This paper deals with the global asymptotic stability of the unique positive equilibrium point and the rate of convergence of positive solutions of the system of two recursive sequences $x_{n+1}=A+(y_{n-m}/y_n)$, $y_{n+1}=A+(x_{n-m}/x_n),$ $n=0,1,\dots ,$ and $m\in {\mathbb{Z}}^{+},$ where $A\in (0,\infty )$, $x_{-i}$ and $y_{-i}$ are arbitrary positive numbers for $i=0,1,\dots ,m.$ Also, we present some results about the general behaviour of solutions of aforementioned system. Finally, some numerical examples are given to demonstrate the effectiveness of the results obtained.

In the present paper, we mainly investigate the qualitative behavior of the solutions of a discrete system of difference equations xn+1 = ? + Pmi =1 xn?i yn , yn+1 = ? + Pmi =1 yn?i xn , n ? N where ?, ? ? (0,?), m ? Z+, x?i and y?i are non-negative real numbers for i ? {0, 1, ...,m}. Namely, we discuss the boundedness character and the asymptotic stability properties of steady states of the mentioned system. Finally, for this system, we give a rate of convergence result which has an important place in the discrete dynamical systems. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.

Hepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number $$\mathcal {R}_0$$R0 , stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when $$\mathcal {R}_0=1$$R0=1 . It is concluded that the endemic equilibrium is globally asymptotically stable when $$\mathcal {R}_0>1$$R0>1 , by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form: , , n = 0,1, …, where the parameters α, β, γ, α₁, β₁, γ₁, p and the initial values u_−i, v_−i for i = 0,1, 2 are positive real numbers.