Dynamic analysis of a modified algae and fish model with aggregation and Allee effect

Area of Science:

• Ecology
• Mathematical Biology
• Aquatic Ecosystems

Background:

• Understanding the complex interactions between algae and fish is crucial for aquatic ecosystem management.
• Existing models often simplify the ecological dynamics, neglecting factors like aggregation and Allee effects.

Purpose of the Study:

• To develop and analyze a modified mathematical model of algae-fish interactions.
• To investigate the influence of aggregation and Allee effects on the model's dynamics.
• To determine conditions for the existence and stability of equilibrium points and bifurcations.

Main Methods:

• Development of a modified mathematical model incorporating aggregation and Allee effects.
• Analysis of critical conditions for the existence and stabilization of equilibrium points.
• Investigation of transcritical, saddle-node, Hopf, and Bogdanov-Takens (B-T) bifurcations.
• Numerical simulations to verify theoretical findings and visualize dynamic relationships.

Main Results:

• Established conditions for the existence and stability of all possible equilibrium points in the model.
• Demonstrated the occurrence of various bifurcation types (transcritical, saddle-node, Hopf, B-T).
• Numerical simulations confirmed theoretical derivations and illustrated the impact of aggregation and Allee effects.

Conclusions:

• The modified model provides a more comprehensive understanding of algae-fish population dynamics.
• Aggregation and Allee effects significantly influence the stability and behavior of the aquatic ecosystem.
• The findings generalize and improve upon existing knowledge of predator-prey models in aquatic environments.