Mathematical modeling and analysis of two-variable system with noninteger-order derivative.

Area of Science:

• Mathematical Biology
• Dynamical Systems Theory
• Fractional Calculus

Background:

• Ecological systems, including predator-prey and commensalism, exhibit complex dynamics.
• Traditional modeling approaches may not fully capture the nonlocal and nonsingular properties inherent in biological interactions.
• The Atangana-Baleanu derivative offers unique mathematical properties suitable for advanced modeling.

Purpose of the Study:

• To apply the Atangana-Baleanu derivative operator to model symbiosis systems.
• To investigate the application of this derivative in describing commensalism and predator-prey processes.
• To analyze the mathematical properties and potential of the Atangana-Baleanu derivative in ecological modeling.

Main Methods:

• Application of the Atangana-Baleanu derivative operator to established ecological models.
• Mathematical analysis of the resulting fractional dynamical systems.
• Investigation of model behavior, including chaotic and spatiotemporal dynamics, for various fractional orders (α).

Main Results:

• Successful application of the Atangana-Baleanu derivative to model ecological symbiosis.
• Demonstration of the derivative's ability to capture nonlocal and nonsingular characteristics in biological systems.
• Observation of chaotic and spatiotemporal patterns, particularly with fractional power α, indicating rich system behavior.

Conclusions:

• The Atangana-Baleanu derivative provides a powerful and versatile tool for modeling complex ecological interactions.
• Its unique properties, including Markovian and non-Markovian aspects, enhance the realism of symbiosis and predator-prey models.
• Further research into fractional ecological models using this derivative can yield deeper insights into ecosystem dynamics.