Threshold dynamics of a delayed reaction diffusion equation subject to the Dirichlet condition

Area of Science:

• Mathematical Biology
• Partial Differential Equations
• Dynamical Systems

Background:

• Reaction-diffusion equations model biological processes.
• Delayed terms introduce complex dynamics.
• Non-monotone reaction terms are common in biological models.

Purpose of the Study:

• To establish the threshold dynamics of a delayed reaction-diffusion equation.
• To analyze the influence of a non-monotone delayed reaction term.
• To provide specific examples illustrating the theoretical results.

Main Methods:

• Analysis of delayed reaction-diffusion equations.
• Application of threshold dynamics principles.
• Investigation of homogeneous Dirichlet boundary conditions.

Main Results:

• The study establishes the threshold dynamics for the considered equation class.
• Demonstration of how non-monotone delays affect system stability and behavior.
• The Nicholson's blowflies diffusion equation serves as a key illustrative example.

Conclusions:

• The findings provide a theoretical framework for understanding delayed reaction-diffusion systems.
• The results are applicable to ecological models with time delays.
• Further research can explore more complex delay functions and boundary conditions.