### Mathematics Colloquium

**Sabrina Streipert
**

McMaster University

**Title:** A Class of Discrete Delay Population Models

**Date:** Friday, January 14, 2022

**Place and Time:** Zoom, 3:05-3:55 pm

**Abstract.**

The continuous Hutchinson model is a delay logistic growth model, where a delay was introduced in the per-capita growth rate. Despite its popularity, this delay differential equation exhibits some questionable behavior as the population persists independent of the delay. One of its discretizations, the so-called Pielou model, can be criticized for the same reason. To formulate an alternative discrete delay population model of logistic growth, we first distinguish the growth and decline processes before introducing a delay solely in the growth term. The obtained model differs from existing discrete delay population models and exhibits realistic behavior. If the delay exceeds a certain critical threshold, then the population goes extinct. On the other hand, if the delay is below that threshold, then the population survives and converges to a positive asymptotically stable equilibrium that decreases as the delay increases. As the next step toward ex- tending our delay discrete population model to interacting species, we derived a discrete predator-prey model without delay. The analysis of this predator-prey model led us to the formulation of a discrete phase plane approach that has been deemed ineffective for planar maps. By considering the direction field, the corresponding nullclines, and our “next iterate root curves associated with the nullclines”, the global dynamics of the discrete predator-prey model are discussed.