Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model
This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inlin...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/21/4015 |
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| author | Abdul Khaliq Tarek F. Ibrahim Abeer M. Alotaibi Muhammad Shoaib Mohammed Abd El-Moneam |
| author_facet | Abdul Khaliq Tarek F. Ibrahim Abeer M. Alotaibi Muhammad Shoaib Mohammed Abd El-Moneam |
| author_sort | Abdul Khaliq |
| collection | DOAJ |
| description | This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>R</mi><mn>3</mn></msup></semantics></math></inline-formula>. In some assertive parametric circumstances, the discrete-time model has eight equilibrium points among which one is a special or unique positive equilibrium point. We have also investigated the local and global behavior of equilibrium points of an achievable three-dimensional discrete-time two predators and one prey Lotka–Volterra model. The conversion of a continuous-type model into its discrete counterpart model has been completed by adopting a dynamically consistent nonstandard difference scheme with the end goal that the equilibrium points are conserved in twin cases. The difficulty lies in how to find all fixed points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>U</mi><mo>,</mo><mi>V</mi></mrow></semantics></math></inline-formula> and the Jacobian matrix and its characteristic polynomial at the unique positive fixed point. For that purpose, we use Mathematica software to find the equilibrium points and all of the Jacobian matrices at those equilibrium points. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of the obtained system about all of its equilibrium points. The discrete Lotka–Volterra model in three dimensions is given by system (3), where parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>ζ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ε</mi><mo>,</mo><mi>υ</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>ω</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> and initial conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> are positive real numbers. Additionally, the rate of convergence of a solution that converges to a unique positive equilibrium point is discussed. To represent theoretical perceptions, some numerical debates are introduced, including phase portraits. |
| first_indexed | 2024-03-09T18:53:15Z |
| format | Article |
| id | doaj.art-d632025ca3694c36b9358d44e54bad1c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T18:53:15Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d632025ca3694c36b9358d44e54bad1c2023-11-24T05:43:35ZengMDPI AGMathematics2227-73902022-10-011021401510.3390/math10214015Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra ModelAbdul Khaliq0Tarek F. Ibrahim1Abeer M. Alotaibi2Muhammad Shoaib3Mohammed Abd El-Moneam4Department of Mathematics, Riphah International University, Lahore Campus, Lahore 54000, PakistanDepartment of Mathematics, Faculty of Sciences and Arts (Mahayel), King Khalid University, Abha 62529, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi ArabiaResearch and Development Department, AZ International Publishing House, Lahore 54000, PakistanDepartment of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi ArabiaThis research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>R</mi><mn>3</mn></msup></semantics></math></inline-formula>. In some assertive parametric circumstances, the discrete-time model has eight equilibrium points among which one is a special or unique positive equilibrium point. We have also investigated the local and global behavior of equilibrium points of an achievable three-dimensional discrete-time two predators and one prey Lotka–Volterra model. The conversion of a continuous-type model into its discrete counterpart model has been completed by adopting a dynamically consistent nonstandard difference scheme with the end goal that the equilibrium points are conserved in twin cases. The difficulty lies in how to find all fixed points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo>,</mo><mi>R</mi><mo>,</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>U</mi><mo>,</mo><mi>V</mi></mrow></semantics></math></inline-formula> and the Jacobian matrix and its characteristic polynomial at the unique positive fixed point. For that purpose, we use Mathematica software to find the equilibrium points and all of the Jacobian matrices at those equilibrium points. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of the obtained system about all of its equilibrium points. The discrete Lotka–Volterra model in three dimensions is given by system (3), where parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>ζ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ε</mi><mo>,</mo><mi>υ</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>ω</mi><mo>∈</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> and initial conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>y</mi><mn>0</mn></msub><mo>,</mo><msub><mi>z</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> are positive real numbers. Additionally, the rate of convergence of a solution that converges to a unique positive equilibrium point is discussed. To represent theoretical perceptions, some numerical debates are introduced, including phase portraits.https://www.mdpi.com/2227-7390/10/21/4015fixed pointsstabilitypredator-prey systemrate of convergenceglobal stabilityboundedness |
| spellingShingle | Abdul Khaliq Tarek F. Ibrahim Abeer M. Alotaibi Muhammad Shoaib Mohammed Abd El-Moneam Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model Mathematics fixed points stability predator-prey system rate of convergence global stability boundedness |
| title | Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model |
| title_full | Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model |
| title_fullStr | Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model |
| title_full_unstemmed | Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model |
| title_short | Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model |
| title_sort | dynamical analysis of discrete time two predators one prey lotka volterra model |
| topic | fixed points stability predator-prey system rate of convergence global stability boundedness |
| url | https://www.mdpi.com/2227-7390/10/21/4015 |
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