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

Adrian Gomez, Cesar Guayasamin

TL;DR

The paper addresses the existence of a monotone heteroclinic connection in a Nicholson-type delayed population model with harvesting and two distinct delays, connecting the equilibria $x_0=0$ and $x_\kappa=\ln(\frac{\rho}{\delta+H})$ under the conditions $1<\frac{\rho}{\delta+H}\le e$ and $\sigma<r$. It develops a theoretical framework by adapting Wu and Zou's monotone iteration method for a first-order functional equation $-u'(t)+f(u_t)=0$, introducing exponential quasi-monotonicity (H2), the profile set $\Gamma$, and a monotone operator $\mathbf{H}$ that yields convergent upper-lower solutions. The main contributions are a rigorous existence proof of a monotone heteroclinic connection under the stated parameter regime, explicit construction of upper and lower solutions, and numerical approximations that corroborate the analytical results. This work extends open questions on global dynamics with multiple delays in Nicholson-type models and demonstrates the viability of the monotone iteration approach for non-diffusive two-delay systems with harvesting.

Abstract

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

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

TL;DR

The paper addresses the existence of a monotone heteroclinic connection in a Nicholson-type delayed population model with harvesting and two distinct delays, connecting the equilibria and under the conditions and . It develops a theoretical framework by adapting Wu and Zou's monotone iteration method for a first-order functional equation , introducing exponential quasi-monotonicity (H2), the profile set , and a monotone operator that yields convergent upper-lower solutions. The main contributions are a rigorous existence proof of a monotone heteroclinic connection under the stated parameter regime, explicit construction of upper and lower solutions, and numerical approximations that corroborate the analytical results. This work extends open questions on global dynamics with multiple delays in Nicholson-type models and demonstrates the viability of the monotone iteration approach for non-diffusive two-delay systems with harvesting.

Abstract

In this paper we prove the existence of monotone heteroclinic solutions for the delayed Nicholson's blowflies model with harvesting: Under the condition , we establish the connection between the equilibria and using the Wu and Zou monotone iteration method adapted for two delays (). The proof combines explicit upper and lower solutions construction with characteristic equation analysis, supported by numerical simulations.
Paper Structure (3 sections, 13 theorems, 72 equations, 5 figures)

This paper contains 3 sections, 13 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Application to the Nicholson Model

Key Result

Theorem 1.1

Let $1<\frac{\rho}{\delta+H} \leq e$, and $\lambda>0$ the positive root of the equation   If the delays $r, \sigma$ satisfy the conditions:  Then the Nicholson equation with delay and linear harvesting term NichH has a monotone heteroclinic solution connecting the equilibria $x_{0}=0$ and $x_{\kappa}=\ln \left(\frac{\rho}{\delta+H}\right)>0$.

Figures (5)

  • Figure 1: Heteroclinic solution types: monotone (left), eventually monotone (center), and slowly oscillatory (right)
  • Figure 2: Graph of upper solution $\bar{\varphi}$, with $\mu=6.71, \lambda=0.3420$, coefficients $\delta=1, H=2$, $\rho=6$, delays $\sigma=0.15, r=1.8$ and its verification.
  • Figure 3: Graph of lower solution $\underline{\varphi}$, with $\alpha=0.5, \epsilon=0.33$, coefficients $\delta=1, H=2$, $\rho=6$, delays $\sigma=0.15, r=1.8$ and its verification.
  • Figure 4: Compatibility of upper solution and lower solution
  • Figure 5: First 4 iterations approximating the monotone heteroclinic solution connecting the equilibria 0 and $\kappa$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 15 more