Exploring the Dynamics of Lotka-Volterra Systems: Efficiency, Extinction Order, and Predictive Machine Learning

TL;DR

A random forest model and a neural network model are developed that are able to predict the number of extinctions that would occur without the need to simulate the dynamics of dynamics on Lotka-Volterra ecological systems.

Abstract

For years, a main focus of ecological research has been to better understand the complex dynamical interactions between species which comprise food webs. Using the connectance properties of a widely explored synthetic food web called the cascade model, we explore the behavior of dynamics on Lotka-Volterra ecological systems. We show how trophic efficiency, a staple assumption in mathematical ecology, produces systems which are not persistent. With clustering analysis we show how straightforward inequalities of the summed values of the birth, death, self-regulation and interaction strengths provide insight into which food webs are more enduring or stable. Through these simplified summed values, we develop a random forest model and a neural network model, both of which are able to predict the number of extinctions that would occur without the need to simulate the dynamics. To conclude, we highlight the variable that plays the dominant role in determining the order in which species go extinct.

Figures (12)

• Figure 1: A random realization of a cascade model food web with 50 species. Nodes represent uniquely labeled species, and arrows point from predators to prey.
• Figure 2: The resulting stable food web after applying Lotka-Volterra dynamics (Equation (\ref{['equ:L-V']})) to the cascade food web depicted in Figure \ref{['fig:cascade']}. Although the original cascade food web had 50 species, the dynamics induced numerous species extinctions. In this reduced, stable food web, only 26 of the original species have survived. The stable food web includes surviving basal species shown in red, top predators in blue, and intermediate predators in green. This extinction phenomena is general, and can be seen for different cascade food webs of different sizes and for different parameter values in the dynamics.
• Figure 3: Average number of extinctions, $\bar{E}$, as a function of efficiency, $e$, for LVCM food webs of 50 species. The average is computed for (a) 30 realizations, and (b) 100 realizations of food webs and associated Lotka-Volterra rates.
• Figure 4: Ordering of the 120 clusters according to the number of extinctions occurring in each cluster. Each row represents an inequality by displaying the order of the absolute rate sums, each of which is associated with a specific color. Clusters with fewer extinctions are located at the top of the figure, and as one descends, the clusters have an increasing number of extinctions.
• Figure 5: Expected value of the efficiency proxy, $e_p$, for each of the 120 clusters. The cluster ordering is the same as was used in Figure \ref{['fig:order']}.