Traveling wave solutions in an integro-differential competition model
Liang Zhang Bingtuan Li
Discrete & Continuous Dynamical Systems - B 2012, 17(1): 417-428 doi: 10.3934/dcdsb.2012.17.417
We study the existence of traveling wave solutions for the two-species Lotka-Volterra competition model in the form of integro-differential equations. The model incorporates asymmetric dispersal kernels that describe long distance dispersal processes of competing species in space. Using lower and upper traveling wave solutions, we show that the model has traveling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.
keywords: competition dispersal kernel traveling wave solution. Integro-differential equation
Threshold dynamics of a reaction-diffusion epidemic model with stage structure
Liang Zhang Zhi-Cheng Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3797-3820 doi: 10.3934/dcdsb.2017191

A time-delayed reaction-diffusion epidemic model with stage structure and spatial heterogeneity is investigated, which describes the dynamics of disease spread only proceeding in the adult population. We establish the basic reproduction number $\mathcal{R}_0$ for the model system, which gives the threshold dynamics in the sense that the disease will die out if $\mathcal{R}_0<1$ and the disease will be uniformly persistent if $\mathcal{R}_0>1.$ Furthermore, it is shown that there is at least one positive steady state when $\mathcal{R}_0>1.$ Finally, in terms of general birth function for adult individuals, through introducing two numbers $\check{\mathcal{R}}_0$ and $\hat{\mathcal{R}}_0$, we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

keywords: Stage structure spatial heterogeneity basic reproduction number persistence and extinction
Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators
Liang Zhang X. H. Tang Yi Chen
Communications on Pure & Applied Analysis 2017, 16(3): 823-842 doi: 10.3934/cpaa.2017039
In this paper, we consider the following perturbed nonlocal elliptic equation
$\left\{ {\begin{array}{*{20}{l}}{ - {{\cal L}_K}u = \lambda u + f(x,u) + g(x,u),\;\;x \in \Omega ,}\\{u = 0,\;\;x \in \mathbb{R}{^N} \setminus \Omega ,}\end{array}} \right. $
where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.
keywords: Broken symmetry nonlocal elliptic equation variational methods
Threshold dynamics of a time periodic and two–group epidemic model with distributed delay
Lin Zhao Zhi-Cheng Wang Liang Zhang
Mathematical Biosciences & Engineering 2017, 14(5&6): 1535-1563 doi: 10.3934/mbe.2017080

In this paper, a time periodic and two–group reaction–diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number $R_0$ for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of $R_0$, that is, the disease is uniformly persistent if $R_0 > 1$, while the disease goes to extinction if $R_0 < 1$. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.

keywords: Two–group epidemic model time periodic distributed delay threshold dynamics Lyapunov functional
A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment
Danhua Jiang Zhi-Cheng Wang Liang Zhang
Discrete & Continuous Dynamical Systems - B 2018, 23(10): 4557-4578 doi: 10.3934/dcdsb.2018176

In this paper, we study the effects of diffusion and advection for an SIS epidemic reaction-diffusion-advection model in a spatially and temporally heterogeneous environment. We introduce the basic reproduction number $\mathcal{R}_{0}$ and establish the threshold-type results on the global dynamics in terms of $\mathcal{R}_{0}$. Some general qualitative properties of $\mathcal{R}_{0}$ are presented, then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number $\mathcal{R}_{0}$ for the special case that $γ(x,t)-β(x,t) = V(x,t)$ is monotone with respect to spatial variable $x$. Our results suggest that if $V_{x}(x,t)≥0,\not\equiv0$ and $V(x, t)$ changes sign about $x$, the advection is beneficial to eliminate the disease, whereas if $V_{x}(x,t)≤0,\not\equiv0$ and $V(x, t)$ changes sign about $x$, the advection is bad for the elimination of disease.

keywords: SIS epidemic model reaction-diffusion-advection spatial heterogeneity temporal periodic

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