Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie–Gower predator–prey model

Li, Shanbing; Wu, Jianhua; Nie, Hua

doi:10.1016/j.camwa.2015.10.017

Cited by 23 publications

(5 citation statements)

References 25 publications

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“…Throughout this paper, we always assume that a(x) and b(x) ∈ C r (Ω) with r ∈ (0, 1) are nonconstant, and that a(x) > 0 and b(x) > 0 in Ω. For system (4) with Ω = (0, π), d 1 = 1, and constant-valued functions a(x) and b(x), Li et al [24] investigated the Hopf bifurcation and steady-state bifurcation by taking d 2 as the bifurcation parameter and described both the global structure of the steady-state bifurcation from simple eigenvalues and the local structure of the steady-state bifurcation from double eigenvalues by using space decomposition and the implicit function theorem.…”

mentioning

confidence: 99%

Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

Zou¹,

Guo²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we are concerned with a diffusive Leslie-Gower predator-prey model in heterogeneous environment. The global existence and boundedness of solutions are shown. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global stability of semi-trivial solutions. The existence of positive steady state solution bifurcating from semi-trivial solutions is obtained by using local bifurcation theory. The stability analysis of the positive steady state solution is investigated in detail. In addition, we explore the asymptotic profiles of the steady state solution for small and large diffusion rates.2010 Mathematics Subject Classification. Primary 92D25; Secondary 35K57.

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mentioning

confidence: 99%

Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

Zou¹,

Guo²

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…Similar to Theorem 3.2, based on the above analysis, we can get the following Theorem 3.3 about Hopf bifurcation from (λ (2) , λ (2) ) with help of Table 3 and Table 4. are defined in (13), then the following statements are true.…”

Section: Definementioning

confidence: 99%

“…This phenomenon has become crucial for population dynamics since in fact it has a surprising number of ramifications towards different branches of ecology [4,25]. Recently, there have been some works concerning the Allee effect in the classical dynamic population models [5,8,13,18,19,26]. Many interesting dynamical properties caused by the Allee effect are found which differ from the original system.…”

Section: Na Min and Mingxin Wangmentioning

confidence: 99%

Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey

Min¹,

Wang²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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It is well known that the Leslie-Gower prey-predator model (without Allee effect) has a unique globally asymptotically stable positive equilibrium point, thus there is no Hopf bifurcation branching from positive equilibrium point. In this paper we study the Leslie-Gower prey-predator model with strong Allee effect in prey, and perform a detailed Hopf bifurcation analysis to both the ODE and PDE models, and derive conditions for determining the steady-state bifurcation of PDE model. Moreover, by the center manifold theory and the normal form method, the direction and stability of Hopf bifurcation solutions are established. Finally, some numerical simulations are presented. Apparently, Allee effect changes the topology structure of the original Leslie-Gower model.

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“…In the past decade, many scholars have shown great interest in predator-prey systems (1.1) based on different functional response models, which has resulted in many outstanding works. Relevant literature can be found in [9,10] and the references cited therein. However, it has been observed that some prey species survive by avoiding predators.…”

Section: Introductionmentioning

confidence: 99%

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

Yang,

Chen

2023

ARJOM

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In this paper, we propose a new Leslie-Gower predator-prey model with predator-dependent prey refuge. Firstly, we obtain the positivity and boundedness of the system solution. Secondly, we prove that the origin is unstable using blow-up method, analyze the existence and local stability of the boundary equilibrium point and positive equilibrium point, and prove that the unique positive equilibrium point of the system is globally asymptotically stable by constructing a suitable Dulac function. Finally, mathematic analysis and numerical simulation show that: (1) when the strength of the predator-dependent prey refuge k = 0 , the dynamics of the predator-prey system without predator-dependent prey refuge are consistent with the results obtained from the traditional Leslie-Gower predator-prey system; (2) when k tends to positive infinity, the predator-dependent refuge lead to prey population densities fall somewhere between without prey refuge and with proportional refuge. However, the predator densities within this new form of the predator-dependent prey refuge is greater than the densities of predators without prey refuge and with proportional refuge; (3) increasing the strength k of the predator-dependent prey refuge can increase the densities of predator and prey populations respectively.

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Steady-state bifurcation and Hopf bifurcation for a diffusive Leslie–Gower predator–prey model

Cited by 23 publications

References 25 publications

Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

Dynamics of a diffusive Leslie-Gower predator-prey model in spatially heterogeneous environment

Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey

The Impact of Predator-dependent Prey Refuge on the Dynamics of a Leslie-Gower Predator-prey Model

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