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Oscillations and Pattern Formation in a Slow-Fast Prey-Predator System.

Pranali Roy Chowdhury1, Sergei Petrovskii2,3, Malay Banerjee4

  • 1Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India.

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Summary

This study reveals hidden dynamics in slow-fast prey-predator models with Allee effects. A weak Allee effect can trigger abrupt shifts from small to large population oscillations, risking species extinction.

Keywords:
Canard cycleRegime shiftRelaxation oscillationSlow–fast timescaleSpatial pattern

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Population Dynamics

Background:

  • Prey-predator systems are fundamental in ecology, but their complex dynamics, especially with Allee effects and multiple timescales, remain underexplored.
  • The Allee effect, where population growth rate decreases at low densities, introduces unique challenges in population modeling.

Purpose of the Study:

  • To investigate the hidden properties of slow-fast prey-predator systems, particularly when prey growth is influenced by a weak Allee effect.
  • To analyze the impact of multiple timescales and spatial dynamics on population oscillations and pattern formation.

Main Methods:

  • Comprehensive mathematical analysis of the non-spatial slow-fast prey-predator system.
  • Investigation of the spatially explicit model to understand pattern formation.
  • Utilized techniques to study regime shifts driven by the interplay of Allee effects and timescales.

Main Results:

  • The interplay between the Allee effect and multiple timescales can induce a regime shift, transitioning population dynamics from small-amplitude to large-amplitude oscillations.
  • In spatially explicit models, reduced timescale ratios can lead to another regime shift, resulting in spatially correlated patterns and increased risk of species extinction due to large oscillations.

Conclusions:

  • Slow-fast prey-predator systems with Allee effects exhibit complex, often overlooked behaviors.
  • Timescale dynamics significantly influence population stability and spatial patterns, with potential for abrupt shifts and extinction risks.