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

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 R3. 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 x0,y0,z0 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.

The Dynamical Behavior of a Three-Dimensional System of Exponential Difference Equations

“…Thus, from (1), there exists m 0 such that (10) holds true. This completes the proof of the theorem.…”

“…Modeling dynamic systems [10]: Difference equations are often used to model dynamic systems, such as population growth, stock prices, or the spread of infectious diseases. By describing the relationship between the values of a sequence over time, these equations help researchers to understand how these systems behave.…”

“…Analyzing the dynamics of solutions to systems of difference equations, and discussing the local and global asymptotic stability of their equilibriums, is interesting [10,12,13,[15][16][17][18]. The authors in [19], obtained the behavior of solutions to the difference equation:…”

Harvested Predator–Prey Models Considering Marine Reserve Areas: Systematic Literature Review

“…However, various mathematical model options can be explored for future investigations. Potential directions for model development include a difference model [36][37][38], fractional difference [39,40], fractional differential [41,42], and partial differential [43][44][45]. Given that the data obtained in applications are discrete in time, the inclusion of a difference- Srinivas et al [24] developed a mathematical model for predator-prey dynamics in marine reserves and harvesting areas considering both immature and mature predator species.…”

Stability and Bifurcation Analysis in a Discrete Predator–Prey System of Leslie Type with Radio-Dependent Simplified Holling Type IV Functional Response

“…Predator-prey models are the building blocks of ecosystems, as biomass grows from resource masses. This topic plays an important role in ecology [4][5][6][7][8][9][10][11][12][13][14][15][16].…”