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Persistence and neutrality in interacting replicator dynamics.

Leonardo Videla1, Mauricio Tejo2, Cristóbal Quiñinao3

  • 1Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile.

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|January 3, 2025
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Summary
This summary is machine-generated.

This study models primordial ecology using stochastic dynamics, proving chaos propagation and system persistence. Fitness equivalence emerges as a condition for persistence, unlike in neutral ecology models.

Keywords:
Emergence of ecologiesInvariant distributionsMcKean–Vlasov equationPropagation of ChaosStochastic persistenceStochastic replicator dynamics

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Area of Science:

  • Mathematical Biology
  • Theoretical Ecology
  • Statistical Physics

Background:

  • Investigates the large-time behavior of systems with mean-field interactions, modeling primordial ecologies.
  • Focuses on replicator-like stochastic dynamics, a framework for population genetics and evolutionary game theory.

Purpose of the Study:

  • To analyze the long-term dynamics and stability of a primordial ecological model.
  • To establish conditions for the strong persistence and emergent properties of N-replicator systems.
  • To explore the connection between fitness equivalence, neutrality, and invariant distributions in ecological models.

Main Methods:

  • Proving the propagation-of-chaos property for the stochastic dynamics.
  • Establishing conditions for strong persistence in the N-replicator system.
  • Analyzing the existence of invariant distributions for associated McKean-Vlasov dynamics.

Main Results:

  • Demonstrated propagation-of-chaos and conditions for strong persistence.
  • Showed that fitness equivalence emerges as a condition for persistence, not a prerequisite.
  • Identified neutrality with a unique Dirichlet invariant probability measure.

Conclusions:

  • The study provides a rigorous mathematical framework for understanding primordial ecological dynamics.
  • Findings challenge assumptions in neutral ecology, highlighting emergent properties like fitness equivalence.
  • Results offer insights into the mathematical underpinnings of ecological stability and diversity.