally interpreted as a particularly striking example of an oscillation from the Lotka -Volterra type, are, according to recent studies, the conditions here much more complicated. High hare populations seem to break together mainly by lack of food. Almost but not food as such is here ( the bunnies eat their habitat rather than bald), but good food with high nutritional value. The grazed plants can form under heavy grazing stomach poisons (toxins ) and are thus less well fressbar for the rabbits. They form this ( energetically costly ) toxins but only if high grazing pressure is. The interaction of the predators snowshoe hare and its prey, the green plants seems to drive the cycle here. The lynx follows therefore after only passive. This example (which would be no way to the last detail actually cleared up! ) Clearly shows that one should guard against simple explanations of complex issues, even if they seem to fit well into the model used to explain.
Other models
Investigations, which the American zoologist and ecologist Paul Errington (1946 ) carried out for the predator-prey relationship between muskrats and mink, show a completely different behavior. Thus, the Mink is indeed the most important predators of the muskrat, the population size of the muskrat, however, less influenced by the number of their predators than by stocking density of the territory. Especially wandering animals without territory or injured animals are prey of mink. So it will be killed preferred the individuals who have already had the slightest probability of survival. The population size of the prey is so limited in this case by the eco-factor robbers on a regulated density, which is determined by the ecofactors food and space for creating building. Comparable cases have been very commonly found in other studies.