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, 22(11): 1-22 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
Local exact controllability to positive trajectory for parabolic system of chemotaxis
Bao-Zhu Guo Liang Zhang
Mathematical Control & Related Fields 2016, 6(1): 143-165 doi: 10.3934/mcrf.2016.6.143
In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani's fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimate for parabolic equations.
keywords: Kakutani's fixed point theorem. Carleman inequality chemotaxis system Local exact controllability
A wedge trust region method with self-correcting geometry for derivative-free optimization
Liang Zhang Wenyu Sun Raimundo J. B. de Sampaio Jinyun Yuan
Numerical Algebra, Control & Optimization 2015, 5(2): 169-184 doi: 10.3934/naco.2015.5.169
Recently, some methods for solving optimization problems without derivatives have been proposed. The main part of these methods is to form a suitable model function that can be minimized for obtaining a new iterative point. An important strategy is geometry-improving iteration for a good model, which needs a lot of calculations. Besides, Marazzi and Nocedal (2002) proposed a wedge trust region method for derivative free optimization. In this paper, we propose a new self-correcting geometry procedure with less computational efforts, and combine it with the wedge trust region method. The global convergence of new algorithm is established. The limited numerical experiments show that the new algorithm is efficient and competitive.
keywords: Derivative-free optimization self-correcting geometry. model-based method unconstrained optimization trust region method
Spatial dynamics of a diffusive predator-prey model with stage structure
Liang Zhang Zhi-Cheng Wang
Discrete & Continuous Dynamical Systems - B 2015, 20(6): 1831-1853 doi: 10.3934/dcdsb.2015.20.1831
In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.
keywords: principal eigenvalues fluctuation methods persistence and extinction Predator-prey model non-local delays global attractivity.
An almost periodic epidemic model with age structure in a patchy environment
Bin-Guo Wang Wan-Tong Li Liang Zhang
Discrete & Continuous Dynamical Systems - B 2016, 21(1): 291-311 doi: 10.3934/dcdsb.2016.21.291
An almost periodic epidemic model with age structure in a patchy environment is considered. The existence of the almost periodic disease-free solution and the definition of the basic reproduction ratio $R_{0}$ are given. Based on those, it is shown that a disease dies out if the basic reproduction number $R_{0}$ is less than unity and persists in the population if it is greater than unity.
keywords: epidemic model age structure threshold dynamics. Almost periodic basic reproduction ratio

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