DCDS
Dynamics for the diffusive Leslie-Gower model with double free boundaries
Mingxin Wang Qianying Zhang
Discrete & Continuous Dynamical Systems - A 2018, 38(5): 2591-2607 doi: 10.3934/dcds.2018109

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to∞$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u, v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.

keywords: Leslie-Gower model free boundary problem spreading-vanishing dichotomy long time behavior criteria for spreading and vanishing
DCDS-B
Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey
Na Min Mingxin Wang
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1721-1737 doi: 10.3934/dcdsb.2018073

In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

keywords: Ratio-dependent prey-predator model reaction-diffusion Allee effect protection zone bifurcation
DCDS-B
Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey
Wenjie Ni Mingxin Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3409-3420 doi: 10.3934/dcdsb.2017172

This paper is devoted to study the dynamical properties of a Leslie-Gower prey-predator system with strong Allee effect in prey. We first gives some estimates, and then study the dynamical properties of solutions. In particular, we mainly investigate the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium.

keywords: Leslie-Gower prey-predator model Allee effect dynamical properties
DCDS-B
The reaction-diffusion system for an SIR epidemic model with a free boundary
Haomin Huang Mingxin Wang
Discrete & Continuous Dynamical Systems - B 2015, 20(7): 2039-2050 doi: 10.3934/dcdsb.2015.20.2039
The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.
keywords: long time behavior. Reaction-diffusion systems SIR model dynamics free boundary
DCDS
Existence, uniqueness, and stability of bubble solutions of a chemotaxis model
Xin Lai Xinfu Chen Mingxin Wang Cong Qin Yajing Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 805-832 doi: 10.3934/dcds.2016.36.805
Existence, uniqueness, and stability of Heaviside function like solutions of a Keller and Segel's minimal chemotaxis model are established when a chemotaxis parameter is large enough. Asymptotic expansions of the solution in terms of the large chemotaxis parameter are also derived.
keywords: chemotaxis. asymptotic expansion stability uniqueness Existence eigenvalue
DCDS
Some remarks for a modified periodic Camassa-Holm system
Guangying Lv Mingxin Wang
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1161-1180 doi: 10.3934/dcds.2011.30.1161
This paper is concerned with a modified two-component periodic Camassa-Holm system. The local well-posedness and low regularity result of solution are established by using the techniques of pseudoparabolic regularization and some priori estimates derived from the equation itself. A wave-breaking for strong solutions and several results of blow-up solution with certain initial profiles are described. In addition, the initial boundary value problem for a modified two-component periodic Camassa-Holm system is also considered.
keywords: Wave breaking. Periodic Two-component Camassa-Holm system Blow-up Local well posedness
DCDS
Properties of blow-up solutions to a parabolic system with nonlinear localized terms
Huiling Li Mingxin Wang
Discrete & Continuous Dynamical Systems - A 2005, 13(3): 683-700 doi: 10.3934/dcds.2005.13.683
This paper deals with blow-up properties of the solution to a semi-linear parabolic system with nonlinear localized sources involved in a product with local terms, subject to the null Dirichlet boundary condition. We investigate the influence of localized sources and local terms on blow-up properties for this system. It will be proved that: (i) when $m, q\leq 1$ this system possesses uniform blow-up profiles. In other words, the localized terms play a leading role in the blow-up profile for this case. (ii) when $m, q>1$, this system presents single point blow-up patterns, or say that, in this time, local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.
keywords: blow-up rate estimate. uniform blow-up profile nonlinear local terms single point blow-up pattern nonlinear localized sources Parabolic system
DCDS-B
Existence, uniqueness and stability of traveling wave fronts of discrete quasi-linear equations with delay
Guangying Lv Mingxin Wang
Discrete & Continuous Dynamical Systems - B 2010, 13(2): 415-433 doi: 10.3934/dcdsb.2010.13.415
This paper is concerned with the existence, uniqueness and asymptotically stability of traveling wave fronts of discrete quasi-linear equations with delay. We first establish the existence of traveling wave fronts by using the super-sub solution and monotone iteration technique. Then we show that the traveling wave front is unique up to a translation. At last, we employ the comparison principle and the squeezing technique to prove that the traveling wave front is globally asymptotic stable with phase shift.
keywords: Existence; Uniqueness; Stability; Traveling wave fronts; Discrete quasi-linear equations; Delay.
DCDS
Qualitative analysis of a diffusive variable-territory prey-predator model
Mingxin Wang Peter Y. H. Pang
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 1061-1072 doi: 10.3934/dcds.2009.23.1061
In this paper, we study the variable-territory prey-predator model. We first establish the global stability of the unique positive constant steady state for the ODE system and the reaction diffusion system, and then prove the existence, uniqueness and stability of positive stationary solutions for the heterogeneous environment case.
keywords: prey-predator model variable-territory heterogeneous environment stationary solution stability
DCDS-B
Existence and uniqueness of solutions of free boundary problems in heterogeneous environments
Mingxin Wang
Discrete & Continuous Dynamical Systems - B 2018, 22(11): 1-7 doi: 10.3934/dcdsb.2018179

In this short paper we study the existence and uniqueness of solutions of free boundary problems coming from ecology in heterogeneous environments.

keywords: Existence and uniqueness free boundary problems heterogeneous environments

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