Global Existence and Uniqueness of Periodic Waves in a Population Model with Density-Dependent Migrations and Allee Effect

Sun, Xiaobing; Yu, Pei; Qin, Bin

doi:10.1142/s0218127417501929

Cited by 17 publications

(6 citation statements)

References 14 publications

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“…Our research results reveal a classical Hopf bifurcation in the studied system, and play an important role for the better understanding of the complex dynamics of QPP model subject to time delay. Finally, we mention that when the delay is continuous and modeled by a convolution, the problem on the periodic phenomenon can be restricted on the critical manifold, the limit cycle can be detected by the zeros of Melnikov function, see [17], [18], [19].…”

Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 99%

Complex dynamics in a quasi-periodic plasma perturbations model

Zhang¹,

Yang²

2021

DCDS-B

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In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag τ between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power h and the relaxation of the instability δ do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient η just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

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Section: Theorem 23 (Existence Of Hopf Bifurcation) Assume That (Hmentioning

confidence: 99%

Complex dynamics in a quasi-periodic plasma perturbations model

Zhang¹,

Yang²

2021

DCDS-B

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“…Recently, many mathematicians are interested in studying traveling wave solutions for the KdV-mKdV equation, and there are many tools to find the traveling wave solutions, such as the inverse scattering method [2], the double subequation method [3], the sine-cosine method [4,5], the Darboux transformation [6], the Exp-expansion method [7,8], and the bifurcation method of dynamic systems [9]. KdV-mKdV equation is one of the most popular soliton equations; therefore, there were widely related investigations in [10][11][12][13][14]. On the contrary, based on the relationship between the solitary wave solution and the homoclinic orbit of the associated ordinary differential equations, Fan and Tian [15] proved that the solitary wave persists the singularly perturbed mKdV-KS equation from the geometric singular perturbation point of view, when the perturbation parameter is suitably small.…”

Section: Introductionmentioning

confidence: 99%

Existence of Solitary Waves in a Perturbed KdV-mKdV Equation

Wei

Lin

2021

Journal of Mathematics

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In this paper, we establish the existence of a solitary wave in a KdV-mKdV equation with dissipative perturbation by applying the geometric singular perturbation technique and Melnikov function. The distance of the stable manifold and unstable manifold is computed to show the existence of the homoclinic loop for the related ordinary differential equation systems on the slow manifold, which implies the existence of a solitary wave for the KdV-mKdV equation with dissipative perturbation.

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“…In particular, many scientists paid their attention to the stability and bifurcation phenomena of the predator-prey system with multiple delays (see, for example, [7][8][9][10][11][12][13]). When the delay is continuous and modeled by a convolution, the problem on the periodic phenomenon can be restricted on the critical manifold, and the limit cycle can be detected by the zeros of Melnikov function, see [14][15][16]. In fact, much commonness is reflected between species that co-evolve in nature and different enterprises that co-exist in economic society, so numerous researchers have widely presented the competition and cooperation model of the enterprises [17,18], which are governed by the following ordinary differential equation: ( ) ( ) denote the output of enterprise x 1 and enterprise x 2 at time t, respectively; x t x t , 1 2 1 1…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a diffusive delayed competition and cooperation system

Wei¹,

Zhang

2020

Open Mathematics

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In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.

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Global Existence and Uniqueness of Periodic Waves in a Population Model with Density-Dependent Migrations and Allee Effect

Cited by 17 publications

References 14 publications

Complex dynamics in a quasi-periodic plasma perturbations model

Complex dynamics in a quasi-periodic plasma perturbations model

Existence of Solitary Waves in a Perturbed KdV-mKdV Equation

Dynamics of a diffusive delayed competition and cooperation system

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