[Paper Review] Extinctions as a vestige of instability: the geometry of stability and feasibility

Stav Marcus, Ari M. Turner+1 more|arXiv (Cornell University)|May 18, 2024

Earth Systems and Cosmic Evolution4 citations

TL;DR

This paper investigates the geometric structure of coexistence in Lotka-Volterra ecosystems with symmetric competitive interactions, showing that feasibility—positive species abundances—is generically the first condition to break during parameter changes, while stability loss occurs only at isolated points. The key finding is that extinction arises from feasibility failure before stability breakdown, revealing feasibility as the more restrictive constraint in coexistence dynamics.

ABSTRACT

Species coexistence is a complex, multifaceted problem. At an equilibrium, coexistence requires two conditions: stability under small perturbations; and feasibility, meaning all species abundances are positive. Which of these two conditions is more restrictive has been debated for many years, with many works focusing on statistical arguments for systems with many species. Within the framework of the Lotka-Volterra equations, we examine the geometry of the region of coexistence in the space of interaction strengths, for symmetric competitive interactions and any finite number of species. We consider what happens when starting at a point within the coexistence region, and changing the interaction strengths continuously until one of the two conditions breaks. We find that coexistence generically breaks through the loss of feasibility, as the abundance of one species reaches zero. An exception to this rule - where stability breaks before feasibility - happens only at isolated points, or more generally on a lower dimensional subset of the boundary. The reason behind this is that as a stability boundary is approached, some of the abundances generally diverge towards minus infinity, and so go extinct at some earlier point, breaking the feasibility condition first. These results define a new sense in which feasibility is a more restrictive condition than stability, and show that these two requirements are closely interrelated. We then show how our results affect the changes in the set of coexisting species when interaction strengths are changed: a system of coexisting species loses a species by its abundance continuously going to zero, and this new fixed point is unique. As parameters are further changed, multiple alternative equilibria may be found. Finally, we discuss the extent to which our results apply to asymmetric interactions.

Motivation & Objective

• To determine which of feasibility or stability is the more restrictive condition for species coexistence in ecological systems.
• To analyze the geometry of the coexistence region in the space of interaction strengths for symmetric Lotka-Volterra models.
• To investigate the sequence in which feasibility and stability break as interaction strengths are varied.
• To explore the implications for community assembly and species loss dynamics under parameter changes.
• To assess the robustness of these findings to asymmetric interactions.

Proposed method

• The authors model ecological communities using symmetric Lotka-Volterra equations with interaction matrices A = I + aM, where M is a random matrix with zero diagonal and off-diagonal elements drawn from a uniform distribution.
• They analyze the coexistence region in parameter space by tracking species abundances as interaction strength a increases along linear trajectories.
• Feasibility is assessed by monitoring whether any species abundance drops to zero; stability is evaluated via the smallest eigenvalue of the interaction matrix A.
• The response stability is quantified by the smallest eigenvalue of the matrix DA, where D is the diagonal matrix of abundances.
• The authors use numerical simulations across 3×3 and 4×4 matrices to compute probabilities of stability breaking before feasibility under various conditions.
• They compare symmetric versus asymmetric interaction matrices to assess the role of matrix structure on the sequence of failure.

Experimental results

Research questions

• RQ1 In symmetric Lotka-Volterra systems, does feasibility or stability break first as interaction strengths are increased?
• RQ2 What is the geometric structure of the coexistence region in the space of interaction strengths?
• RQ3 How common is the scenario where stability breaks before feasibility in random systems?
• RQ4 Can species loss occur through continuous decline to zero abundance, or are abrupt transitions possible?
• RQ5 How do asymmetric interactions affect the relative timing of feasibility and stability breakdown?

Key findings

• Feasibility breaks before stability in the vast majority of cases, with stability loss occurring only at isolated points or on lower-dimensional subsets of the boundary.
• For symmetric matrices, the smallest eigenvalue of A becomes zero at a critical a, but abundances diverge to negative infinity before this point, causing feasibility to break earlier.
• In asymmetric systems, stability can break before feasibility, but this occurs with low probability—only 0.36% for 3×3 matrices and 0.43% for 4×4 matrices with real leading eigenvalues.
• When stability breaks before feasibility in asymmetric systems, it is due to complex eigenvalues that prevent abundance divergence, allowing feasibility to persist longer.
• Species are lost continuously via abundance approaching zero, and the resulting fixed point is unique, indicating a smooth transition in species composition.
• For asymmetric matrices with positive real leading eigenvalues, abundances remain finite and positive for all finite a, and feasibility is never violated, though abundances tend to zero as a→∞.