Methods of quantifying interactions among populations using Lotka-Volterra models

Abstract

The Lotka-Volterra (LV) model was introduced in the early 20th Century to describe predator-prey systems. Since then, the model has been expanded to capture the dynamics of numerous types of interacting populations and to include the effects of external factors from the environment. Despite many simplifying assumptions, the LV approach has proven to be a very valuable tool for gaining insights into the dynamics of diverse biological interaction systems. In particular, recognizing the critical importance of microbiomes for human and environmental heath, LV systems have become effective tools of analysis and, indeed, the default for quantitatively assessing interactions within these large microbial communities. Here we present an overview of parameter inference methods for LV systems, specifically addressing individuals entering the field of biomathematical modeling, who have a modest background in linear algebra and calculus. The methods include traditional local and global strategies, as well as a recently developed inference method based strictly on linear algebra. We compare the different strategies using both lab-acquired and synthetic time series data. We also address a recent debate within the scientific community of whether it is legitimate to compose large models from information inferred for the dynamics of subpopulations. In addition to parameter estimation methods, the overview includes preparatory aspects of the inference process, including data cleaning, smoothing, and the choice of an adequate loss function. Our comparisons demonstrate that traditional fitting strategies, such as gradient descent optimization and differential evolution, tend to yield low residuals but sometimes overfit noisy data and incur high computation costs. The linear-algebra-based method produces a satisfactory solution much faster, generally without overfitting, but requires the user to estimate slopes from the time series, which can introduce undue error. The results also suggest that composing large models from information regarding sub-models can be problematic. Overall, there is no clear “always-best method” for inferring parameters from data, and prudent combinations may be the best strategy.