Qamar Din

Dynamics of a discrete Lotka-Volterra model

AbstractIn this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, and global behavior of equilibrium points of a discrete Lotka-Volterra model given by xn+1=αxn−βxnyn1+γxn,yn+1=δyn+ϵxnyn1+ηyn, where parameters α,β,γ,δ,ϵ,η∈R+, and initial conditions x0, y0 are positive real numbers. Moreover, the rate of convergence of a solution that converges to the unique positive equilibrium point is discussed. Some numerical examples are given to verify our theoretical results.MSC:39A10, 40A05.

Dynamics of a fourth-order system of rational difference equations

AbstractIn this paper, we study the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form xn+1=αxn−3β+γynyn−1yn−2yn−3,yn+1=α1yn−3β1+γ1xnxn−1xn−2xn−3,n=0,1,… , where the parameters α, β, γ, α1, β1, γ1 and initial conditions x0, x−1, x−2, x−3, y0, y−1, y−2, y−3 are positive real numbers. Some numerical examples are given to verify our theoretical results.MSC:39A10, 40A05.

Neimark-Sacker bifurcation and chaos control in Hassell-Varley model

Abstract We investigate the complex behaviour of a modified Nicholson–Bailey model. The modification is proposed by Hassel and Varley taking into account that interaction between parasitoids is taken in such a way that the searching area per parasitoid is inversely proportional to the m-th power of the population density of parasitoids. Under certain parametric conditions the unique positive equilibrium point of system is locally asymptotically stable. Moreover, it is proved that system undergoes Neimark-Sacker bifurcation for small range of parameters by using standard mathematical techniques of bifurcation theory. In order to control Neimark-Sacker bifurcation, we apply simple feedback control strategy and pole-placement technique which is a modification of OGY method. Moreover, the hybrid control methodology is also implemented for chaos controlling. Numerical simulations are provided to illustrate theoretical discussion.

Qualitative behavior of a host-pathogen model

In this paper, we study the qualitative behavior of a discrete-time host-pathogen model for spread of an infectious disease with permanent immunity. The time-step is equal to the duration of the infectious phase. Moreover, the local asymptotic stability, the global behavior of unique positive equilibrium point, and the rate of convergence of positive solutions is discussed. Some numerical examples are given to verify our theoretical results.MSC:39A10, 40A05.

Global stability and Neimark-Sacker bifurcation of a host-parasitoid model

ABSTRACT We investigate qualitative behaviour of a density-dependent discrete-time host-parasitoid model. Particularly, we study boundedness of solutions, existence and uniqueness of positive steady-state, local and global asymptotic stability of the unique positive equilibrium point and rate of convergence of modified host-parasitoid model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation with the help of bifurcation theory. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behaviour over a broad range of parameters. The computation of the maximum Lyapunov exponents confirm the presence of chaotic behaviour in the model.

Global dynamics of some systems of higher-order rational difference equations

In the present work, we study the qualitative behavior of two systems of higher-order rational difference equations. More precisely, we study the local asymptotic stability, instability, global asymptotic stability of equilibrium points and rate of convergence of positive solutions of these systems. Our results considerably extend and improve some recent results in the literature. Some numerical examples are given to verify our theoretical results.MSC:39A10, 40A05.

Asymptotic behavior of a Nicholson-Bailey model

AbstractIn this paper, we study the qualitative behavior of a Nicholson-Bailey host-parasitoid model given by xn+1=Rxne−ayn,yn+1=xn(1−e−ayn), where R, a and initial conditions x0, y0 are positive real numbers. More precisely, we investigate the local asymptotic stability of the unique positive equilibrium point of this system. Some numerical examples are given to verify our theoretical results.MSC:39A10, 40A05.

Qualitative Behaviour of Generalised Beddington Model

Abstract This work is related to the dynamics of a discrete-time density-dependent generalised Beddington model. Moreover, we investigate the existence and uniqueness of positive equilibrium point, boundedness character, local and global behaviours of unique positive equilibrium point, and the rate of convergence of positive solutions that converge to the unique positive equilibrium point of this model. Numerical examples are provided to illustrate theoretical discussion.

Bifurcation analysis and chaos control in discrete-time glycolysis models

In this paper, the qualitative behavior of two discrete-time glycolysis models is discussed. The discrete-time models are obtained by implementing forward Euler’s scheme and nonstandard finite difference method. The parametric conditions for local asymptotic stability of positive steady-states are investigated. Moreover, we discuss the existence and directions of period-doubling and Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. OGY feedback control and hybrid control methods are implemented in order to control chaos in discrete-time glycolysis model due to emergence of period-doubling and Neimark–Sacker bifurcations. Numerical simulations are provided to illustrate theoretical discussion.

On A System of Rational Difference Equation

Abstract In this paper, we study local asymptotic stability, global character and periodic nature of solutions of the system of rational difference equations given by xn+1= , yn=, n=0, 1,…, where the parameters a; b; c; d; e; f ∊ (0; ∞), and with initial conditions x0; y0 ∊ (0; ∞). Some numerical examples are given to illustrate our results.