Unbounded and blow-up solutions for a delay logistic equation with positive feedback

TL;DR

This paper investigates the behavior of solutions to a delay logistic equation, revealing conditions for unbounded, blow-up, and stable solutions, and highlighting the coexistence of stability and blow-up phenomena.

Contribution

It provides new criteria for the existence of blow-up and unbounded solutions without assuming dominant instantaneous feedback.

Findings

Existence of exponential (unbounded) solutions.

Necessary and sufficient conditions for blow-up solutions.

Coexistence of local stability and blow-up solutions.

Abstract

We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear problem, and in this case the positive equilibrium is always unstable. We obtain a necessary and sufficient condition for the existence of blow-up solutions, and characterize a wide class of such solutions. There is a parameter set such that the non-trivial equilibrium is locally stable but not globally stable due to the co-existence with blow-up solutions.