A Discrete Model of Three Species Prey- Predator System

ABSTRACT: This paper investigates an eco-system with two prey species and a predator. The model equations constitute a set of three first order non-linear coupled difference equations. All possible positive equilibrium points of the model are computed and criteria for the stability of all equilibrium states are established. Simulations are performed in order to illustrate the theoretical results and gain further insight into the behaviour of the system

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, using the Hilfer operator. Hybrid fixed point theorem is considered to study the existence of solutions, and the uniqueness of the solution is proved using Banach contraction theorem. Stability results in the sense of Ulam and Hyers, and its generalized form of stability for the considered initial value problem are established and we depict numerical simulations to demonstrate the impact of the fractional order on stability

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations

Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term

The paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Δψλzηλ+pλzηλ+qλF∑s=λ0λ−1+μ λ−s−1−μys=0, where zλ=aλ+bλΔμyλ, Δμ stands for the fractional difference operator in Riemann-Liouville settings and of order μ, 0<μ≤1, and η≥1 is a quotient of odd positive integers and λ∈ℕλ0+1−μ. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples

Bifurcation and Stability Analysis of a Discrete Time Sir Epidemic Model with Vaccination

In this paper, we study the qualitative behavior of a discrete-time epidemic model with vaccination. Analysis of the model shows forth that the Disease Free Equilibrium (DFE) point is asymptotically stable if the basic reproduction number R0 is less than one, while the Endemic Equilibrium (EE) point is asymptotically stable if the basic reproduction number R0 is greater than one. The results are reinforced with numerical simulations and enhanced with graphical representations like time trajectories, phase portraits and bifurcation diagrams for different sets of parameter values