scholarly journals Global Asymptotic Stability and Hopf Bifurcation in a Homogeneous Diffusive Predator-Prey System with Holling Type II Functional Response

2020 ◽  
Vol 11 (05) ◽  
pp. 389-406
Author(s):  
Shuangte Wang ◽  
Hengguo Yu ◽  
Chuanjun Dai ◽  
Min Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Type Ii ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

scholarly journals A Predator-Prey Model with Functional Response and Stage Structure for Prey

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Xiao-Ke Sun ◽  
Hai-Feng Huo ◽  
Xiao-Bing Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Stage Structure ◽  
Transcendental Equation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Type Ii Functional Response

A predator-prey system with Holling type II functional response and stage structure for prey is presented. The local and global stability are studied by analyzing the associated characteristic transcendental equation and using comparison theorem. The existence of a Hopf bifurcation at the positive equilibrium is also studied. Some numerical simulations are also given to illustrate our results.

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.

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.

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

scholarly journals Dynamics of a Predator-Prey System with Beddington-DeAngelis Functional Response and Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Nai-Wei Liu ◽  
Ting-Ting Kong
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Global Asymptotic Stability ◽  
Stage Structure ◽  
Necessary Condition ◽  
Positive Equilibrium ◽  
Sufficient Condition ◽  
Predator Prey ◽  
Prey System ◽  
Hyperbolic Curve

We consider a predator-prey system with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered. First, we give a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov function, we prove that the positive equilibrium is locally asymptotically stable under a sufficient condition. Finally, by using comparison theorem and theω-limit set theory, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the global asymptotic stability.

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