Heteroclinic Connection in a Nicholson's delayed model with Harvesting term

TL;DR

This paper proves the existence of monotone heteroclinic solutions in a delayed Nicholson's blowflies model with harvesting, using a combination of analytical methods and numerical simulations.

Contribution

It establishes the existence of heteroclinic connections in a delayed population model with harvesting, extending previous methods to two delays.

Findings

Existence of heteroclinic solutions under specific parameter conditions

Construction of explicit upper and lower solutions

Numerical simulations supporting theoretical results

Abstract

In this paper we prove the existence of monotone heteroclinic solutions for the delayed Nicholson's blowflies model with harvesting: \[ x'(t) = -\delta x(t) - Hx(t-\sigma) + \rho x(t-r)e^{-x(t-r)}. \] Under the condition $1 < \dfrac{\rho}{\delta+H} \leq e$, we establish the connection between the equilibria $0$ and $\ln(\rho/(\delta+H))$ using the Wu and Zou monotone iteration method adapted for two delays ($\sigma \neq r$). The proof combines explicit upper and lower solutions construction with characteristic equation analysis, supported by numerical simulations.