On the Cushing–Henson conjecture, delay difference equations and attenuant cycles

Abstract

We consider the second part of the Cushing–Henson conjecture (the cycle's average is less than the average of carrying capacities; the first part of the conjecture deals with the existence and global stability of periodic cycles) for a periodic delay difference equation

Sufficient conditions on f and h _i are obtained, when the second part of the conjecture is valid. We demonstrate the sharpness of these conditions by presenting several counterexamples. In addition, sufficient global attractivity conditions are deduced for the Pielou equation.

Keywords:

• Periodic difference equations
• Average population density
• Beverton–Holt equation
• Pielou equation
• Global asymptotic stability

Keywords:

• 39A11
• 92D25

Acknowledgements

The authors are grateful to Profs. S. Elaydi and R.J. Sacker for useful discussions and to the anonymous referee for valuable remarks and comments.

Notes

^¶ Partially supported by the NSERC Research Grant and the AIF Research Grant.

^§ The work was partially implemented at the University of Calgary and supported by the AIF Research Grant.

Additional information

Notes on contributors

E. Braverman

¶ ¶ Partially supported by the NSERC Research Grant and the AIF Research Grant.

S.H. Saker

§ § The work was partially implemented at the University of Calgary and supported by the AIF Research Grant.