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Permanence and Extinction of a Diffusive Predator-Prey Model with Robin Boundary Conditions.

M A Aziz-Alaoui1, M Daher Okiye2, A Moussaoui3

  • 1Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, BP 540, 76600, Le Havre, France. aziz.alaoui@univ-lehavre.fr.

Acta Biotheoretica
|May 30, 2018
PubMed
Summary

This study analyzes a predator-prey system with diffusion, using Holling-type-II and modified Leslie-Gower functional responses. It establishes criteria for the system's permanence and predator extinction under specific boundary conditions.

Keywords:
ExtinctionPermanencePredator–preyReaction–diffusionRobin boundary conditions

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Area of Science:

  • Mathematical Biology
  • Ecology
  • Differential Equations

Background:

  • Predator-prey models are fundamental in ecology.
  • Diffusion and functional responses significantly influence population dynamics.
  • Robin boundary conditions are relevant for spatially extended systems.

Purpose of the Study:

  • To investigate the dynamic behavior of a diffusive predator-prey system.
  • To incorporate Holling-type-II and modified Leslie-Gower functional responses.
  • To analyze system permanence and predator extinction under Robin boundary conditions.

Main Methods:

  • Utilizing the comparison principle to study system dissipativeness.
  • Deriving mathematical criteria for population permanence.
  • Establishing conditions for predator extinction.

Main Results:

  • The study establishes criteria for the permanence of the predator-prey system.
  • Conditions for the extinction of the predator population are derived.
  • The analysis considers the impact of diffusion and specific functional responses.

Conclusions:

  • The research provides insights into the stability and persistence of predator-prey interactions in spatially heterogeneous environments.
  • The derived criteria can predict population outcomes under different ecological scenarios.
  • This work contributes to understanding complex ecological dynamics through mathematical modeling.