The Beverton–Holt q-difference equation

Abstract

The Beverton–Holt model is a classical population model which has been considered in the literature for the discrete-time case. Its continuous-time analogue is the well-known logistic model. In this paper, we consider a quantum calculus analogue of the Beverton–Holt equation. We use a recently introduced concept of periodic functions in quantum calculus in order to study the existence of periodic solutions of the Beverton–Holt q-difference equation. Moreover, we present proofs of quantum calculus versions of two so-called Cushing–Henson conjectures.

Keywords:

• Beverton–Holt equation
• Cushing–Henson conjecture
• time scale
• dynamic equation
• logistic equation
• Jensen inequality

AMS Subject Classifications:

• 39A10
• 39A11
• 39A12
• 39A20
• 34C25

Notes

Note that, as Jim Cushing points out, the terminology ‘discrete logistic equation’ differs, due to Robert May [11], in a slight but essential way, from the discrete model (2).