# Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description of the Fractional Discrete System

**Theorem**

**1**

**.**The solution of the following system

**Remark**

**1.**

## 3. Nonlinear Dynamics of the Fractional Discrete-Time Predator–Prey Leslie–Gower Model

#### 3.1. Commensurate Order FDNN Model

#### 3.2. Incommensurate Fractional Discrete System

## 4. The 0–1 Test for Chaos and the Complexity Analysis of the Model

#### 4.1. The 0–1 Test of the Model

#### 4.2. The ApEn of the Model

#### 4.3. The ${C}_{0}$ Complexity of the Model

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ali, I.; Saleem, M.T. Spatiotemporal Dynamics of Reaction–Diffusion System and Its Application to Turing Pattern Formation in a Gray–Scott Model. Mathematics
**2023**, 11, 1459. [Google Scholar] [CrossRef] - Atici, F.M.; Eloe, P. Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ.
**2009**, 3, 1–12. [Google Scholar] [CrossRef] - Anastassiou, G.A. Principles of delta fractional calculus on time scales and inequalities. Math. Comput. Model.
**2010**, 52, 556–566. [Google Scholar] [CrossRef] - Abdeljawad, T. On Riemann and Caputo fractional differences. Comput. Math. Appl.
**2011**, 62, 1602–1611. [Google Scholar] [CrossRef][Green Version] - Hanif, A.; Kashif Butt, A.I.; Ahmad, W. Numerical approach to solve Caputo-Fabrizio-fractional model of Corona pandemic with Optimal Control Design and analysis. Math. Methods Appl. Sci.
**2023**, 46, 9751–9782. [Google Scholar] [CrossRef] - He, Z.Y.; Abbes, A.; Jahanshahi, H.; Alotaibi, N.D.; Wang, Y. Fractional-order discrete-time SIR epidemic model with vaccination: Chaos and complexity. Mathematics
**2022**, 10, 165. [Google Scholar] [CrossRef] - Shatnawi, M.T.; Abbes, A.; Ouannas, A.; Batiha, I.M. A new two-dimensional fractional discrete rational map: Chaos and complexity. Phys. Scr.
**2022**, 98, 015208. [Google Scholar] [CrossRef] - Vignesh, D.; Banerjee, S. Dynamical analysis of a fractional discrete-time vocal system. Nonlinear Dyn.
**2023**, 111, 4501–4515. [Google Scholar] [CrossRef] - Abbes, A.; Ouannas, A.; Shawagfeh, N.; Khennaoui, A.A. Incommensurate Fractional Discrete Neural Network: Chaos and complexity. Eur. Phys. J. Plus
**2022**, 137, 235. [Google Scholar] [CrossRef] - Shatnawi, M.T.; Abbes, A.; Ouannas, A.; Batiha, I.M. Hidden multistability of fractional discrete non-equilibrium point memristor based map. Phys. Scr.
**2023**, 98, 035213. [Google Scholar] [CrossRef] - Abbes, A.; Ouannas, A.; Shawagfeh, N.; Jahanshahi, H. The fractional-order discrete COVID-19 pandemic model: Stability and chaos. Nonlinear Dyn.
**2023**, 111, 965–983. [Google Scholar] [CrossRef] [PubMed] - Batiha, I.M.; Alshorm, S.; Jebril, I.; Zraiqat, A.; Momani, Z.; Momani, S. Modified 5-point fractional formula with Richardson extrapolation. AIMS Math.
**2023**, 8, 9520–9534. [Google Scholar] [CrossRef] - Butt, A.I.K.; Imran, M.; Batool, S.; Nuwairan, M.A. Theoretical Analysis of a COVID-19 CF-Fractional Model to Optimally Control the Spread of Pandemic. Symmetry
**2023**, 15, 380. [Google Scholar] [CrossRef] - Abbes, A.; Ouannas, A.; Shawagfeh, N. The incommensurate fractional discrete macroeconomic system: Bifurcation, chaos and complexity. Chin. Phys. B
**2023**, 32, 030203. [Google Scholar] [CrossRef] - Khennaoui, A.A.; Ouannas, A.; Bendoukha, S.; Grassi, G.; Lozi, R.P.; Pham, V.T. On fractional–order discrete–time systems: Chaos, stabilization and synchronization. Chaos Solitons Fractals
**2019**, 119, 150–162. [Google Scholar] [CrossRef] - Ouannas, A.; Khennaoui, A.A.; Momani, S.; Grassi, G.; Pham, V.T. Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization. AIP Adv.
