Spatiotemporal Dynamics of a Mussel-Algae Model on the Square Domain.

Area of Science:

• Mathematical Biology
• Ecology
• Dynamical Systems

Background:

• Mussel-algae interactions are crucial in marine ecosystems.
• Understanding pattern formation in these systems is key to ecological dynamics.
• Previous models often simplified spatial dimensions, limiting insights.

Purpose of the Study:

• To investigate the complex spatiotemporal dynamics of a non-local mussel-algae model.
• To analyze pattern formation mechanisms in a two-dimensional symmetric domain with time delays.
• To classify the resulting spatiotemporal patterns and their ecological implications.

Main Methods:

• Analysis of well-posedness and eigenvalue multiplicity.
• Application of equivariant Hopf bifurcation theory and normal form methods.
• Phase space decomposition and center manifold reduction.
• Numerical simulations to validate theoretical findings.

Main Results:

• Established existence of Hopf and equivariant Hopf bifurcations.
• Derived third-order truncated normal forms.
• Classified ten distinct spatiotemporal patterns, including standing waves, rotating waves, stripes, and spots.
• Demonstrated more complex dynamics in 2D compared to 1D models.

Conclusions:

• The 2D mussel-algae model exhibits rich spatiotemporal dynamics.
• Theoretical predictions are supported by numerical simulations and field observations.
• The model provides insights into how mussel aggregation behavior drives pattern formation.