Multiple periodic solutions for a non-autonomous delayed predator-prey model with harvesting terms
type: Item
instance of: scholarly article
title: Multiple periodic solutions for a non-autonomous delayed predator-prey model with harvesting terms
zbMATH Open document ID: 1237.34140
author: Linlin Wang
author: Yong-Hong Fan
published in: Applied Mathematics and Computation
publication date: 2011-07-22T00:00:00Z
review text: The authors study the following non-autonomous predator-prey model with harvest efforts for both the prey and the predator population and a time delay due to the gestation of the predator \[ \begin{aligned} \dot x(t)=&x(t)(r(t)-a(t)x(t))-\frac{b(t)x(t)}{m+x(t)}y(t)-h_1(t),\\ \dot y(t)=&y(t)(\frac{c(t)x(t-\tau(t))}{m+x(t-\tau(t))}-d(t))-h_2(t), \end{aligned}\tag{1} \] where \(x(t)\) and \(y(t)\) represent the densities of the prey and the predator populations, respectively; \( a, b, c, d, r, \tau, h_1\) and \(h_2\) are all nonnegative continuously periodic functions with period \(\omega>0\), \(m>0\) is a constant. Here, the prey population \(x\) grows logistically with the intrinsic growth rate \(r(t)\) and carrying capacity \(r(t)/a(t)\), \(d(t)\) is the death rate of the predator, \(b(t)\) is the capturing rate of the predator, \(c(t)/b(t)\) is the conversion rate, \(m>0\) is the half saturation rate of the predator. Sufficient conditions are derived for the existence of at least two positive periodic solutions to system (1) by using the generalized continuation theorem developed by Gaines and Mawhin [\textit{R. E. Gaines} and \textit{J. L. Mawhin}, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag. (1977; Zbl 0339.47031)].
reviewed by: Rui Xu
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5931311
zbMATH Keywords: predator-prey model
zbMATH Keywords: functional response
zbMATH Keywords: periodic solution
zbMATH Keywords: generalized continuation theorem
MaRDI profile type: Publication
full work available at URL: j.amc.2011.04.010
OpenAlex ID: W2063827646
Wikidata QID: Q115598549
cites work: Q3895297
cites work: Q3994920
cites work: Existence of positive periodic solutions for a generalized prey-predator model with harvesting term
cites work: Non-selective harvesting in prey--predator models with delay
cites work: Periodic solutions of a periodic delay predator-prey system
cites work: Periodic solutions of periodic delay Lotka-Volterra equations and systems
cites work: Multiplicity of positive periodic solutions to a generalized delayed predator-prey system with stocking
cites work: Positive periodic solutions in delayed Gause-type predator-prey systems
cites work: Periodic solutions in non autonomous predator prey system with delays
cites work: Multiple periodic solutions of delayed predator--prey systems with type IV functional responses
cites work: Periodic solutions in a delayed predator-prey model with nonmonotonic functional response
cites work: Periodic solution of a delayed predator-prey system with Michaelis-Menten type functional response.
cites work: Existence and global stability of positive periodic solutions of a predator-prey system with delays
cites work: Periodic solutions for a predator-prey model with Holling-type functional response and time delays
cites work: Existence of multiple positive periodic solutions of delayed predator-prey models with functional responses
cites work: Coincidence degree, and nonlinear differential equations
cites work: Coincidence degree and periodic solutions of neutral equations
DOI: 10.1016/J.AMC.2011.04.010
Recommended article: Periodic solutions for a delayed nonautonomous stage-structured predator-prey model
Recommended article: On a non-selective harvesting prey-predator model with Hassell-Varley type functional response
Recommended article: A Lotka-Volterra type predator-prey model in a two-patchy environment.
Recommended article: Q2972890
Recommended article: Existence of positive periodic solutions for a generalized prey-predator model with harvesting term
Recommended article: Multiple positive periodic solutions for a generalized predator-prey system with exploited terms
Recommended article: Q3014059
Recommended article: Q4996632
Recommended article: Four periodic solutions of a generalized delayed predator-prey system on time scales
Recommended article: Dynamic of a non-autonomous predator-prey system with infinite delay and diffusion
description: scientific article; zbMATH DE number 5931311