scholarly journals A Time Delay Predator-Prey System with Three-Stage-Structure

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qiaoqin Gao ◽  
Zhen Jin
Keyword(s):  
Coincidence Degree ◽  
Degree Theory ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Sufficient Condition ◽  
Discrete Delay ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Three Stages

A predator-prey system was studied that has a discrete delay, stage-structure, and Beddington-DeAngelis functional response, where predator species has three stages, immature, mature, and old age stages. By using of Mawhin's continuous theorem of coincidence degree theory, a sufficient condition is obtained for the existence of a positive periodic solution.

PERIODICITY IN AN STAGE-STRUCTURED THREE-SPECIES PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS AND HOLLINF IV FUNCTIONAL RESPONSE

2012 ◽  
Vol 05 (02) ◽  
pp. 1250031
Author(s):  
Changjin Xu ◽  
Maoxin Liao
Keyword(s):  
Functional Response ◽  
Lyapunov Functional ◽  
Global Attractivity ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Sufficient Condition ◽  
Positive Periodic Solutions ◽  
Predator Prey ◽  
Prey System

In this paper, by using the continuation theorem of coincidence degree theory, a sufficient condition of existence of positive periodic solutions is obtained for an stage-structured three-species predator–prey system with Beddington–DeAngelis and Holling IV functional response. By constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Our result is a good complement to the earlier publications.

scholarly journals Periodic solutions of a two-species ratio-dependent predator-prey system with time delay in a two-patch environment

2003 ◽  
Vol 45 (2) ◽  
pp. 233-244 ◽  
Author(s):  
Zhengqiu Zhang ◽  
Zhicheng Wang
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Periodic Solutions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Prey System ◽  
System With Time Delay

AbstractBy using the continuation theorem of coincidence degree theory, a sufficient condition is obtained for the existence of a positive periodic solution of a predator-prey diffusion system.

scholarly journals Periodicity in a ratio-dependent predator-prey system with stage structure for predator

2005 ◽  
Vol 2005 (2) ◽  
pp. 153-169 ◽  
Author(s):  
Fengde Chen
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Positive Periodic Solutions ◽  
Priori Estimate ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio-dependent predator-prey system with stage structure for predator. The approach involves some new technique of priori estimate. For the system without delay, by constructing a suitable Lyapunov function, some sufficient conditions which guarantee the existence of a unique global attractive positive periodic solution are obtained. Those results have further applications in population dynamics.

POSITIVE PERIODIC SOLUTION FOR A GAUSE-TYPE RATIO-DEPENDENT PREDATOR-PREY SYSTEM WITH DIFFUSION AND TIME DELAY

2008 ◽  
Vol 01 (03) ◽  
pp. 339-354 ◽  
Author(s):  
XIAOQUAN DING ◽  
YUANYUAN WANG
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Continuation Theorem ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent ◽  
System With Time Delay

A two-species Gause-type ratio-dependent predator-prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of periodic solution for the system. As corollaries, some applications are listed. In particular, our results extend and improve some known results.

Periodic Solution of a Class Predator-Prey System with Rate Stocking and Time Delay

2013 ◽  
Vol 291-294 ◽  
pp. 2412-2415
Author(s):  
Hui Li ◽  
Yi Fei Wang
Keyword(s):  
Periodic Solution ◽  
Time Delay ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Positive Periodic Solution ◽  
Coincidence Degree Theory ◽  
Predator Prey ◽  
Prey System

In this paper, we investigate of a class of predator-prey system with rate stocking and time delay, the existence positive periodic solution by using coincidence degree theory. We obtain the sufficient conditions which guarantee existence of the positive periodic solution of the periodic system. Some new results obtained.

scholarly journals Global Stability Analysis of a Nonautonomous Stage-Structured Competitive System with Toxic Effect and Double Maturation Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Chao Liu ◽  
Yuanke Li
Keyword(s):  
Global Stability ◽  
Toxic Effect ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Stage Structure ◽  
Positive Periodic Solution ◽  
Competitive System ◽  
Comparison Arguments ◽  
Double Time

We investigate a nonautonomous two-species competitive system with stage structure and double time delays due to maturation for two species, where toxic effect of toxin liberating species on nontoxic species is considered and the inhibiting effect is zero in absence of either species. Positivity and boundedness of solutions are analytically studied. By utilizing some comparison arguments, an iterative technique is proposed to discuss permanence of the species within competitive system. Furthermore, existence of positive periodic solutions is investigated based on continuation theorem of coincidence degree theory. By constructing an appropriate Lyapunov functional, sufficient conditions for global stability of the unique positive periodic solution are analyzed. Numerical simulations are carried out to show consistency with theoretical analysis.

scholarly journals Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

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.

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.

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