EXISTENCE AND GLOBAL ATTRACTIVITY OF A POSITIVE PERIODIC SOLUTION FOR A NON-AUTONOMOUS PREDATOR-PREY MODEL UNDER VIRAL INFECTION

2009 ◽  
Vol 02 (04) ◽  
pp. 419-442 ◽  
Author(s):  
FENGYAN ZHOU
Keyword(s):  
Numerical Simulation ◽  
Periodic Solution ◽  
Lyapunov Function ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Positive Periodic Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

A new non-autonomous predator-prey system with the effect of viruses on the prey is investigated. By using the method of coincidence degree, some sufficient conditions are obtained for the existence of a positive periodic solution. Moreover, with the help of an appropriately chosen Lyapunov function, the global attractivity of the positive periodic solution is discussed. In the end, a numerical simulation is used to illustrate the feasibility of our results.

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.

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 Permanence and Global Attractivity of Discrete Predator-Prey System with Hassell-Varley Type Functional Response

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Runxin Wu ◽  
Lin Li
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Difference Equation ◽  
Functional Response ◽  
Comparison Theorem ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Prey System

By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley type functional response are obtained. Example together with its numerical simulation shows that the main results are verifiable.

scholarly journals Permanence and Global Attractivity of the Discrete Predator-Prey System with Hassell-Varley-Holling III Type Functional Response

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Runxin Wu ◽  
Lin Li
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Difference Equation ◽  
Functional Response ◽  
Comparison Theorem ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Prey System

By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling III type functional response are obtained. An example together with its numerical simulation shows that the main results are verifiable.

scholarly journals Global Attractivity and Periodic Solution of a Discrete Multispecies Cooperation and Competition Predator-Prey System

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Zheyan Zhou
Keyword(s):  
Periodic Solution ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Periodic Case ◽  
Predator Prey ◽  
Cooperation And Competition ◽  
Prey System ◽  
Nonautonomous Case ◽  
Globally Stable

We propose a discrete multispecies cooperation and competition predator-prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

scholarly journals Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Caifeng Du
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

AbstractIn this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

scholarly journals Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Li Zu ◽  
Daqing Jiang ◽  
Fuquan Jiang
Keyword(s):  
Numerical Simulation ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Ergodic Property ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

We consider a predator-prey model in which the preys disperse amongnpatches (n≥2) with stochastic perturbation. We show that there is a unique positive solution and find out the sufficient conditions for the extinction to the system with any given positive initial value. In addition, we investigate that there exists a stationary distribution for the system and it has ergodic property. Finally, we illustrate the dynamic behavior of the system withn=2via numerical simulation.

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.

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 Dynamic Behavior of Positive Solutions for a Leslie Predator-Prey System with Mutual Interference and Feedback Controls

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Cong Zhang ◽  
Nan-jing Huang ◽  
Chuan-xian Deng
Keyword(s):  
Global Attractivity ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Mutual Interference ◽  
Periodic Case ◽  
Feedback Controls ◽  
Predator Prey ◽  
Prey System ◽  
Inequality Theory ◽  
Nonautonomous Case

We consider a Leslie predator-prey system with mutual interference and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain some sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain some sufficient conditions which guarantee the existence, uniqueness, and stability of a positive periodic solution.

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