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Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system.

G Gambino1, M C Lombardo1, M Sammartino2

  • 1Department of Mathematics, University of Palermo, via Archirafi 34, 90123 Palermo, Italy.

Physical Review. E
|February 17, 2018
PubMed
Summary
This summary is machine-generated.

Nonlinear cross-diffusion in predator-prey models can unexpectedly create complex dynamics. This study reveals how subharmonic resonance drives oscillating patterns and chaos, even from stable states.

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Area of Science:

  • Mathematical Biology
  • Theoretical Ecology
  • Nonlinear Dynamics

Background:

  • Predator-prey models are crucial for understanding ecological interactions.
  • Reaction-diffusion systems often exhibit complex spatial-temporal patterns.
  • Cross-diffusion can significantly alter system dynamics beyond linear diffusion.

Purpose of the Study:

  • Investigate complex dynamics from cross-diffusion in a predator-prey system.
  • Analyze the subharmonic destabilization of Turing modes.
  • Explore pattern formation and transitions to chaos.

Main Methods:

  • Utilized a two-species Lotka-Volterra model with linear and nonlinear cross-diffusion.
  • Performed linear analysis to identify instabilities.
  • Studied bifurcations from stationary patterns to oscillating solutions.
  • Examined pattern formation on 2D domains.

Main Results:

  • Nonlinear cross-diffusion induced subharmonic resonance, destabilizing stationary Turing modes.
  • Out-of-phase oscillating solutions (subT modes) emerged due to spatial resonance.
  • Varying parameters led to secondary instabilities and chaotic dynamics.
  • 2D simulations showed oscillating square and
  • twinkling-eyes
  • patterns.

Conclusions:

  • Cross-diffusion can generate complex dynamics not predicted by linear analysis.
  • Subharmonic resonance is a key mechanism for pattern formation and chaos in this system.
  • The model provides insights into emergent complex behaviors in ecological systems.