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A fractional model for predator-prey with omnivore
E Bonyah1, A Atangana2, A A Elsadany3
1Department of Mathematics Education, University of Education, Winneba (Kumasi Campus), Kumasi, Ghana.
Chaos (Woodbury, N.Y.)
|February 3, 2019
Summary
This study explores predator-prey-omnivore models with diverse waiting time distributions, including power law and Mittag-Leffler. New fractional derivative models reveal novel attractors, enhancing ecological system analysis.
Area of Science:
- Ecological Dynamics
- Mathematical Biology
- Fractional Calculus
Background:
- Predator-prey interactions are fundamental in ecology.
- Incorporating omnivory and complex waiting time distributions enhances ecological model realism.
- Fractional calculus offers advanced tools for modeling memory effects in dynamic systems.
Purpose of the Study:
- To investigate predator-prey-omnivore models with three distinct waiting time distributions: power law, exponential decay (linked to Poisson), and Mittag-Leffler.
- To analyze the existence and uniqueness of solutions using the fixed-point Picard method.
- To explore novel attractors and enhance modeling capabilities using fractional derivatives.
Main Methods:
- Utilized power law, exponential decay, and Mittag-Leffler distributions for waiting times.
- Applied the fixed-point Picard method to establish conditions for unique exact solutions.
- Employed a numerical scheme to solve fractional differential equations and generate simulations.
- Investigated the Atangana-Baleanu fractional derivative and modified models for attractor analysis.
Main Results:
- Established conditions for unique solutions in models with different waiting time distributions.
- Identified a new attractor combining Brownian motion and power law in the Atangana-Baleanu fractional derivative model.
- Developed a modified model yielding more attractors, even in fractional differential cases.
- Demonstrated that non-local, non-singular kernel fractional derivatives capture more complex dynamics and physical problems.
Conclusions:
- Fractional calculus, particularly with non-local kernels, provides a richer framework for ecological modeling.
- The proposed models and numerical methods offer enhanced capabilities for understanding complex ecological interactions and system dynamics.
- The study highlights the importance of waiting time distributions and fractional derivatives in revealing emergent behaviors like novel attractors.
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