scholarly journals Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Wanlin Chen ◽  
Yuhua Lin
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.

scholarly journals A Stochastic Predator-Prey System with Stage Structure for Predator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shufen Zhao ◽  
Minghui Song
Keyword(s):  
Global Existence ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Stability In Probability ◽  
Prey System ◽  
Theoretical Results

The authors introduce stochasticity into a predator-prey system with Beddington-DeAngelis functional response and stage structure for predator. We present the global existence and positivity of the solution and give sufficient conditions for the global stability in probability of the system. Numerical simulations are introduced to support the main theoretical results.

scholarly journals Global dynamics of a predator-prey system with Holling type II functional response

2011 ◽  
Vol 16 (2) ◽  
pp. 242-253 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Global Dynamics ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Prey System ◽  
Coexistence Equilibrium ◽  
Type Ii Functional Response

In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The existence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium.

scholarly journals Qualitative Analysis for a Predator Prey System with Holling Type III Functional Response and Prey Refuge

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Xia Liu ◽  
Yepeng Xing
Keyword(s):  
Qualitative Analysis ◽  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Limit Cycle ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Holling Iii Functional Response

A predator prey system with Holling III functional response and constant prey refuge is considered. By using the Dulac criterion, we discuss the global stability of the positive equilibrium of the system. By transforming the system to a Liénard system, the conditions for the existence of exactly one limit cycle for the system are given. Some numerical simulations are presented.

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 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.

DYNAMIC COMPLEXITIES OF HOLLING TYPE II FUNCTIONAL RESPONSE PREDATOR–PREY SYSTEM WITH DIGEST DELAY AND IMPULSIVE HARVESTING ON THE PREY

2009 ◽  
Vol 02 (02) ◽  
pp. 229-242 ◽  
Author(s):  
JIANWEN JIA ◽  
HUI CAO
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Impulsive Differential Equation ◽  
Global Attractivity ◽  
Sufficient Condition ◽  
Type Ii ◽  
Predator Prey ◽  
Prey System ◽  
Impulsive Harvesting ◽  
Type Ii Functional Response

In this paper, we introduce and study Holling type II functional response predator–prey system with digest delay and impulsive harvesting on the prey, which contains with periodically pulsed on the prey and time delay on the predator. We investigate the existence and global attractivity of the predator-extinction periodic solutions of the system. By using the theory on delay functional and impulsive differential equation, we obtain the sufficient condition with time delay and impulsive perturbations for the permanence of the system.

scholarly journals Average Conditions for the Permanence of a Bounded Discrete Predator-Prey System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Yong-Hong Fan ◽  
Lin-Lin Wang
Keyword(s):  
Functional Response ◽  
Lower Bounds ◽  
Population Models ◽  
Upper And Lower Bounds ◽  
Multiple Delays ◽  
Type Ii ◽  
Predator Prey ◽  
Prey System ◽  
Bounded System ◽  
Type Ii Functional Response

Average conditions are obtained for the permanence of a discrete bounded system with Holling type II functional responseu(n+1)=u(n)exp{a(n)-b(n)u(n)-c(n)v(n)/(u(n)+m(n)v(n))},v(n+1)=v(n)exp{-d(n)+e(n)u(n)/(u(n)+m(n)v(n))}.The method involves the application of estimates of uniform upper and lower bounds of solutions. When these results are applied to some special delay population models with multiple delays, some new results are obtained and some known results are generalized.

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