[Paper Review] Many-species ecological fluctuations as a jump process from the brink of extinction

Thibaut Arnoulx de Pirey, Guy Bunin|arXiv (Cornell University)|Jun 23, 2023

Ecosystem dynamics and resilienceEnvironmental Science3 citations

TL;DR

This paper proposes a non-Markovian jump-diffusion framework to explain large-scale ecological fluctuations in diverse ecosystems driven by species interactions, showing that rare species undergo sudden population jumps to abundance before gradual decline. In the limit of low migration, the system exhibits a scaling regime where species turnover emerges as intermittent jumps, with abundance distributions and diversity metrics that deviate from equilibrium predictions and remain below the linear stability bound.

ABSTRACT

Highly-diverse ecosystems exhibit a broad distribution of population sizes and species turnover, where species at high and low abundances are exchanged over time. We show that these two features generically emerge in the fluctuating phase of many-variable model ecosystems with disordered species interactions, when species are supported by migration from outside the system at a small rate. We show that these and other phenomena can be understood through the existence of a scaling regime in the limit of small migration, in which large fluctuations and long timescales emerge. We construct an exact analytical theory for this asymptotic regime, that provides scaling predictions on timescales and abundance distributions that are verified exactly in simulations. In this regime, a clear separation emerges between rare and abundant species at any given time, despite species moving back and forth between the rare and abundant subsets. The number of abundant species is found to lie strictly below a well-known stability bound, maintaining the system away from marginality. At the same time, other measures of diversity, which also include some of the rare species, go above this bound. In the asymptotic limit where the migration rate goes to zero, trajectories of individual species abundances are described by non-Markovian jump-diffusion processes, which proceeds as follows: A rare species remains so for some time, then experiences a jump in population sizes after which it becomes abundant (a species turnover event) and later sees its population size gradually decreasing again until rare, due to the competition with other species. The asymmetry of abundance trajectories under time-reversal is maintained at small but finite migration rate. These features may serve as fingerprints of endogenous fluctuations in highly-diverse ecosystems.

Motivation & Objective

• To understand the origin of broad species abundance distributions and large population fluctuations in highly diverse ecosystems.
• To identify universal scaling behaviors in the fluctuating phase of many-species ecological systems with disordered interactions.
• To develop an analytical framework for the asymptotic regime of small migration rates, where large timescales and rare events dominate.
• To explain the persistence of diversity and the separation between rare and abundant species despite continuous turnover.
• To characterize the non-Markovian nature of species abundance trajectories, including time-reversal asymmetry.

Proposed method

• Formulates a rescaled dynamics in the limit of vanishing migration rate, enabling exact analytical treatment of the asymptotic regime.
• Uses dynamical mean-field theory (DMFT) to self-consistently solve for the statistics of interaction strengths and population trajectories.
• Applies a non-Markovian jump-diffusion process to model individual species abundance evolution: long periods of rarity followed by sudden jumps to abundance.
• Derives scaling predictions for abundance distributions and timescales that are verified exactly in large-scale simulations.
• Introduces a filtering procedure to identify top species based on invasion growth rates and dynamic stability, separating them from transient fluctuations.
• Employs numerical simulations of the full Lotka-Volterra model with random interactions and a small migration rate, validated against DMFT results.

Experimental results

Research questions

• RQ1 How do large-scale population fluctuations and broad abundance distributions emerge in diverse ecosystems with only internal species interactions?
• RQ2 What is the role of low migration in generating long timescale fluctuations and intermittent species turnover?
• RQ3 Why do diversity measures in the fluctuating phase exceed the linear stability bound, while the number of abundant species remains below it?
• RQ4 How does the time-reversal asymmetry of species abundance trajectories manifest in the small-migration limit?
• RQ5 Can the dynamics of individual species be described as a non-Markovian jump process, and what are the scaling laws governing this behavior?

Key findings

• In the small-migration limit, species abundance trajectories are best described by a non-Markovian jump-diffusion process, with rare species remaining so for extended periods before sudden jumps to high abundance.
• The number of abundant species is strictly below the linear stability bound, with a value of φ_top = 0.28, which is 1% below the theoretical bound of φ = 0.31.
• The asymptotic abundance distribution is distinct from both the instantaneous distribution and the truncated Gaussian seen in the fixed-point phase, confirming a unique dynamical phase.
• The system maintains time-reversal asymmetry in species abundance trajectories, even at small but finite migration rates, indicating endogenous, non-equilibrium dynamics.
• The DMFT-based analytical framework accurately predicts scaling laws for timescales and abundance distributions, validated against full simulations of large systems (S = 20,000).
• Filtering species based on invasion growth rate and dynamic stability recovers a distribution close to the asymptotic one, confirming the robustness of the top species identification in the fluctuating regime.