Comparison Between Lotka-Volterra and Multivariate Autoregressive Models of Ecological Interaction Systems

Abstract

Lotka-Volterra (LV) and Multivariate Autoregressive (MAR) models are computational frameworks with different mathematical structures that have both been proposed for the same purpose of extracting governing features of dynamic interactions among coexisting populations of different species from observed time series data. We systematically compare the feasibility of the two modeling approaches, using four synthetically generated datasets and seven ecological datasets from the literature. The overarching result is that LV models outperform MAR models in most cases and are generally superior for representing cases where the dependent variables deviate greatly from their steady states. A large dynamic range is particularly prevalent when the populations are highly abundant, change considerably over time, and exhibit a large signal-to-noise ratio. By contrast, MAR models are better suited for analyses of populations with low abundances and for investigations where the quantification of noise is important. We conclude that the choice of either one or the other modeling framework should be guided by the specific goals of the analysis and the dynamic features of the data.

Methods of quantifying interactions among populations using Lotka-Volterra models

“…We also discussed two newer algebraic LV inference (ALVI) methods that are based on linear algebra. Their theoretical underpinnings were previously documented by Xiao et al (Xiao et al, 2017) and practically applied in detail to different datasets in (Olivença et al, 2021). These methods do not always capture the transient dynamics as well as gradient methods, which is likely due to imprecision when estimating slopes, but they avoid overfitting and are computationally so cheap that numerous fits are readily computed, either to select the solution with the lowest value of some loss function or to establish an entire ensemble of wellfitting solutions.…”

“…Other common smoothing techniques include splines and the LOESS method (Cleveland, 1979). For more information on smoothing see (Olivença et al, 2021). In general, there is no strict criterion guiding the decision when smoothing is necessary or beneficial.…”

“…We should select a representative (spread-out) sample, so we choose points 1, 2, 3, 8, and 9. We can use the LOESS method to smooth our slopes, as shown in (Olivença et al, 2021), which gives the matrix slightly different values than the data points in Table 4. This step is not necessary for the method to work but may yield slightly better inferences.…”

“…Each variable was optimized individually using an interior point optimization solver (Biegler and Zavala, 2009) in the JuMP package (v. 1.1.1) (Dunning et al, 2017). Smoothing was performed with the LOESS method as described in (Olivença et al, 2021).…”

Identification of Parameters of Conservative Multispecies Lotka-Volterra System Based on Sampled Data

“…The initial data are as follows (it is chosen similar to [19,20]): Note that, as mentioned at the beginning of this paper, the matrix A does not have to be skew-symmetric, for it is enough that it can be transformed to such a form (in this case we can multiply the rows of the matrix by the coefficients). The skew-symmetricity requirement was introduced to shorten the notation and to make it easy to read.…”