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Area of Science:

  • Mathematical Biology
  • Ecological Dynamics
  • Nonlinear Systems

Background:

  • Predator-prey models are crucial for understanding ecological interactions.
  • Allee effects in prey populations introduce complex dynamics.
  • Functional responses and density-dependent mortality shape population stability.

Purpose of the Study:

  • Investigate the dynamics of a diffusive predator-prey system with Allee effects.
  • Analyze stability, bifurcations, and pattern formation.
  • Identify the role of predator density-dependent death rate in ecological instability.

Main Methods:

  • Mathematical modeling of predator-prey interactions.
  • Analysis of steady states and stability.
  • Hopf bifurcation analysis.
  • Investigation of Turing instability and pattern formation via numerical simulations.

Main Results:

  • Established conditions for dissipation and persistence.
  • Determined stability of constant steady states.
  • Found Hopf bifurcation at the positive constant solution.
  • Demonstrated Turing instability leading to pattern formation (spots, stripes, holes).
  • Identified predator density-dependent death rate as the inducer of Turing instability.

Conclusions:

  • The model exhibits rich dynamics, including pattern formation driven by diffusion.
  • Predator density-dependent death rate plays a critical role in inducing spatial instability.
  • The findings contribute to understanding complex ecological pattern formation in predator-prey systems.