Abstract
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
x(n+1) = y(n)y(n-1)/x(n) (+/- 1 +/- y(n)y(n-1)), y(n+1) = x(n)x(n-1)/y(n)(+/- 1 +/- x(n)x(n-1)),
with positive initial conditions x(-1), x(0); y(-1) and y(0) are nonzero real numbers. If we let u(n) = x(n)x(n-1) and v(n) = YnYn-1, then these systems can be viewed as special cases of the system of the form
u(n+1) = f(v(n)), v(n+1) = g(u(n)).
This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let w(n) = v(2n), we have w(n+1) = f (g(w(n))) equivalent to h(w(n)). At a nonzero steady state w* of the last difference equation, we have
vertical bar h '*)vertical bar = vertical bar f '(g(w*))g '(w*)vertical bar = 1,
indicating that the system is degenerate at this steady state.