Global dynamics of three-dimensional Lotka-Volterra competition models with seasonal succession: II. Uniqueness of positive fixed points

TL;DR

This paper analyzes the uniqueness of positive fixed points in a 3D seasonal Lotka-Volterra model, revealing classes with multiple fixed points and complex dynamics, contrasting with simpler models.

Contribution

It establishes conditions for non-uniqueness of positive fixed points in certain classes, highlighting richer dynamical behaviors in the model.

Findings

Classes 26 and 27 can have multiple positive fixed points.

Classes 19-25 and 28-33 have unique fixed points with trivial dynamics.

Classes 26 and 27 exhibit complex dynamics including invariant curves and quasi-periodic solutions.

Abstract

In this second part of the series, we investigate the uniqueness of positive fixed points of the Poincare map associated with the 3-dimensional Lotka-Volterra competition model with seasonal succession. Building on our first part of the series on the classification of 33 dynamical equivalence classes (regardless of the uniqueness of positive fixed points), we demonstrate in this paper that classes 26 and 27 may indeed exhibit multiple positive fixed points. This reveals a fundamental distinction from both its 2-dimensional analogue and the classical 3-dimensional competitive Lotka-Volterra model. More concretely, by focusing on the model with identical growth and death rates, we establish an equivalent characterization for the (non)uniqueness of positive fixed points. Based on this characterization, we further show that classes 19-25 and 28-33 admit a unique positive fixed point and exhibit trivial dynamics: all trajectories converge to some fixed point (corresponding to harmonic solutions). In contrast, classes 26 and 27 possess richer dynamical scenarios: there can contain a continuum of invariant closed curves, on which orbits may be periodic (corresponding to subharmonic solutions), dense (corresponding to quasi-periodic solutions), or may even consist entirely of positive fixed points (which exhibits the nonuniqueness of positive fixed points).