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Universality in random matrix theory fails for ecological communities. Non-Gaussian statistics of species interactions are crucial for predicting community stability using eigenvalue spectra.

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Area of Science:

  • Ecology
  • Statistical Physics
  • Mathematical Biology

Background:

  • Random matrix theory often assumes universality, where eigenvalue spectra depend only on moments, not distributions.
  • This universality principle is frequently applied without rigorous proof in various scientific fields.
  • Ecological community stability is often assessed using random matrix approaches.

Purpose of the Study:

  • To investigate the validity of the universality principle in random matrix theory within ecological contexts.
  • To determine if non-Gaussian interaction statistics affect ecological community stability predictions.
  • To provide a counterexample to the universality principle in the generalized Lotka-Volterra equations.

Main Methods:

  • Application of dynamic mean-field theory to derive interaction statistics in evolved ecological communities.
  • Analysis of the eigenvalue spectrum of interaction matrices.
  • Comparison of results with Gaussian ensemble predictions.

Main Results:

  • The eigenvalue spectrum and stability of ecological communities depend on full interaction statistics, not just moments.
  • A counterexample to the universality principle was demonstrated in the generalized Lotka-Volterra model.
  • Emergent non-Gaussian statistics of species interactions are critical for accurate stability prediction.

Conclusions:

  • The universality principle of random matrix theory does not hold for all ecological systems.
  • Accurate prediction of ecological community stability requires considering non-Gaussian interaction statistics.
  • Random matrix theory can inform ecological stability analysis when emergent interaction properties are incorporated.