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Approximate message passing for block-structured ecological systems

Maxime Clenet, Mohammed-Younes Gueddari

Abstract

Ecological interaction networks are rarely homogeneous: species naturally form communities with distinct interaction structures, resulting in block-structured variance and correlation profiles in the interaction matrix. We study the equilibrium properties of generalized Lotka-Volterra systems whose interaction matrices are random and non-symmetric with variance and correlation profiles. Based on recent advances in approximate message passing (AMP) for heterogeneous and correlated random matrices, we derive a set of self-consistent fixed-point equations that, in the large-$n$ limit, characterize the equilibrium abundance distribution. In particular, we show that this limiting distribution is an explicit mixture of truncated Gaussian, driven by the variance and correlation profiles. We then illustrate the ecological implications of this result through three applications involving two interacting communities. First, we show that local changes in the correlation profile within a single community induce system-wide responses in species persistence, revealing the non-local nature of persistence dynamics. Second, we find that communities dominated by mutualistic or competitive interactions are more robust to increasing inter-community coupling, whereas communities structured by predator-prey interactions are more prone to collapse. Third, we demonstrate that asymmetric interaction variance alone, in the complete absence of correlation, can generate feedback loop between communities.

Approximate message passing for block-structured ecological systems

Abstract

Ecological interaction networks are rarely homogeneous: species naturally form communities with distinct interaction structures, resulting in block-structured variance and correlation profiles in the interaction matrix. We study the equilibrium properties of generalized Lotka-Volterra systems whose interaction matrices are random and non-symmetric with variance and correlation profiles. Based on recent advances in approximate message passing (AMP) for heterogeneous and correlated random matrices, we derive a set of self-consistent fixed-point equations that, in the large- limit, characterize the equilibrium abundance distribution. In particular, we show that this limiting distribution is an explicit mixture of truncated Gaussian, driven by the variance and correlation profiles. We then illustrate the ecological implications of this result through three applications involving two interacting communities. First, we show that local changes in the correlation profile within a single community induce system-wide responses in species persistence, revealing the non-local nature of persistence dynamics. Second, we find that communities dominated by mutualistic or competitive interactions are more robust to increasing inter-community coupling, whereas communities structured by predator-prey interactions are more prone to collapse. Third, we demonstrate that asymmetric interaction variance alone, in the complete absence of correlation, can generate feedback loop between communities.
Paper Structure (17 sections, 1 theorem, 52 equations, 4 figures)

This paper contains 17 sections, 1 theorem, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Model and assumptions
  4. Outline of the article
  5. Main result
  6. Model setup
  7. Main result
  8. Example
  9. The general case of non structured matrices
  10. Proof outline of Theorem \ref{['thm:main_block']}
  11. Asymptotics of the empirical measure
  12. Existence of a solution to the fixed point equations
  13. Applications
  14. Effect of correlation profile
  15. Opposing correlation structure
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $\|S\|_\infty < \frac{1}{4}$ and consider the interaction matrix $\Sigma$ defined above with block-structured variance and correlation profiles. Then the empirical measure of the equilibrium vector $u^\star$ satisfies where and $\xi_1,\dots,\xi_K \stackrel{i.i.d.}{\sim} \mathcal{N}(0,1)$. The vector $(\delta_1^\star, \cdots, \delta_K^\star, \sigma_1^\star, \cdots, \sigma_K^\star )$ i

Figures (4)

  • Figure 1: Empirical equilibrium abundance distributions (histograms) compared with the truncated normal densities predicted by the fixed-point equations \ref{['eq:truncated_gaussian']} (dashed lines). Colors denote the two blocks of the interaction matrix, $S = 0.25000.64, \quad R = -0.9000, \quad \alpha = (0.5, 0.5), \quad n = 10^4$. Blue and red correspond to communities 1 and 2, respectively.
  • Figure 2: Effects of correlation $\rho_{11}$ in community 1 on the properties of species abundance distribution. (a) Solid and dashed lines show the proportion of persisting species in each community as a function of the intra-community correlation $\rho_{11}$ in community 1. (b) Solid and dashed lines show the variance of equilibrium abundances in each community, under the same variation of $\rho_{11}$. While persistence responds globally and symmetrically, abundance variance exhibits a more localized and asymmetric response to the correlation $\rho_{11}$.
  • Figure 3: Effects of increasing inter-community interaction variance ($s^2 = s^2_{12} = s^2_{21}$) on species persistence and equilibrium abundance variance, under opposing intra-community correlation structures: $\rho_{11} = 0.8$ (mutualistic/competitive) in community 1 and $\rho_{22} = -0.8$ (predator–prey) in community 2. In panels (a) and (b), the solid line corresponds to community 1 and the dashed line to community 2.
  • Figure 4: Heatmap of the proportion of persisting species in community 1 ($\gamma_1$) as a function of the inter-community interaction variance $s^2_{12}$ and $s^2_{21}$, with zero correlation structure ($R = 0$). The intra-community interaction variance are fixed at $s^2_{11} = s^2_{22} = 0.5$.

Theorems & Definitions (2)

  • Theorem 1
  • Conjecture 2