In this article, we study some properties of the solutions of the following difference equation:, n = 0, 1, ... where the initial conditions x −4 , x −3 , x −2 , x −1 , x 0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give specific form of the solutions of four special cases of this equation.

We study the qualitative behavior of a predator-prey model, where the carrying capacity of the predators environment is proportional to the number of prey. The considered system is given by the following rational difference equations:where the initial conditions x−2, x−1, x0, y−2, y−1, y0 are arbitrary positive real numbers. Also, we give specific form of the solutions of some special cases of this equation. Some numerical examples are given to verify our theoretical results.

We use Lyapunov functionals combined with the z-transform and obtain boundedness results regarding the solutions of the nonlinear Volterra system of difference equations y(n + 1) = f (y(n)) + n s=0 C(n, s)h(y(s)) + g(n).

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In this paper, we study some results on the following rational recursive sequences:xn+1 = xn−9 / ±1 ± xn−1xn−3xn−5xn−7xn−9, n = 0, 1, · · · ,where the initial conditions are arbitrary real numbers.