An augmented phase plane approach for discrete planar maps: Introducing next-iterate operators

TL;DR

This paper introduces an augmented phase plane method for analyzing planar discrete maps by incorporating next-iterate operators, enhancing the understanding of global dynamics through augmented phase portraits.

Contribution

It presents a novel approach using next-iterate operators to augment phase portraits, providing clearer insights into the global behavior of discrete dynamical systems.

Findings

Augmented phase portraits help identify invariant regions.

The method effectively analyzes Lotka-Volterra and other ecological models.

Limitations and potential applications are discussed.

Abstract

The next-iterate operators and corresponding next-iterate root-sets and root-curves associated with the nullclines of a planar discrete map are introduced. How to augment standard phase portraits that include the nullclines and the direction field, by including the signs of the root-operators associated with their nullclines, thus producing an augmented phase portrait, is described. The sign of a next-iterate operator associated with a nullcline determines whether a point is mapped above or below the corresponding nullcline and can, for example, identify positively invariant regions. Using a Lotka-Volterra type competition model, we demonstrate how to construct the augmented phase portrait. We show that the augmented phase portrait provides an elementary, alternative approach for determining the complete global dynamics of this model. We further explore the limitations and potential of the augmented phase portrait by considering a Ricker competition model, a model involving mutualism, and a predator-prey model.