Turing patterns and long-time behavior in a three-species food-chain model

Area of Science:

• Ecology
• Mathematical Biology
• Theoretical Ecology

Background:

• Investigates complex ecological dynamics in a three-species food chain.
• Focuses on generalist top predator, specialist middle predator, and prey interactions.
• Examines long-time dynamics within a spatially explicit framework.

Purpose of the Study:

• To analyze the long-time dynamics and stability of a spatially explicit three-species food chain model.
• To identify and characterize emergent spatial patterns, including Turing patterns.
• To explore chaotic dynamics and estimate the fractal dimension of the system's attractor.

Main Methods:

• Linear stability analysis to detect Turing instability.
• Numerical simulations in 1D and 2D to generate and visualize Turing patterns.
• Nonlinear time series analysis to reconstruct and analyze the chaotic attractor.

Main Results:

• Demonstrated the existence of a finite-dimensional global attractor in L(2)(Ω).
• Confirmed the occurrence of Turing instability and diffusion-induced chaos.
• Generated diverse Turing patterns (stripes, spots, labyrinth, etc.) and explored parameter spaces.
• Estimated the fractal dimension of the low-dimensional chaotic attractor.

Conclusions:

• The model exhibits complex spatio-temporal dynamics, including pattern formation and chaos.
• Turing instability and diffusion-induced chaos are key mechanisms driving pattern diversity.
• The study provides a lower bound for the fractal dimension of the spatially explicit model's attractor.