# Abdul Qadeer Khan's research while affiliated with Central University of Kashmir and other places

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## Publications (26)

In this paper, we explore local dynamics with topological classifications, bifurcation analysis, and chaos control in a discrete‐time COVID‐19 epidemic model in the interior of ℝ+4$$ {\mathbb{R}}_{+}^4 $$. It is explored that for all involved parametric values, discrete‐time COVID‐19 epidemic model has boundary equilibrium solution and also it has...

Phytoplankton and zooplankton’s interconnection coordinate in various dynamical processes that occur in ecological population and make it a fascinating subject matter to explore. Time delay is an additional factor that plays an imperative part in the dynamical frameworks. For a stable aquatic ecosystem, the growth of both zooplankton and phytoplank...

The local dynamics, chaos, and bifurcations of a discrete Brusselator system are investigated. It is shown that a discrete Brusselator system has an interior fixed point P 1 , r if r > 0 . Then, by linear stability theory, local dynamical characteristics are explored at interior fixed point P 1 , r . Furthermore, for the discrete Brusselator system...

In this work, we explore the boundedness and local and global asymptotic behavior of the solutions to a second-order difference formula of the exponential type ξn+1=a+bξn−1+cξn−1e−ρξn, where a,c,ρ∈(0,∞), b∈(0,1) and the initials ξ0,ξ−1 are non-negative real numbers. Some other special cases are given. We provide two concrete numerical examples to c...

In this paper, we obtain the solution forms of fifth order systems of rational difference equations P n + 1 = P n 4 S n 2 Q n / S n − 3 Q n − 1 1 ± P n − 4 S n − 2 Q n , Q n + 1 = Q n − 4 P n − 2 S n / P n − 3 S n − 1 1 ± Q n − 4 P n − 2 S n , and S n + 1 = S n − 4 Q n − 2 P n / Q n − 3 P n − 1 1 ± S n − 4 Q n − 2 P n . Where the initial values are...

In this paper, we deal with the form and the periodicity of the solutions of the max-type system of difference equations u n + 1 = max A n / v n − 1 , u n − 1 , v n + 1 = max B n / w n − 1 , w n + 1 = max C n / u n − 1 , w n − 1 where the initial conditions u − 1 , u 0 ∈ 0 , ∞ , v − 1 , v 0 ∈ 0 , ∞ , w − 1 , w 0 ∈ 0 , ∞ and A n n ∈ N 0 , B n n ∈ N...

Most nonlinear difference equations have exact solutions that are not always possible to obtain theoretically. As a result, a large number of researchers investigate several qualitative aspects of difference equations in order to predict their lengthy behavior. The goal of our research is to obtain the solutions of a tenth-order difference equation...

In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete activator-inhibitor system. More specifically, it is proved that discrete-time activator-inhibitor system has an interior equilibrium solution. Then, by using linear stability theory, local dynamics with different topological classifications for the in...

In this paper, we explore the global dynamical characteristics, boundedness, and rate of convergence of certain higher-order discrete systems of difference equations. More precisely, it is proved that for all involved respective parameters, discrete systems have a trivial fixed point. We have studied local and global dynamical characteristics at tr...

In this work, we derive the solution formulas and study their behaviors for the difference equations and with real initials and positive parameters. We show that there exist periodic solutions for the second equation under certain conditions when . Finally, we give some illustrative examples.
1. Introduction
In [1–5], the first author ([1] togethe...

In this paper, we study the solution of the difference equation , where the initials are positive real numbers.
1. Introduction
Difference equations appear naturally as discrete analogues in many sciences such as biology, ecology, and physics. In recent years, many authors studied the solution form of difference equations. For instance, Cinar [1–3...

The local behavior with topological classifications, bifurcation analysis, chaos control, boundedness, and global attractivity of the discrete-time Kolmogorov model with piecewise-constant argument are investigated. It is explored that Kolmogorov model has trivial and two semitrival fixed points for all involved parameters, but it has an interior f...

We explore existence of fixed points, topological classifications around fixed points, existence of periodic points and prime period, and bifurcation analysis of a three-species discrete food chain model with harvesting. Finally, theoretical results are numerically verified.
1. Introduction
Many different types of interactions exist in nature betw...

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. yi+...

Mathematical Problems in Engineering

In this paper, we explore local stability, attractor, periodicity character, and boundedness solutions of the second-order nonlinear difference equation. Finally, obtained results are verified numerically.
1. Introduction
For decades, the qualitative analysis of difference equations has been steadily increasing. This is due to the fact that differ...

A convex polytope is the convex hull of a finite set of points in the Euclidean space R n. By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. e detour index is defined t...

The principle purpose of this article is to examine some stability properties for the fixed point of the below rational difference equation where , and are arbitrary real numbers. Moreover, solutions for some special cases of the proposed difference equation are introduced.
1. Introduction
In recent years, many researchers have tended to use diffe...

In this paper, we explore the bifurcations and hybrid control in a $3\times3$ discrete-time predator-prey model in the interior of $\mathbb{R}_+^3$. It is proved that $3\times3$ model has four boundary fixed points: $P_{000}(0,0,0)$, $P_{0y0}\left(0,\frac{r-1}{r},0\right)$, $P_{0yz}\left(0,\frac{d}{f},\frac{rf-f-dr}{cf}\right)$, $P_{x0z}\left(\frac...

