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This study models predator-prey dynamics with a delayed interaction, introducing an intermediate species. The research reveals how delay and diffusion create complex patterns like spirals and turbulence in these systems.

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Area of Science:

  • Mathematical Biology
  • Dynamical Systems
  • Epidemiology

Background:

  • Predator-prey models are standard for species interaction dynamics.
  • Previous models often used delays for maturation, not interaction.
  • This work introduces a delayed interaction, potentially involving an intermediate species.

Purpose of the Study:

  • To analyze a delayed predator-prey model with an intermediate species.
  • To investigate the effects of spatial diffusion on the system's dynamics.
  • To explore interdisciplinary applications of the model.

Main Methods:

  • Analysis of a delayed predator-prey model, including a compartment model analogy for epidemiology.
  • Mathematical analysis using Hopf instability and weakly nonlinear expansion.
  • Derivation of the Ginzburg-Landau normal form to study diffusion effects.

Main Results:

  • A Hopf instability emerges beyond a critical delay time, leading to a stable limit cycle.
  • Diffusion introduces phase diffusion equations with stable turbulent phase solutions.
  • Defect-mediated patterns like spirals and weakly turbulent structures arise from delay and diffusion interplay.

Conclusions:

  • The delayed predator-prey model with diffusion exhibits rich pattern formation.
  • The findings have implications for mathematical biology and interdisciplinary fields like chemistry, social science, and economics.
  • The interplay of delay and diffusion is crucial for emergent complex behaviors.