Study of memory-dependent dynamics of a predator-prey model with hyperbolic mortality

Area of Science:

• Mathematical Biology
• Fractional Calculus
• Ecology

Background:

• Ecological models often simplify complex species interactions.
• Fractional calculus offers a powerful framework to model memory effects in biological systems.
• Hyperbolic mortality and memory effects are crucial for realistic prey-predator dynamics.

Purpose of the Study:

• To analyze a fractional-order prey-predator model with hyperbolic mortality and memory effects.
• To establish the well-posedness (existence, uniqueness, positivity, boundedness) of the model's solutions.
• To investigate the local stability and parameter sensitivity of the ecological system.

Main Methods:

• Application of the Caputo fractional derivative.
• Fixed-point theory for proving solution properties.
• Jacobian matrix and fractional-order stability criteria for local stability analysis.
• Numerical simulations and error analysis for validation.
• Sensitivity analysis to assess parameter impact.

Main Results:

• The fractional-order prey-predator model is proven to be well-posed.
• Local stability of the system is rigorously analyzed.
• Numerical simulations confirm theoretical findings.
• The numerical scheme demonstrates convergence and precision.
• Parameter sensitivity analysis reveals key drivers of system dynamics.

Conclusions:

• The fractional-order model with memory effects provides a more realistic representation of prey-predator interactions.
• The established theoretical framework ensures reliable analysis of the model.
• Numerical methods effectively validate the model's behavior and stability.
• Understanding parameter sensitivity is vital for ecological management and prediction.