gauseR: Simple methods for fitting Lotka-Volterra models describing Gause's "Struggle for Existence"

Abstract

Point 1: 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 1930s. Although well known, the data from these experiments are not widely available and are often difficult to analyze using standard statistical and computational tools. Point 2: 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. Point 3: 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. Point 4: 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.

Keywords: competition; differential equation; growth rate; logistic growth; optimization; predator‐prey.

FIGURE 1

Figure comparing per‐capita growth rate (following equation 1b) versus abundance for Paramecium aurelia grown in monoculture. Note that when plotted in this manner, the parameter values for the Lotka‐Volterra equations can be easily “read” off the graph, which r as the y‐intercept, K as the x‐intercept, and a_ii as the slope. In multi‐species systems, interaction coefficients a_ij (i.e., effect of species j on species i) can be similarly computed based on the slope of dN_i/N_idt versus N_j

FIGURE 2

Logistic growth for size standardized volume of Paramecium caudatum grown in monoculture on Osterhout medium over 24 days, from Gause (1934a). Points show observations, and line shows fitted logistic growth curve, calculated with the get_log() function, that can be used to fit the logistic growth function to time series of an individual species. Goodness of fit shows results from the tests_goodness_of_fit() function. Recall that values near 1 imply a close correspondence between observations and predictions. See Table 2a for parameter values

FIGURE 3

Lotka‐Volterra competition for size standardized volume of Paramecium caudatum and Paramecium aurelia grown in mixed population over 24 days, from Gause (1934a). Points show observations, and lines show growth curves fitted using the lv_interaction() function, which can be used to simulate dynamics in multi‐species mixtures following the classic Lotka‐Volterra equations. See Table 2a for parameter values

FIGURE 4

Predator‐prey interactions between Didinium nasutum and Paramecium caudatum grown in mixture over 17 days, from Gause (1934a). Additional individuals of both species were added to the mixture periodically to prevent local extinction. Points show observations, and lines show fitted growth curves. Top panel shows results for model fitted using linear regressions of species abundances versus per‐capita growth rate, whereas bottom panel shows results for parameters fitted using the simulated differential equations using the lv_optim() function. Note that goodness of fit is varies substantially between the two methods. See Table 2c for parameter values

FIGURE 5

Multi‐trophic dynamics for wolves, moose, and fir trees on Isle Royale from 1960 to 1994, from McLaren and Peterson (1994). Points show observations, and lines show growth curves fitted using the lv_optim() function. The left axis shows wolf and moose abundances, separated by a ten‐fold scaling difference for easier visualization, whereas the right axis shows tree growth increments. Two methods for testing goodness of fit are shown. In the legend, values show univariate tests for each species, whereas “Total GOF” shows fit when considered across all observations simultaneously. Note that these two methods of comparison can lead to very different conclusions. See Table 2d for parameter values

FIGURE 6

Predator‐prey interactions between Eotetranychus sexmaculatus and Typhlodromus occidentalis in a spatially structured experiment carried out on a grid of oranges over 60 weeks, from Huffaker et al. (1963). Points show observations, and lines show growth curves fitted using the lv_optim() function. Top panel shows dynamics for a system where neither species directly inhibits its own growth (i.e., a_ii = a_jj =0), whereas bottom panel is for a system where only the predator directly inhibits its own growth. Note that neither option is able to fully capture the realized dynamics, and that goodness of fit is relatively low in both panels. See Table 2e for parameter values