Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting

Area of Science:

• Mathematical Biology
• Ecological Dynamics
• Nonlinear Systems

Background:

• Predator-prey models are crucial for understanding ecological interactions.
• Turing instability explains pattern formation in biological systems.
• Cross-diffusion effects in ecological models are increasingly studied.

Purpose of the Study:

• To investigate Turing instability and pattern formation in a super cross-diffusion predator-prey system.
• To analyze the role of Michaelis-Menten type predator harvesting.
• To determine the conditions under which cross-diffusion induces spatial patterns.

Main Methods:

• Linear stability analysis of equilibrium points.
• Weakly nonlinear theory to derive amplitude equations.
• Numerical simulations to validate theoretical findings.

Main Results:

• Cross-diffusion can induce instability of equilibria in predator-prey systems.
• Cross-diffusion is identified as the key mechanism for spatial pattern formation.
• Amplitude equations reveal structural transitions and stability of various Turing patterns.

Conclusions:

• Cross-diffusion plays a critical role in generating spatial complexity in ecological models.
• The study provides theoretical and numerical insights into pattern formation driven by cross-diffusion.
• Findings contribute to a deeper understanding of ecological dynamics and stability.