Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations

For nonnegative real numbers α, β, γ , A, B, and C such that B C > 0 and α β γ > 0, the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . . . has a unique positive equilibrium. A proof is given here for the following statements: 1 For every choice of positive parameters α, β, γ , A, B, and C, all solutions to the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . . . , x−1, x0 ∈ 0,∞ converge to the positive equilibrium or to a prime period-two solution. 2 For every choice of positive parameters α, β, γ , B, and C, all solutions to the difference equation xn 1 α βxn γxn−1 / Bxn Cxn−1 , n 0, 1, 2, . . . , x−1, x0 ∈ 0,∞ converge to the positive equilibrium or to a prime period-two solution.