Technique to derive discrete population models with delayed growth

Abstract

We provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles that a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of, what we consider, the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to obtain two models, a Beverton–Holt adult model and a Ricker adult model and discuss the global dynamics of both models.

Keywords:

• Delay difference equations
• global stability
• persistence
• critical delay threshold
• Beverton–Holt
• Ricker

Acknowledgments

We thank the handling editor and the reviewers for their suggestions that have lead to an improvement in the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Based on [17], the positive equilibrium is unstable if $\frac{\alpha -1}{\alpha }>2\cos \left(\frac{k\pi }{2k+1}\right)$, where the right-hand side approaches zero as $k\to \mathrm{\infty }$.

Additional information

Funding

The research of Gail S. K. Wolkowicz was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant.