The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response
type: Item
instance of: scholarly article
title: The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response
zbMATH Open document ID: 1218.34099
author: Yuanfu Shao
author: Binxiang Dai
published in: Nonlinear Analysis. Real World Applications
publication date: 2010-10-14T00:00:00Z
review text: The authors consider the following stage-structured predator-prey model with impulsive effect \[ \begin{cases} \dot x(t)=x(t)(a-bx(t))-\frac{cx(t)y_2(t)}{\alpha+x(t)+\beta y_2(t)}, &t\neq nT, \\ \dot y_1(t)=\lambda\frac{cx(t)y_2(t)}{\alpha+x(t)+\beta y_2(t)}-d_1y_1(t)-\lambda e^{-d_2\tau}\frac{cx(t-\tau)y_2(t-\tau)}{\alpha+x(t-\tau)+\beta y_2(t-\tau)}, &t\neq nT,\\ \dot y_2(t)=\lambda e^{-d_2\tau}\frac{cx(t-\tau)y_2(t-\tau)}{\alpha+x(t-\tau)+\beta y_2(t-\tau)}-d_2y_2(t)-ry_2^2(t), &t\neq nT,\\ x(t^+)=(1-p)x(t),\;y_1(t^+)=y_1(t)+\mu, \;y_2(t^+)=y_2(t), &t=nT, \end{cases}\tag{1} \] where \(x(t), y_1(t), y_2(t)\) represent the densities of prey, immature predator and mature predator populations at time \(t\), respectively. \(a\) is the intrinsic growth rate of the prey, \(b\) is the intra-specific competition rate of the prey, \(c\) is the capturing rate of the predator, \(\alpha\) is a saturation constant, \(\beta\) scales the impact of the predator interference, \(\lambda\) is the conversion coefficient, \(d_1, d_2\) are the mortality rates of the immature and mature predator, respectively, \(r\) is the intra-specific competition rate of the mature predator, \(\tau\) represents a constant time to maturity for the immature predator, \(p\) \((0\leq p<1)\) represents the partial impulsive harvest of the prey by catching or pesticide, \(\mu\geq 0\) is the amount of immature predators released at fixed time \(t=nT\). By using the discrete dynamical system determined by the stroboscopic map, the existence and global attractivity of the predator-extinction periodic solution are obtained. Based on the theory of impulsive delay differential equations, the permanence of the system is studied.
reviewed by: Rui Xu
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 34K45
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 5799993
zbMATH Keywords: attractivity
zbMATH Keywords: delay
zbMATH Keywords: impulsive effect
zbMATH Keywords: extinction
zbMATH Keywords: permanence
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2010.01.004
OpenAlex ID: W2046432690
cites work: Q4325523
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DOI: 10.1016/J.NONRWA.2010.01.004
description: scientific article; zbMATH DE number 5799993