Solutions and properties of some degenerate systems of difference equations

This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational difference equations of order two (Formula Presented), with positive initial conditions x₋₁, x₀, y₋₁ and y₀ are nonzero real numbers. If we let u_n = x_nx_n−1 and vn = y_ny_n−1, then these systems can be viewed as special cases of the system of the form un+1 = f (v_n), v_n+1 = g(un). This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let w_n = u_2n, we have w_n+1 = f (g(w_n)) ≡ h(w_n). At a nonzero steady state w^∗ of the last difference equation, we have |h′^∗)| = |f′ (g(w^∗))g′ (w^∗)| = 1, indicating that the system is degenerate at this steady state.