Associate Professor University of Illinois at Urbana-Champaign Urbana, Illinois, United States
Abstract: Many studies in theoretical ecology focus on communities of species with a stable and feasible global equilibrium. But increased attention has recently been paid to systems with many stable equilibria, each containing a subset of the original species pool. Such systems can arise, for example, when existing ecosystems experience large perturbations through species extinction or immigration. There are many basic questions about these ecological communities: Under what circumstances does a system have multiple equilibria? How many stable equilibria can there be? What is the relative likelihood of different equilibria given uncertain initial conditions? This work explores these questions in the context of competitive Lotka-Volterra models. Cases where competition is based around species location on a niche axis and interactions are (near-)symmetric form the primary focus of our work; analysis of these models is motivated by basic questions about niche clustering/differentiation and coexistence on a niche axis. We also consider exclusionary models with single-species stable equilibria, and models with random competitive interactions.
We use a combination of theoretical analyses (eigenvalue spectra, perturbative analysis, large N methods, geometric analyses for small systems), numerical methods (solutions of ODEs for larger numbers of species, automated algebraic analysis of all possible equilibria for systems with up to 30 species), and physics methods drawn from statistical mechanics. For all models studied, and each of these different methodologies, we provide evidence that equilibria with larger total biomass generally have larger basins of attraction, and conclude that states with larger biomass are more likely to be found as equilibrium states observed in natural systems. We identify two interesting limits of niche systems for a large species pool. A continuous limit has fixed numbers of species in limiting equilibria. In the other limit the number of species in equilibria grows linearly and the number of equilibria grows exponentially in the number of species, and the equilibria can be described by a simple statistical mechanical model. In the latter case, domain sizes are described by a Boltzmann-type distribution, where the probability of an equilibrium is proportional to the exponential of that equilibrium's total biomass. We conclude with initial exploration of the robustness of these outcomes for systems with more general non-symmetric and/or random interactions, and show preliminary evidence for the broader validity of the conclusion that larger biomass equilibria are more likely to occur than smaller biomass equilibria in competitive Lotka-Volterra communities.
