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From Lotka–Volterra to Arditi–Ginzburg: 90 Years of Evolving Trophic Functions

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Abstract

The trophic function, often called a “functional response,” determines qualitative properties in models of predator–prey dynamics. Many of the theoretical and empirical studies have demonstrated the problem of choosing and fitting a trophic function to experimental data. Notably the publication by Arditi and Ginzburg (1989) has stimulated a lively debate in scientific literature regarding the application of trophic functions. The authors highlighted contradictions between the observed dynamics of natural ecosystems and the qualitative properties of predator–prey models using Holling-type trophic functions. They suggested revising the theoretical models by means of the trophic functions that depend on the ratio of prey to predator abundances. By comparing this and other proposed trophic functions, we demonstrate that the Arditi–Ginzburg function offers the simplest way of accounting for mutual interference in predator–prey models. This trophic function effectively resolves contradictions between theoretical models and natural systems, including the paradox of enrichment, paradox of biological control, and the paradoxical enrichment response mediated by trophic cascades. We review the debate regarding the Arditi–Ginzburg model and present recent results obtained with continuous and individual-based models of animal foraging behaviour in predator–prey systems, which explain predator interference as an emerging property.

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ACKNOWLEDGMENTS

The authors are grateful to L.R. Ginzburg and D.O. Logofet for insightful comments and stimulating discussions. We also express our gratitude to Roger Arditi for his friendship, for the long-lasting fruitful collaboration, and for attracting our interest to studying of mechanisms causing emergence of the predator interference in natural trophic systems.

Funding

The research was funded by the project 0256-2019-0038 (state reg. no. 01201363188) of SSC RAS “Development of GIS-based methods of modelling marine and terrestrial ecosystems” (for Tyutyunov), by the basic part of the state assignment research, project 1.5169.2017/8.9 of the Southern Federal University “Fundamental and applied problems of mathematical modelling” (for Titova), and Russian Foundation for Basic Research, project no. 18-01-00453 A “Multistable spatiotemporal scenarios for population models” (for Tyutyunov).

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Authors and Affiliations

  1. Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS), 344006, Rostov-on-Don, Russia

    Yu. V. Tyutyunov

  2. Southern Federal University (SFedU), 344090, Rostov-on-Don, Russia

    Yu. V. Tyutyunov & L. I. Titova

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  1. Yu. V. Tyutyunov
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Correspondence to Yu. V. Tyutyunov or L. I. Titova.

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EDITOR’S AFTERWORD

EDITOR’S AFTERWORD

Term “trophic function” has naturally appeared in mathematical theory of trophic chains by Yu.M. Svirezhev (see Ch. 5 in Svirezhev and Logofet, 1978, 1983) as the designation of function \(g\) (\(V\) in the original notations) in system (1) of differential equations that describes how the predator/consumer consumption rate depends on the prey/resource abundance (\(N\)). In English literature of those years, this function had an already established name “functional response” of predator (to the prey population density, Solomon, 1949). The result of predator’s trophic activity in terms of predator population had a name “numerical response” (Solomon, 1949), designating, as it might seem, an independent process. According to the opinion of Ginzburg and Arditi (Ginzburg, 1998; Arditi and Ginzburg, 2012) and of the authors of this review, the difference of these principal attributes had later caused the models to appear that violated physical conservation laws.

In the theory of Yu.M. Svirezhev, whose background was physics, the conservation law holds by default: the same trophic function (with the conversion coefficient \(k\)) enters the consument equation so that quantity \(\left( {1 - k} \right)\) represents the energetic loss in consumption. However, it is not the conservation law that makes Svirezhev’s theory original—his innovation was rather in replacing the growth term in the primary resource (\(R\) in original notations) equation with a constant flux (\(Q\)) of nutrients entering the system of several trophic levels.

The existence and stability conditions for equilibria in the trophic chain model were then analysed as functions of \(Q\) and other parameters (Svirezhev and Logofet, 1978, 1983). It was found that if there existed a stable positive equilibrium \(\left( {R^{*},~N_{1}^{*},N_{2}^{*}, \ldots ,~N_{n}^{*}} \right)\) corresponding to a trophic chain of n levels, then all equilibria with the number of levels less than \(n\) became unstable. Furthermore, there are no partially positive equilibria with zero population densities below the nth level. Moreover, there are threshold values of parameter \(Q\): \(Q_{1}^{*},Q_{2}^{*},Q_{3}^{*}, \ldots ~\) subdividing the virtual Q-axis into a set of non-overlapping intervals within which there exist stable chains with one, two, three,… trophic levels (Fig. 7). In other words, if in a stably functioning trophic chain the flux of nutrients reduces for some reason to a value of \(Q\) from a neighboring interval, then the top trophic level loses its stability and goes to extinction; if the flux \(Q\) increases in similar manner, then the higher trophic level can be occupied by an appropriate top predator.

Fig. 7.
figure 7

Results of mathematical theory of trophic chains by Yu.M. Svirezhev. See text for details.

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Thus, the mathematical results comply with biological understanding of trophic chain structures and functioning, and the theory of Yu.M. Svirezhev is an exempt from “trophic” paradoxes mentioned in this review. The theory was built for trophic functions depending on the prey/resource abundance or density, N. The question is whether the theory will retain its elegance with the RD trophic functions.

Dmitrii O. Logofet

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Tyutyunov, Y.V., Titova, L.I. From Lotka–Volterra to Arditi–Ginzburg: 90 Years of Evolving Trophic Functions. Biol Bull Rev 10, 167–185 (2020). https://doi.org/10.1134/S207908642003007X

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