**2020**, 10, 045310. [Google Scholar] [CrossRef] - Ouannas, A.; Khennaoui, A.A.; Batiha, I.M.; Pham, V.T. Synchronization between fractional chaotic maps with different dimensions. In Fractional-Order Design; Radwan, A.G., Khanday, F.A., Said, L.A., Eds.; Volume 3 in Emerging Methodologies and Applications in Modelling; Academic Press: Cambridge, MA, USA, 2022; pp. 89–121. [Google Scholar]
- Saadeh, R.; Abbes, A.; Al-Husban, A.; Ouannas, A.; Grassi, G. The Fractional Discrete Predator–Prey Model: Chaos, Control and Synchronization. Fractal Fract.
**2023**, 7, 120. [Google Scholar] [CrossRef] - Rahmi, E.; Darti, I.; Suryanto, A.; Trisilowati. A modified Leslie–Gower model incorporating Beddington–deangelis functional response, double Allee effect and memory effect. Fractal Fract.
**2021**, 5, 84. [Google Scholar] [CrossRef] - Lin, S.; Chen, F.; Li, Z.; Chen, L. Complex dynamic behaviors of a modified discrete Leslie–Gower Predator–prey system with fear effect on prey species. Axioms
**2022**, 11, 520. [Google Scholar] [CrossRef] - Mondal, N.; Barman, D.; Roy, J.; Alam, S.; Sajid, M. A modified Leslie–Gower fractional order prey-predator interaction model incorporating the effect of fear on prey. J. Appl. Anal. Comput.
**2023**, 13, 198–232. [Google Scholar] [CrossRef] - Yuan, L.G.; Yang, Q.G. Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system. Appl. Math. Model.
**2015**, 39, 2345–2362. [Google Scholar] [CrossRef] - Ajaz, M.B.; Saeed, U.; Din, Q.; Ali, I.; Siddiqui, M.I. Bifurcation analysis and chaos control in discrete-time modified Leslie–Gower prey harvesting model. Adv. Differ. Equ.
**2020**, 2020, 45. [Google Scholar] [CrossRef] - Din, Q. Complexity and chaos control in a discrete-time prey-predator model. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 49, 113–134. [Google Scholar] [CrossRef] - Khan, A.Q.; Bukhari, S.A.H.; Almatrafi, M.B. Global dynamics, Neimark–Sacker bifurcation and hybrid control in a Leslie’s prey-predator model. Alex. Eng. J.
**2022**, 61, 11391–11404. [Google Scholar] [CrossRef] - Vinoth, S.; Sivasamy, R.; Sathiyanathan, K.; Unyong, B.; Vadivel, R.; Gunasekaran, N. A novel discrete-time Leslie–Gower model with the impact of Allee effect in predator population. Complexity
**2022**, 2022, 6931354. [Google Scholar] [CrossRef] - Allee, W.C. Animal Aggregations, a Study in General Sociology; The University of Chicago Press: Chicago, IL, USA, 1931. [Google Scholar]
- Aziz-Alaoui, M.A.; Okiye, M.D. Boundedness and global stability for a predator-prey model with modified Leslie–Gower and Holling-type II schemes. Appl. Math. Lett.
**2003**, 16, 1069–1075. [Google Scholar] [CrossRef][Green Version] - Wu, G.C.; Baleanu, D. Discrete fractional logistic map and its chaos. Nonlinear Dyn.
**2014**, 75, 283–287. [Google Scholar] [CrossRef] - Wu, G.C.; Baleanu, D. Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 22, 95–100. [Google Scholar] [CrossRef] - Gottwald, G.; Melbourne, I. The 0–1 test for chaos: A review. In Chaos Detection and Predictability; Springer: Berlin/Heidelberg, Germany, 2016; pp. 221–247. [Google Scholar]
- Pincus, S. Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA
**1991**, 88, 2297–2301. [Google Scholar] [CrossRef][Green Version] - Shen, E.; Cai, Z.; Gu, F. Mathematical foundation of a new complexity measure. Appl. Math. Mech.
**2005**, 26, 1188–1196. [Google Scholar] - He, S.; Sun, K.; Wang, H. Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system. Entropy
**2015**, 17, 8299–8311. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Bifurcations of (3) where $m\in (0,1)$ for (

