Stabilization of macroscopic dynamics by fine-grained disorder in many-species ecosystems

Abstract

A central feature of complex systems is the relevance and entanglement of different levels of description. For instance, the dynamics of ecosystems can be alternatively described in terms of large ecological processes and classes of organisms, or of individual species and their relations. Low-dimensional heuristic 'macroscopic' models that are widely used to capture ecological relationships -- and commonly evidence out-of equilibrium regimes -- implicitly assume that species-level 'microscopic' heterogeneity can be neglected. Here, we address the stability of such macroscopic descriptions to the addition of disordered microscopic interactions. We find that increased heterogeneity can stabilize collective as well as species fluctuations -- contrary to the well-known destabilizing effect of disorder on fixed points. We analytically find the conditions for the existence of heterogeneity-driven equilibria, and relate their stability to a mismatch in microscopic time scales. This may shed light onto the empirical observation that many-species ecosystems often appear stable at aggregated levels despite highly diverse interactions and large fluctuations at the species level.

Figures (5)

• Figure 1: Phase diagram of the microscopic RPS, displaying the three meaningful phases described in the text (SO in orange, FP in green and AF in blue) as well as the physically unrealistic Unbounded Growth phase (yellow) Bunin2017. The SO/FP transition line was obtained using eq. (\ref{['eq: woodbury']}), while the other two thick lines have already been studied in Galla2018. While our analysis does not apply to the SO/AF transition, a calculation assuming the fixed point solution in the blue phase yields a reasonable approximation (dashed line).
• Figure 2: Trajectories for the microscopic RPS system in the SO phase. Thin lines are a subset of species's trajectories, color-coded as the groups they belong to, whose average abundance is represented as thick lines. As heterogeneity increases from $\sigma = 0.25$ (a) to $0.75$ (b), synchronous trajectories become less coherent, the amplitude of the group oscillations decreases and the transient becomes longer. Finally, the system develops a stable fixed point, seen here at $\sigma=1.1$ (c). In this simulation, $\kappa=5$ and $S_g=300$.
• Figure 3: Spectrum of the Lotka-Volterra Jacobian $J$ (left) and the pseudo-Jacobian of eq. (\ref{['eq:Jcal']}) (right) for the RPS system with parameters $\kappa=5$ and $\sigma=0.75$. The red circles are predictions using Woodbury's identity eq. (\ref{['eq: woodbury']}). On top of such outliers, the pseudo-Jacobian features a bulk of stable eigenvalues, corresponding to its diagonal component.
• Figure 4: Flow of eigenvalues for the RPS system, as predicted by eq. (\ref{['eq:lambda_evol']}). The arrows correspond to the flow at $\sigma=0$. The initial eigenvalue $\lambda_0$ (whose location depends on $\kappa$) moves left-wards as disorder increases, thus enhancing stability. The black line shows the evolution of the eigenvalues until stability is lost, for $\sigma=0.5$. The value of $\kappa=3.5$ was chosen so that the impact of extinctions in eq. (\ref{['eq: woodbury']}) is negligible. Dynamical simulations (diamonds) display an excellent agreement with the predictions of eq. (\ref{['eq:lambda_evol']}). The red and blue regions denote eigenvalues $\lambda_0$ that are stabilized or not, respectively, when $\sigma\leq0.5$.
• Figure 5: Poincaré map of a system with chaotic macroscopic dynamics (Appendix A) when changing heterogeneity. For small $\sigma$ (e.g.$\sigma=0.04$), both the microscopic and macroscopic systems have chaotic dynamics. Chaos is lost through an inverse Feigenbaum cascade, it becomes cyclic (e.g.$\sigma=0.2$) and ultimately reaches a fixed point through a Hopf bifurcation.