scholarly journals Existence and Uniqueness of Positive Periodic Solutions for a Delayed Predator-Prey Model with Dispersion and Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-21
Author(s):  
Zhenguo Luo ◽  
Liping Luo ◽  
Liu Yang ◽  
Zhenghui Gao ◽  
Yunhui Zeng
Keyword(s):  
Periodic Solutions ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Predator Population ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

An impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays is investigated, where we assume the model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches. By applying the continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system. Some known results subject to the underlying systems without impulses are improved and generalized. As an application, we also give two examples to illustrate the feasibility of our main results.

scholarly journals Positive Periodic Solutions for Neutral Delay Ratio-Dependent Predator-Prey Model with Holling-Tanner Functional Response

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Guirong Liu ◽  
Sanhu Wang ◽  
Jurang Yan
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Neutral Delay ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional responsex'(t)=x(t)[r(t)-a(t)x(t-σ(t))-b(t)x'(t-σ(t))]-c(t)x(t)y(t)/h(t)y(t)+x(t),y'(t)=y(t)d(t)-f(t)y(t-τ(t))/x(t-τ(t)).

scholarly journals Mathematical Analysis for a Discrete Predator-Prey Model with Time Delay and Holling II Functional Response

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Dehong Ding ◽  
Kui Fang ◽  
Yang Zhao
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Mawhin’S Continuation Theorem ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Holling Ii Functional Response

This paper is concerned with a discrete predator-prey model with Holling II functional response and delays. Applying Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, we obtain some sufficient conditions for the existence global asymptotic stability of positive periodic solutions of the model.

scholarly journals Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Tursuneli Niyaz ◽  
Ahmadjan Muhammadhaji
Keyword(s):  
Periodic Solutions ◽  
Continuous Time ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Continuation Theorem ◽  
Positive Periodic Solutions ◽  
Feedback Controls ◽  
Volterra Systems

This paper studies a class of periodicnspecies cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established.

scholarly journals Multiple Positive Periodic Solutions for a Gilpin-Ayala Competition Predator-Prey System with Harvesting Terms

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Yongzhi Liao ◽  
Yongkun Li ◽  
Xiaoyan Dou
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Continuation Theorem ◽  
Coincidence Degree Theory ◽  
Positive Periodic Solutions ◽  
Mawhin’S Continuation Theorem ◽  
Predator Prey ◽  
Prey System

By applying Mawhin’s continuation theorem of coincidence degree theory, we study the existence of multiple positive periodic solutions for a Gilpin-Ayala competition predator-prey system with harvesting terms and obtain some sufficient conditions for the existence of multiple positive periodic solutions for the system under consideration. The result of this paper is completely new. An example is employed to illustrate our result.

scholarly journals Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Huilan Wang ◽  
Zhengqiu Zhang ◽  
Weiping Zhou
Keyword(s):  
Periodic Solutions ◽  
Topological Degree ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Prey System ◽  
System With Time Delay ◽  
Function Matrix

By using continuation theorem of coincidence degree theory, sufficient conditions of the existence of positive periodic solutions are obtained for a generalized predator-prey system with diffusion and delays. In this paper, we construct a V-function to make the prior estimation for periodic solutions, which makes the discussion more concise. Moreover, to compute the mapping's topological degree, a polynomial function matrix is constructed straightforwardly as a homotopic mapping for the generalized one, which improves the methods of computation on topological degree for a generalized mapping.

EXISTENCE OF PERIODIC SOLUTIONS OF NONAUTONOMOUS PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE AND TIME DELAY

2021 ◽  
Vol 10 (5) ◽  
pp. 2641-2652
Author(s):  
S. Mahalakshmi ◽  
V. Piramanantham
Keyword(s):  
Time Delay ◽  
Periodic Solutions ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence Of Periodic Solutions

In this paper we establish some easily verifiable sufficient conditions for the existence of periodic solutions of nonautonomous Predator-Prey Model with Beddington-DeAngelis Functional response and time delay using Mowhins Coincidence degree method.

scholarly journals Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zhenguo Luo ◽  
Liping Luo ◽  
Yunhui Zeng
Keyword(s):  
Periodic Solutions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Continuation Theorem ◽  
Coincidence Degree Theory ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Multiple Periodic Solutions

We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria.

scholarly journals Periodic solutions of a delayed predator-prey model with stage structure for predator

2004 ◽  
Vol 2004 (3) ◽  
pp. 255-270 ◽  
Author(s):  
Rui Xu ◽  
M. A. J. Chaplain ◽  
F. A. Davidson
Keyword(s):  
Periodic Solutions ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Time Dependent ◽  
Volterra Type ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Proposed Model

A periodic time-dependent Lotka-Volterra-type predator-prey model with stage structure for the predator and time delays due to negative feedback and gestation is investigated. Sufficient conditions are derived, respectively, for the existence and global stability of positive periodic solutions to the proposed model.

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