Stability of Ecological Systems: A Theoretical Review

Abstract

The stability of ecological systems is a fundamental concept in ecology, which offers profound insights into species coexistence, biodiversity, and community persistence. In this article, we provide a systematic and comprehensive review on the theoretical frameworks for analyzing the stability of ecological systems. Notably, we survey various stability notions, including linear stability, sign stability, diagonal stability, D-stability, total stability, sector stability, structural stability, and higher-order stability. For each of these stability notions, we examine necessary or sufficient conditions for achieving such stability and demonstrate the intricate interplay of these conditions on the network structures of ecological systems. Finally, we explore the future prospects of these stability notions.

Figures (15)

• Figure 1: Pairwise interactions versus higher-order interactions in ecological systems. (a) Pairwise interspecies interactions. (b) Higher-order interspecies interactions. (c) Consumer-resource interactions naturally implies higher-order interspecies interactions.
• Figure 2: Geometrical implication of stability. For nonlinear systems $\dot{\textbf{x}}(t) = \textbf{f} (\textbf{x}(t), t)$, due to the complex and rich behavior of nonlinear dynamics, various types of stability, e.g., stability, asymptotic stability, and global asymptotic stability, can be discussed. Intuitively and roughly speaking, if all solutions of the system that start out near an equilibrium point ${\bf x}^*$ stay near ${\bf x}^*$ forever, then ${\bf x}^*$ is stable (in the sense of Lyapunov). More strongly, if ${\bf x}^*$ is stable and all solutions that start out near ${\bf x}^*$ converge to ${\bf x}^*$ as $t \to \infty$, then ${\bf x}^*$ is asymptotically stable. ${\bf x}^*$ is marginally stable if it is stable but not asymptotically stable. This figure was redrawn from Slotine-Book-91.
• Figure 3: Eigenvalue distributions of 10 community matrices (color) with $-d=-1$ on the diagonal and off-diagonal elements following the random (a), predator-prey (b), and mixture of competition and mutualism interactions (c), respectively. $S=250$, $C=0.25$, $\sigma=1$. The black ellipses are analytical results. Panels a, b, and c were redrawn from Allesina-Nature-2012. Impact of degree heterogeneity on the stability of ecological systems with random interactions (d), predator-prey (e), and mixture of competition and mutualism interactions (f). The dots represent the results from numerical simulations on 3-modal networks with different connectances. Higher $\text{Re}[\lambda_1]$ indicates lower stability. Each error bar represents the standard deviation of 100 independent runs. $S=1200$, $\sigma=1$, and $d=0$. Panel d was drawn in the log-log scale. Panels d, e, and f were redrawn from yan2017degree.
• Figure 4: Eigenvalue distributions of the three matrices M, Q, and J. The off-diagonal elements of A are sampled from a normal bivariate distribution with identical marginal $\mu_A=5/S$, $\sigma_A=5/{\sqrt{S}}$, and correlation $\rho_A=-0.5$. The diagonal elements of A are fixed to $\mu_d = -1$, while the diagonal elements of X are sampled from a uniform distribution on [0,1]. The two matrices Q and J are defined as $\textbf{Q}=\textbf{X}[(\mu _d-\mu_A)\textbf{I}+\mu_A \textbf{1}]$ and $\textbf{J}=\textbf{X}(\mu _d\textbf{I}+\textbf{B})$, respectively, such that the bulk of the eigenvalue distributions of M are the same as these of J, and the outlier eigenvalue of M is the same as that of Q. Here, B is a random matrix with zero diagonal elements whose off-diagonal elements have mean zero and variance $\sigma_A^2$. $S=500$. This figure was redrawn from gibbs2018effect.
• Figure 5: Eigenvalue distributions with time delay, where the blue teardrop-shaped area represents the stability region with time delay. (a) The eigenvalues (represented by red dots) are within the teardrop-shaped area, indicating that the corresponding GLV model with time delay is locally stable. (b) The eigenvalues are outside the teardrop-shaped area, indicating that the corresponding GLV model with time delay is unstable. $\tau=1$. The figure was redrawn from yang2023time.