Blow Up of a Cycle in Lotka-Volterra Type Equations with Competition-Cooperation Terms and Quasi-Linear Systems

TL;DR

This paper investigates how certain biological and mathematical systems exhibit unbounded growth of cycles due to parameter changes, highlighting phenomena in Lotka-Volterra and quasi-linear systems.

Contribution

It demonstrates the blow-up of cycles in specific Lotka-Volterra and quasi-linear systems, providing explicit examples and analysis of this behavior.

Findings

Cycle blow-up occurs over finite parameter ranges.

Examples include planar Lotka-Volterra systems with competition-cooperation.

Higher-order quasi-linear equations also exhibit this blow-up.

Abstract

We consider systems where a cycle born via the Hopf bifurcation blows up to infinity as a parameter ranges over a finite interval. Two examples demonstrating this effect are presented: planar Lotka-Volterra type systems with a competition-cooperation term and quasi-linear higher order equations.