BIFURCATIONS OF A HOLLING-TYPE II PREDATOR–PREY SYSTEM WITH CONSTANT RATE HARVESTING

2009 ◽  
Vol 19 (08) ◽  
pp. 2499-2514 ◽  
Author(s):  
GUOJUN PENG ◽  
YAOLIN JIANG ◽  
CHANGPIN LI
Keyword(s):  
Hopf Bifurcation ◽  
Type Ii ◽  
Saddle Node Bifurcation ◽  
Predator Prey ◽  
Prey System ◽  
Bifurcation Phenomena ◽  
Constant Rate ◽  
Generalized Hopf Bifurcation ◽  
Subcritical Hopf Bifurcation ◽  
Dynamical Properties

The objective of this paper is to study the dynamical properties of a Holling-type II predator–prey system with constant rate harvesting. It is shown that the model has at most three equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the degenerate Bogdanov–Takens bifurcation of codimension 3, the supercritical and subcritical Hopf bifurcation, the generalized Hopf bifurcation. These results reveal far richer dynamics than that of the model with no harvesting.

scholarly journals Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xia Liu ◽  
Yepeng Xing
Keyword(s):  
Hopf Bifurcation ◽  
Backward Bifurcation ◽  
Heteroclinic Bifurcation ◽  
Saddle Node Bifurcation ◽  
Predator Prey ◽  
Prey System ◽  
Saddle Node ◽  
Bifurcation Phenomena ◽  
Ratio Dependent

The bifurcation properties of a predator prey system with refuge and constant harvesting are investigated. The number of the equilibria and the properties of the system will change due to refuge and harvesting, which leads to the occurrence of several kinds bifurcation phenomena, for example, the saddle-node bifurcation, Bogdanov-Takens bifurcation, Hopf bifurcation, backward bifurcation, separatrix connecting a saddle-node and a saddle bifurcation and heteroclinic bifurcation, and so forth. Our main results reveal much richer dynamics of the system compared to the system with no refuge and harvesting.

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.

Stability and Bifurcation Analysis in a Predator–Prey System with Constant Harvesting and Prey Group Defense

2017 ◽  
Vol 27 (11) ◽  
pp. 1750179 ◽  
Author(s):  
Jianfeng Luo ◽  
Yi Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Maximum Sustainable Yield ◽  
Sustainable Yield ◽  
Defensive Strategy ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Prey System ◽  
Constant Rate ◽  
The Given ◽  
Existence Of Equilibria

In this paper, we study a predator–prey system that the prey population gathers in herds to defend its predator and both are harvested by constant rate. The defensive strategy of the gathered prey makes the individuals at the border of the herd mostly suffer from the attacks of the predators. This behavior can be described by a modified Holling-type II functional response in mathematics. Notably, we consider harvesting under two cases: prey harvesting only and predator harvesting only. We investigate the existence of equilibria for both cases, and then find there exists the maximum sustainable yield for two cases to guarantee predator and prey to coexist. Moreover, both species can coexist under some conditions and initial values through investigation of stability of the interior equilibrium in the given system. These results demonstrate that, when hunting the prey or predator for economic interest, harvesting rate must be chosen at a suitable value (not merely less than the maximum sustainable yield) to maintain the coexistence of the predator and prey as well as ecological balance. Finally, we analyze the saddle-node bifurcation and Hopf bifurcation, and determine the direction of Hopf bifurcation by calculating the first Lyapunov number for both cases. In particular, Bogdanov–Takens bifurcation occurs only in the given system with predator harvesting.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

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 Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Kankan Sarkar ◽  
Subhas Khajanchi ◽  
Prakash Chandra Mali ◽  
Juan J. Nieto
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Prey Population ◽  
Uniform Persistence ◽  
Functional Responses ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

Sign in / Sign up

Export Citation Format

Share Document