Global Dynamics of Higher-Order Transcendental-Type Generalized Beverton–Holt Equations
We investigate generalized Beverton–Holt difference equations of order k of the form xn+1 = af(xn, xn−1, . . . , xn+1−k) 1 + f(xn, xn−1, . . . , xn+1−k) , n = 0, 1, . . . , k ≥ 1, where f is a function nondecreasing in all arguments, a > 0, and x0, . . . , x1−k ≥ 0 such that the solution is defined. We will discuss several interesting examples of such equations involving transcendental functions and present some general theory. In particular, we will analyze the global dynamics of the class of difference equations for which f(x, . . . , x) is chosen to be a concave function. Moreover, we give sufficient conditions to guarantee this equation has a unique positive and globally attracting fixed point. AMS Subject Classifications: 39A20, 39A28, 39A30.