Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model

Abdul Khaliq (), Tarek F. Ibrahim, Abeer M. Alotaibi, Muhammad Shoaib and Mohammed Abd El-Moneam
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Abdul Khaliq: Department of Mathematics, Riphah International University, Lahore Campus, Lahore 54000, Pakistan
Tarek F. Ibrahim: Department of Mathematics, Faculty of Sciences and Arts (Mahayel), King Khalid University, Abha 62529, Saudi Arabia
Abeer M. Alotaibi: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Muhammad Shoaib: Research and Development Department, AZ International Publishing House, Lahore 54000, Pakistan
Mohammed Abd El-Moneam: Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia

Mathematics, 2022, vol. 10, issue 21, 1-30

Abstract: 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 R 3 . 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 O , P , Q , R , S , T , U , V 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 α , β , γ , δ , ζ , η , μ , ε , υ , ρ , σ , ω ∈ R + and initial conditions x 0 , y 0 , z 0 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.

Keywords: fixed points; stability; predator-prey system; rate of convergence; global stability; boundedness; Lotka–Volterra model; three-species model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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