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This study analyzes a diffusive predator-prey model, establishing conditions for species permanence and global stability. It also investigates the absence of nontrivial steady states under specific parameters.

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Area of Science:

  • Mathematical Biology
  • Ecology
  • Differential Equations

Background:

  • Predator-prey models are fundamental in ecology.
  • Understanding population dynamics requires analyzing stability and permanence.
  • Nonlinear functional responses and nonconstant death rates add complexity.

Purpose of the Study:

  • To analyze a diffusive predator-prey model with a nonconstant death rate and general nonlinear functional response.
  • To determine conditions for the permanence of both predator and prey populations.
  • To investigate the global asymptotic stability of the positive equilibrium and the existence of nontrivial steady states.

Main Methods:

  • Stability analysis of the equilibrium for the reduced ordinary differential equation (ODE) system.
  • Derivation of sufficient and necessary conditions for species permanence.
  • Application of Lyapunov function methods to establish global asymptotic stability.
  • Analysis of steady-state solutions for specific parameter configurations.

Main Results:

  • Conditions for the permanence of predator and prey populations were established.
  • The global asymptotic stability of the unique positive equilibrium was proven using Lyapunov functions.
  • It was demonstrated that no nontrivial steady-state solutions exist for certain parameter values.

Conclusions:

  • The study provides a comprehensive analysis of a complex predator-prey model.
  • The findings contribute to understanding population dynamics and ecological stability.
  • The research highlights the impact of nonconstant death rates and nonlinear functional responses on ecosystem stability.