A two or three compartments hyperbolic reaction-diffusion model for the aquatic food chain

Area of Science:

• Mathematical Biology
• Ecology
• Theoretical Physics

Background:

• Reaction-diffusion systems are crucial for modeling ecological dynamics.
• Parabolic models exhibit instantaneous diffusion, a biologically unrealistic phenomenon.
• Extended Thermodynamics offers a framework for hyperbolic systems.

Purpose of the Study:

• To develop and analyze hyperbolic reaction-diffusion models for aquatic food chains.
• To investigate spatio-temporal dynamics in two- and three-compartment systems.
• To address the instantaneous diffusion paradox inherent in parabolic models.

Main Methods:

• Formulation of two hyperbolic reaction-diffusion models within Extended Thermodynamics.
• Analysis of phytoplankton-zooplankton and nutrient-phytoplankton-zooplankton interactions.
• Application of linear stability analysis to study steady states, traveling waves, and Hopf bifurcations.
• Numerical integration of governing equations to validate analytical findings.

Main Results:

• Hyperbolic models eliminate the instantaneous diffusion paradox, demonstrating finite-velocity wave propagation.
• Characterization of steady states and traveling wave solutions.
• Identification of conditions for Hopf bifurcations, indicating complex population dynamics.
• Numerical simulations confirm analytical predictions and provide further insights into population dynamics.

Conclusions:

• Hyperbolic reaction-diffusion models provide a more realistic framework for aquatic food chain dynamics.
• The finite-velocity wave propagation is a key feature distinguishing these models from parabolic counterparts.
• The study offers valuable analytical and numerical tools for understanding ecological pattern formation and stability.