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Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting  [PDF]
Yang Ni, Yan Meng, Yiming Ding
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.37094
Abstract:

In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a delayed predator-prey system with harvesting. We show that the positive equilibrium point is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Furthermore, using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation.

Hopf Bifurcation Analysis on General Gause-Type Predator-Prey Models with Delay
Shuang Guo,Weihua Jiang
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/363051
Abstract: A class of three-dimensional Gause-type predator-prey model with delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation is obtained via employing the polynomial theorem by analyzing the distribution of the roots of the associated characteristic equation. Secondly, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are determined by applying the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the obtained results.
Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay
Yuzhen Bai,Xiaopeng Zhang
Abstract and Applied Analysis , 2011, DOI: 10.1155/2011/463721
Abstract: This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form theory and center manifold reduction for partial functional differential equations.
带有非线性收获和妊娠时滞的微分代数捕食者-食饵系统的稳定性及Hopf分支
STABILITY AND HOPF BIFURCATION FOR A DIFFERENTIAL ALGEBRAIC PREDATOR-PREY SYSTEM WITH NONLINEAR HARVESTING AND GESTATIONAL TIME DELAY
 [PDF]

作者,李蒙,陈伯山,李必文
- , 2016,
Abstract: 本文研究了一类同时带有非线性食饵收获和捕食者妊娠时滞的微分代数捕食者-食饵系统的稳定性及Hopf分支问题.利用了分支理论和稳定性理论,以捕食者妊娠时滞作为系统的分支参数,获得了所提出的新系统在正平衡点处系统稳定性的相关判据条件和Hopf分支的产生条件.推广了一般带有线性收获和时滞的微分代数捕食者-食饵系统的结论.
In this paper, we investigate the stability and Hopf bifurcation for a new differential algebraic predator-prey system which combined with nonlinear harvesting in prey and gestational time delay of predator. Using bifurcation theorem and stability theorem, through considering gestational time delay of predator as bifurcation parameter, we obtain the interrelated stability criterion and the related conditions of producing Hopf bifurcation at the positive equilibrium point of the proposed system, which popularize the conclusions of the general differential algebraic predator-prey system which combined with linear harvesting and time delay
一类带无选择性捕获和时滞的捕食食饵系统的Hopf分支分析
HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY
 [PDF]

作者,李震威,李必文,刘炜,汪淦
- , 2017,
Abstract: 本文主要研究了一个改进的带时滞和无选择捕获函数的捕食-食饵生态经济系统的稳定性和Hopf分支.利用微分代数系统的稳定性理论和分支理论,得到了系统正平衡点稳定性的条件,以及当时滞τ作为分支参数时系统产生Hopf分支的条件.对Leslie-Gower捕食-食饵模型进行了一定程度的完善,使得建立的模型更符合实际情况,因此得到的结论也更加科学.
In this paper,we mainly study the Hopf bifurcation and the stability of modified predator-prey biological economic system with nonselective harvesting and time delay.By using the stability and bifurcation theory of differential-algebraic system,the conditions for stability of the positive equilibrium point are obtained,let time delay as bifurcation parameter,the existence of Hopf bifurcation and direction of Hopf bifurcation are obtained.We have improved the Leslie-Gower predator-prey system,make the system which we established more practical,so the conclusions are made more scientific
Stability and Hopf bifurcation in a symmetric Lotka-Volterra predator-prey system with delays
Jing Xia,Zhixian Yu,Rong Yuan
Electronic Journal of Differential Equations , 2013,
Abstract: This article concerns a symmetrical Lotka-Volterra predator-prey system with delays. By analyzing the associated characteristic equation of the original system at the positive equilibrium and choosing the delay as the bifurcation parameter, the local stability and Hopf bifurcation of the system are investigated. Using the normal form theory, we also establish the direction and stability of the Hopf bifurcation. Numerical simulations suggest an existence of Hopf bifurcation near a critical value of time delay.
一类具有时滞的比率依赖捕食者-食饵模型的全局Hopf分支
GLOBAL HOPF BIFURCATION OF A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH TIME DELAY
 [PDF]

作者,赵汇涛
- , 2016,
Abstract: 本文研究了一类比率依赖的捕食者-食饵模型的Hopf分支问题,运用吴建宏等人利用等变拓扑度理论建立起的一般泛函微分方程的全局分支理论,得到了由系统的正平衡点分支出来的周期解的全局存在性,最后利用数值模拟验证了理论分析的正确性.
This paper is concerned with a ratio-dependent predator-prey model with time delay. By using a global Hopf bifurcation result of general functional differential equations due to Wu Jianhong etc., the global existence results of periodic solutions bifurcating from Hopf bifurcations are established. Finally, numerical simulations are also included to support the theoretic analysis
Hopf Bifurcation of a Predator-Prey System with Delays and Stage Structure for the Prey  [PDF]
Zizhen Zhang,Huizhong Yang
Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/282908
Abstract: This paper is concerned with a Holling type III predator-prey system with stage structure for the prey population and two time delays. The main result is given in terms of local stability and bifurcation. By choosing the time delay as a bifurcation parameter, sufficient conditions for the local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained. In particular, explicit formulas that can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form method and center manifold theorem. Finally, numerical simulations supporting the theoretical analysis are also included. 1. Introduction Predator-prey dynamics continues to draw interest from both applied mathematicians and ecologists due to its universal existence and importance. Many kinds of predator-prey models have been studied extensively [1–6]. It is well known that there are many species whose individual members have a life history that takes them through immature stage and mature stage. To analyze the effect of a stage structure for the predator or the prey on the dynamics of a predator-prey system, many scholars have investigated predator-prey systems with stage structure in the last two decades [7–15]. In [7], Wang considered the following predator-prey system with stage structure for the predator and obtained the sufficient conditions for the global stability of a coexistence equilibrium of the system: where represents the density of the prey at time . and represent the densities of the immature predator and the mature predator at time , respectively. For the meanings of all the parameters in system (1.1), one can refer to [7]. Considering the gestation time of the mature predator, Xu [8] incorporated the time delay due to the gestation of the mature predator into system (1.1) and considered the effect of the time delay on the dynamics of system (1.1). There has also been a significant body of work on the predator-prey system with stage structure for the prey. In [12], Xu considered a delayed predator-prey system with a stage structure for the prey: where and denote the population densities of the immature prey and the mature prey at time , respectively. denotes the population density of the predator at time . All the parameters in system (1.2) are assumed positive. is the birth rate of the immature prey. is the transformation rate from immature individual to mature individuals. is the intraspecific competition coefficient of
Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters
Changjin Xu
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/264870
Abstract: A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.
Harvesting and Hopf Bifurcation in a prey-predator model with Holling Type IV Functional Response
Manju Agarwal,Rachana Pathak
International Journal of Mathematics and Soft Computing , 2012,
Abstract: This paper aims to study the effect of Harvesting on predator species with time-delay on a Holling type-IV prey-predator model. Harvesting has a strong impact on the dynamic evolution of a population. Two delays are considered in the model of this paper to describe the time that juveniles of prey and predator take to mature. Dynamics of the system is studied in terms of local and Hopf bifurcation analysis. Finally, numerical simulation is done to support the analytical findings.
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