Intraspecific variation stabilizes classic predator-prey dynamics

In 1920, Alfred J. Lotka found that, to his "considerable surprise", the dynamics of a simple predator-prey model he had devised led "to undamped, and hence indefinitely continued, oscillations"---which he thought epitomized the "rhythm of Nature" dear to the Victorians. In 1926, the same model was proposed independently by mathematician Vito Volterra, who was inspired by the work of his son-in-law, fish biologist Umberto D'Ancona. For over a century, the equations that now bear their names have served as a template for the development of sophisticated models for population dynamics. Coexistence in this classic predator-prey model is fragile---stochasticity or temporal variability in parameter values result in extinctions. The dynamics can be stabilized by intraspecific competition or other forms of self-regulation, but the prevalence of these processes in large food webs has been questioned. Here we show that when we consider populations characterized by intraspecific variability, dynamics are stable---despite the absence of any direct self-regulation. Our results can be generalized further, defining a new class of consumer-resource models. By accounting for intraspecific variation, which is manifest in all biological populations, we obtain dynamics that differ qualitatively and quantitatively from those found for homogeneous populations---challenging a central assumption of many ecological models.

gauseR: Simple methods for fitting Lotka-Volterra models describing Gause’s “Struggle for Existence”

The ecological models of Alfred J. Lotka and Vito Volterra have had an enormous impact on ecology over the past century. Some of the earliest – and clearest – experimental tests of these models were famously conducted by Georgy Gause in the 1930’s. Although well known, the data from these experiments are not widely available, and are often difficult to analyze using standard statistical and computational tools. Here, we introduce the gauseR package, a collection of tools for fitting Lotka-Volterra models to time series data of one or more species. The package includes several methods for parameter estimation and optimization, and includes 42 datasets from Gause’s species interaction experiments and related work. Additionally, we include with this paper a short blog post discussing the historical importance of these data and models, and an R vignette with a walk-through introducing the package methods. The package is available for download at github.com/adamtclark/gauseR. To demonstrate the package, we apply it to several classic experimental studies from Gause, as well as two other well-known datasets on multi-trophic dynamics on Isle Royale, and in spatially structured mite populations. In almost all cases, models fit observations closely, and fitted parameter values make ecological sense. Taken together, we hope that the methods, data, and analyses that we present here provide a simple and user-friendly way to interact with complex ecological data. We are optimistic that these methods will be especially useful to students and educators who are studying ecological dynamics, as well as researchers who would like a fast tool for basic analyses.

Economic and Mathematical Model for Size and Structure Optimisation of Predator and Prey Populations

The paper proposes an original economic and mathematical model for size and structure optimisation of Predator and Prey populations. The most well-known mathematical model in biology for periodical dynamics of antagonistic animal species was developed independently by Alfred Lotka and Vito Volterra. This classical mathematical Predator-Prey model is known as the Lotka-Volterra model.

MATHEMATICS IS THE LANTERN: VITO VOLTERRA, LÉON WALRAS, AND IRVING FISHER ON THE MATHEMATIZATION OF ECONOMICS

The interdisciplinary project to unite the field of mathematics with the social and biological sciences marks the work of Vito Volterra, one of Italy’s most prominent mathematicians of the twentieth century. This paper explores the connections between Volterra’s 1901 inaugural address at the Accademia dei Lincei in Rome and the work of two of his contemporaries, Léon Walras and Irving Fisher. All three thinkers were ardent advocates of the mathematical turn in economic thinking. This paper argues, however, that it is the previously unexplored relationship between Volterra and Fisher that sheds the most light on the way in which mechanical physics contributed to the project of mathematization within economics more generally. Furthermore, it explores the way in which mathematical inquiry postulated a new and coherent abstraction of the economy, at the same time that it gave epistemological authority to the economist.