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Three-species predator-prey model with respect to Caputo and Caputo-Fabrizio fractional operators

Leila Eftekhari, Moein Khalighi, Soleiman Hosseinpour, Leo Lahti

TL;DR

This work analyzes a three-species Lotka–Volterra system under Caputo and Caputo–Fabrizio fractional operators to capture distributed lag effects. It develops a corrected Adams–Bashforth–type scheme tailored to the CF operator and derives equilibrium points along with local stability criteria, comparing results to the Caputo case. The study reveals that the operator type and fractional order $\alpha$ significantly shape stability regions and long-term dynamics, with CF often yielding different stable equilibria than Caputo. The findings offer a practical framework for modeling memory-rich ecological interactions and guide the choice of fractional operator in ecological modeling. The proposed numerical method enhances stability and efficiency for CF-based simulations, enabling robust exploration of CF vs Caputo dynamics in multi-species systems.

Abstract

We study distributed lag effects in three-dimensional Lotka-Volterra systems by applying the concept of fractional calculus. We derive a new numerical method that provides enhanced stability for the Caputo-Fabrizio operator based on Adams-Bashforth method, considering non-singular kernel in the definition of Caputo-Fabrizio operator. We investigate the stability conditions of this system with comparisons to the Caputo fractional derivative. Numerical results show that the type of differential operators and the value of orders significantly influence the stability of the numerical solution, and dynamics of the Lotka-Volterra system.

Three-species predator-prey model with respect to Caputo and Caputo-Fabrizio fractional operators

TL;DR

This work analyzes a three-species Lotka–Volterra system under Caputo and Caputo–Fabrizio fractional operators to capture distributed lag effects. It develops a corrected Adams–Bashforth–type scheme tailored to the CF operator and derives equilibrium points along with local stability criteria, comparing results to the Caputo case. The study reveals that the operator type and fractional order significantly shape stability regions and long-term dynamics, with CF often yielding different stable equilibria than Caputo. The findings offer a practical framework for modeling memory-rich ecological interactions and guide the choice of fractional operator in ecological modeling. The proposed numerical method enhances stability and efficiency for CF-based simulations, enabling robust exploration of CF vs Caputo dynamics in multi-species systems.

Abstract

We study distributed lag effects in three-dimensional Lotka-Volterra systems by applying the concept of fractional calculus. We derive a new numerical method that provides enhanced stability for the Caputo-Fabrizio operator based on Adams-Bashforth method, considering non-singular kernel in the definition of Caputo-Fabrizio operator. We investigate the stability conditions of this system with comparisons to the Caputo fractional derivative. Numerical results show that the type of differential operators and the value of orders significantly influence the stability of the numerical solution, and dynamics of the Lotka-Volterra system.

Paper Structure

This paper contains 18 sections, 4 theorems, 48 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Fractional calculus
  4. Stability of the fractional-order system
  5. Modeling and its stability
  6. Fractional Lotka-Volterra model
  7. Stability of the model
  8. The first equilibrium
  9. The second equilibrium
  10. The third equilibrium
  11. The fourth equilibrium
  12. The fifth equilibrium
  13. Numerical algorithm
  14. Numerical implementation
  15. Example 1
  16. ...and 3 more sections

Key Result

Theorem 2.1

The linear autonomous system e11 with Caputo fractional derivative for $0< \alpha< 1$ is asymptotically stable if and only if $\left| \mathrm{arg} (\mathrm{spec}(A))\right| >\dfrac{\alpha \pi}{2}$, $\mathrm{spec}(A)$ is the spectrum (set of all eigenvalues) of $A$Matignon96stabilityresults.

Figures (5)

  • Figure 1: Comparison stability and unstability domain of Caputo and CF operators
  • Figure 2: (left) Comparing the behavior of Caputo and CF operators for system \ref{['e1']} with the parameters of \ref{['tab:table']}, (right) converging to $\epsilon_3$ with CF operator for $\alpha\leq0.66$ and $(x_0,y_0,z_0)=(1.6,1.9,0)$.
  • Figure 3: System \ref{['e1']} with the parameters of \ref{['tab:table']} and $(x_0,y_0,z_0)=(2,2,3)$ is asymptotically stable for Caputo (left) and unstable for CF (right).
  • Figure 4: System \ref{['e1']} with the parameters of the \ref{['tab:table']} and $(x_0,y_0,z_0)=(0.5,0.1,5)$ is asymptotically stable for both Caputo (left) and CF (right) at $\epsilon_2$ with different oscillations.
  • Figure 5: System \ref{['e1']} with the parameters of the \ref{['tab:table']} and $(x_0,y_0,z_0)=(3,8,0)$ is unstable for Caputo (left) and asymptotically stable for CF (right).

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof