Dynamics of a free boundary problem modelling species invasion with impulsive harvesting

Yue Meng; Jing Ge; Zhigui Lin

doi:10.3934/dcdsb.2022060

Article Contents

2022, Volume 27, Issue 12: 7689-7720. Doi: 10.3934/dcdsb.2022060

This issue Previous Article The isothermal limit for the compressible Euler equations with damping Next Article Global solvability to the 3D incompressible magneto-micropolar system with vacuum

Dynamics of a free boundary problem modelling species invasion with impulsive harvesting

1.
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2.
School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, China

^* Corresponding author: Email: zglin68@hotmail.com (Z. Lin)
^* Corresponding author: Email: zglin68@hotmail.com (Z. Lin)

Received: October 2021

Revised: February 2022

Early access: March 2022

Published: December 2022

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Abstract

To understand the role of impulsive harvesting in dynamics of the invasive species, we explore an impulsive logistic equation with free boundaries. The criteria whether the species spreads or vanishes are given, and some sufficient conditions based on threshold values are established. We then discuss the spreading speeds of moving fronts when the species spreads. Our numerical simulations reveal that impulsive harvesting can reduce the spreading speed of the species, and a large impulsive harvesting is unfavorable for persistence of the species. Moreover, when impulsive harvesting is moderate, the species occurs spreading or vanishing depending on its expanding capability or initial number, that is, the species will die out with a small expanding capability or small initial number and spread with a large expanding capability. Note: KYCX22_3446 is added in Acknowledgments.

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Mathematics Subject Classification: Primary: 35R12; 35R35; Secondary: 92D25.

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Figure 1. A simulation of the invasive species without impulses. Graph (a) show that species will persist and spread in the whole space, graph (b) is its sections at $ t = 0, 3, 6, 10 $. Graph (c) is the right view of (a) and projection of $ u $ on the $ t-u- $plane

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Figure 2. The distribution of the species with a large harvesting rate, where $ H(u) $ takes Beverton-Holt function with $ \beta = 2 $ and $ \alpha = 10 $. The spatial distribution of the species in Graph (a) indicates that the species presents eventual extinction, and the contour one in Graph (b) clearly shows that $ u(t, x) $ decays to zero (the purple line). Graph (c) shows that the number of the species drops sharply when harvesting is implemented every time $ \tau = 2 $

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Figure 3. The distribution of the species with a small harvesting rate, where $ H(u) $ takes Beverton-Holt function with $ \beta = 8 $ and $ \alpha = 10 $. Graphs (a)-(c) indicate that the species persists eventually. The impact of impulsive harvesting is shown in Graph (c)

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Figure 4. The distribution of the species when $ H(u) $ is chosen as Beverton-Holt function with $ \beta = 4 $ and $ \alpha = 10 $. The expanding capability $ \mu $ for Graphs (1a)-(1c) and (2a)-(2c) are $ \mu = 1 $ and $ \mu = 20 $, respectively. For the small expanding capacity $ \mu = 1 $, the fact that the species goes to extinction as time increases is shown in Graphs (1a)-(1c). The small expanding capability of the species is exhibited in Graph (1b). The projection of $ u $ on the $ t-u- $plane in Graph (1c) shows that harvesting takes place every time $ \tau = 2 $. For the large expanding capacity $ \mu = 20 $, the species will persist, which is shown in Graphs (2a)-(2c)

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