Trophic function

A trophic function is defined in the Kolmogorov predator-prey system of equations, also called the Kolmogorov predator-prey model. It generalizes the linear case of predator-prey interaction firstly described by Volterra and Lotka in the Lotka-Volterra equation. A trophic function represents the power of the predator to consume the prey assuming a given number of predators. The trophic function (also referred to as the functional response) was widely applied in chemical kinetics, biophysics, mathematical physics and economics. In economics, "predator" and "prey" become various economic parameters such as price and output (number of produced goods), outputs of the various linked sectors (such as processing and supply sectors). These relationships, in turn, were found to have a similar character as the relationships of magnitudes in chemical kinetics, where the chemical analogues of predators and prey actually interact with each other.

The universal character of the findings in various sciences has established that the trophic functions, as well as the predator-prey models themselves, have an inter-disciplinary essence. In other words, it is remarkable that the dynamics of objects of different natures obey the same principles in their temporally-evolving dynamics, and may be analyzed using results already discovered in other sciences. Among the features of the trophic function which have attracted scholars' attention is its ability to serve as a tool in forecasting temporarily stable conditions (limit cycles and/or attractors) of the coupled dynamics of predator and prey. According to the Pontryagin L.S. theorem on the inflection points of the trophic function, there exists a limit cycle in the coupleded dynamics of predator and prey.

The importance of the trophic function seems high in chaos circumstances when one has numerous temporally changing magnitudes and objects, as is particularly true in global economics. To define and forecast the dynamics in this case becomes highly difficult and mostly impossible with linear methods. The trophic function is a tool which allows one to use non-linear dynamic analysis and discover limit cycles or attractors in the dynamics of the objects. Since in nature there exist only temporarily stable objects (otherwise there would be no dynamics) then there must be limit cycles and attractors in the dynamics of naturally existing and observed objects (chemistry, flora and fauna, economics, universe dynamics). It leads to the conclusion that we haven't yet found these specific features of these objects, but they are nevertheless behind the dynamics of the various objects surrounding us.

Despite the success already achieved in the research on trophic functions, the work in this direction is only just beginning, while their importance in applications is without doubt. Global economics, for instance, needs tools to be developed in order to be able to forecast the dynamics of output and price over a scale of at least 3–5 years, in order to have stable demand and not over-produce and to exclude circumstances such as those that created the 2008 financial crisis.

References

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