We explore the local dynamics, N‐S bifurcation, and hybrid control in a discrete‐time Lotka‐Volterra predator‐prey model in R+2. It is shown that ∀ parametric values, model has two boundary equilibria: P00(0,0) and Px0(1,0), and a unique positive equilibrium point: Pxy+dc,rc−dbc if c>d. We explored the local dynamics along with different topologica...

We study the dynamical properties about fixed points, the existence of prime period and periodic points, and transcritical bifurcation of a one‐dimensional laser model in R+. For the special case, we explore the global dynamics about fixed points, boundedness of positive solution, construction of invariant rectangle, existence of prime period‐2 sol...

Abstract We study the local dynamics and bifurcations of a two-dimensional discrete-time predator–prey model in the closed first quadrant R+2 $\mathbb{R}_{+}^{2}$. It is proved that the model has two boundary equilibria: O(0,0) $O(0,0)$, A(α1−1α1,0) $A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$ and a unique positive equilibrium B(1α2,α1α2−α1−α2α2) $B...

In this paper, we study the global dynamics and bifurcations of a two-dimensional discrete time host–parasitoid model with strong Allee effect. The existence of fixed points and their stability are analysed in all allowed parametric region. The bifurcation analysis shows that the model can undergo fold bifurcation and Neimark–Sacker bifurcation. As...

In this paper, stability and bifurcation of a two-dimensional ratio-dependence predator–prey model has been studied in the close first quadrant R + 2 . It is proved that the model undergoes a period-doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium...

In this paper, bifurcations of a two dimensional discrete time plant-herbivore system formulated by Allen et al. (1993) have been studied. It is proved that the system undergoes a transcritical bifurcation in a small neighborhood of a boundary equilibrium and a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium. A...

## Citations

... Now, it is important here to mention that discrete-time models described by difference equations are more appropriate than continuous-time models described by differential equations, and also discrete-time models provide more efficient computational results for numerical simulation [9]. For instance, in recent years, many mathematicians have investigated the dynamical characteristics of discrete-time biological models instead of continuous-time models [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. So, motivated by the aforementioned studies, the purpose of this paper is to investigate the dynamical characteristics of an activator-inhibitor system that is a discrete analogue of the continuous-time model (1.1), by using a non-standard finite difference scheme [28]. ...

... Difference equations display naturally as discrete peer and as numerical solutions of differential equations having more applications in ecology, biology, physics, economy, and so forth. For all that the difference equations are quite simple in expressions, it is frequently difficult to realize completely the dynamics of their solutions see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the related references therein. ...

... In these previous studies, Bača et al. [16,17] stated an antiprism A n , convex polytopes D n and R n have a magic labeling of type (1, 1, 0), wherease Tarawneh et al. [18] proved the edge irregularity strength on C n P m for n ≥ 4 and m = 2, 3. For other existing studies of convex polytopes, we can refer to [19][20][21][22][23]. Eventually, we obtained the exact reflexive edge strength for these graphs. ...

... For example, a discrete-time model with a single species can display chaos and more complex dynamical behavior, but chaos requires at least three species in a continuous-time model (see literature [3][4][5][6][7][8] ). Discrete-time and continuous-time models that regulate population systems can be found in previous studies [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references listed therein. Many of these studies have used the logistic map to represent the growth in the prey population. ...

... Moreover, Ahmed et al. [5], found new solutions and investigated the dynamical analysis for some nonlinear difference relations of fifteenth order. The authors in [6] obtained novel structures for the solutions of a rational recursive relation. The local and global stability, boundedness, periodicity and solutions of a second order difference equation were investigated in [7]. ...

... For more linked results on this side can be found in [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. ...

... Among the features that are discussed in the study of the dynamical behavior of discrete-time models, model bifurcations are the most important. Most of the codim-1 bifurcations have been studied in recent years, [1][2][3][4] but little attention has been paid to the codim-2 bifurcations; see previous works. 5 -16 In ecology, population dynamics are generally determined by both discrete-time and continuous-time dynamics. ...

... They are the discrete analogues of differential equations and arise whenever an independent variable can have only discrete values. Besides their theoretical importance, their applications are wide-spread, ranging from modeling to discretizations (see [3,5,7,13] and the references therein). methods and results for autonomous systems are no longer applicable for non-autonomous equations, their qualitative study is much more complicated, and they require special attention. ...

... It is generally recognised that when there are non-overlapping generations in populations, discrete-time models defined by difference equations are more useful and trustworthy than continuous-time models. Furthermore, as compared to continuous models, these models give efficient computing results for numerical simulations as well as richer dynamical properties [1][2][3][4][5][6][7] . Many fascinating works on the stability, bifurcation and chaotic occurrences in discrete temporal models have appeared in the literature in recent years [8][9][10][11][12][13][14][15] . ...

... Among the features that are discussed in the study of the dynamical behavior of discrete-time models, model bifurcations are the most important. Most of the codim-1 bifurcations have been studied in recent years, [1][2][3][4] but little attention has been paid to the codim-2 bifurcations; see previous works. 5 -16 In ecology, population dynamics are generally determined by both discrete-time and continuous-time dynamics. ...