**a**) $\vartheta =0.99$, (

**b**) $\vartheta =0.995$, (

**c**) $\vartheta =0.998$.

**Figure 3.**Time evolution of the states of (3) (${z}_{1}\left(r\right)$ (blue line) and ${z}_{2}\left(r\right)$ (red line)) for ${\delta}_{1}=3.3$, ${\delta}_{2}=1$, $\alpha =3.3$, ${h}_{1}=1.2$, ${h}_{2}=0.525$, $b=2.5$, $c=0.5$, $m=0.9$ and I.C $({z}_{1}\left(0\right),{z}_{2}\left(0\right))=(1,2)$.

**Figure 4.**Phase portraits of (3) for ${\delta}_{1}=3.3$, ${\delta}_{2}=1$, $\alpha =3.3$, ${h}_{1}=1.2$, ${h}_{2}=0.525$, $b=2.5$, $c=0.5$, $m=0.9$ and I.C $({z}_{1}\left(0\right),{z}_{2}\left(0\right))=(1,2)$.

**Figure 5.**Phase portraits of (15) for ${\delta}_{1}=3.3$, ${\delta}_{2}=1$, $\alpha =3.3$, ${h}_{1}=1.2$, ${h}_{2}=0.525$, $b=2.5$, $c=0.5$, $m=0.9$ and I.C $({z}_{1}\left(0\right),{z}_{2}\left(0\right))=(1,2)$.

**Figure 6.**Bifurcation of (15) versus $\delta $ for (

**a**) $({\vartheta}_{1},{\vartheta}_{2})=(0.96,0.15)$ (

**b**) $({\vartheta}_{1},{\vartheta}_{2})=(1,0.15)$ (

**c**) $({\vartheta}_{1},{\vartheta}_{2})=(1,0.7)$.

**Figure 7.**(

**a**) Bifurcation of (15) versus ${\vartheta}_{2}$ for ${\vartheta}_{1}=1$ (

**b**) The corresponding $L{E}_{max}$.

**Figure 8.**(

**a**) Bifurcation of (15) versus ${\vartheta}_{1}$ for ${\vartheta}_{2}=0.15$ (

**b**) The corresponding $L{E}_{max}$.

**Figure 9.**The $(p-q)$ plots of the commensurate fractional discrete predator–prey Leslie–Gower model with an Allee effect on the predator population (3) for (

**a**) $\vartheta =0.99$ (

**b**) $\vartheta =0.995$ (

**c**) $\vartheta =0.998$.

**Figure 10.**The $(p-q)$ plots of the incommensurate fractional discrete predator–prey Leslie–Gower model with an Allee effect on the predator population (15) for (

**a**) $({\vartheta}_{1},{\vartheta}_{2})=(0.96,0.15)$ (

**b**) $({\vartheta}_{1},{\vartheta}_{2})=(1,0.15)$ (

**c**) $({\vartheta}_{1},{\vartheta}_{2})=(1,0.7)$.

**Figure 11.**The ApEn of the fractional discrete predator–prey Leslie–Gower model with an Allee effect on the predator population for ${\delta}_{1}=3.3$, ${\delta}_{2}=1$, $\alpha =3.3$, ${h}_{1}=1.2$, ${h}_{2}=0.525$, $b=2.5$, $c=0.5$, $m=0.9$ and I.C $({z}_{1}\left(0\right),{z}_{2}\left(0\right))=(1,2)$ (

**a**) versus $\vartheta $, (

**b**) versus ${\vartheta}_{1}$ for ${\vartheta}_{2}=0.15$, (

**c**) versus ${\vartheta}_{2}$ for ${\vartheta}_{1}=1$.

**Figure 12.**The ${C}_{0}$ complexity of the fractional discrete predator–prey Leslie–Gower model with an Allee effect on the predator population for ${\delta}_{1}=3.3$, ${\delta}_{2}=1$, $\alpha =3.3$, ${h}_{1}=1.2$, ${h}_{2}=0.525$, $b=2.5$, $c=0.5$, $m=0.9$ and I.C $({z}_{1}\left(0\right),{z}_{2}\left(0\right))=(1,2)$ (

**a**) versus $\vartheta $, (

**b**) versus ${\vartheta}_{1}$ for ${\vartheta}_{2}=0.15$, (

**c**) versus ${\vartheta}_{2}$ for ${\vartheta}_{1}=1$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hamadneh, T.; Abbes, A.; Falahah, I.A.; AL-Khassawneh, Y.A.; Heilat, A.S.; Al-Husban, A.; Ouannas, A.
Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders. *Axioms* **2023**, *12*, 561.
https://doi.org/10.3390/axioms12060561

**AMA Style**

Hamadneh T, Abbes A, Falahah IA, AL-Khassawneh YA, Heilat AS, Al-Husban A, Ouannas A.
Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders. *Axioms*. 2023; 12(6):561.
https://doi.org/10.3390/axioms12060561

**Chicago/Turabian Style**

Hamadneh, Tareq, Abderrahmane Abbes, Ibraheem Abu Falahah, Yazan Alaya AL-Khassawneh, Ahmed Salem Heilat, Abdallah Al-Husban, and Adel Ouannas.
2023. "Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders" *Axioms* 12, no. 6: 561.
https://doi.org/10.3390/axioms12060561