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The existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for the time periodic Lotka-Volterra competition-diffusion system have already been studied in \begin{document}$ \Bbb{R}^{N} $\end{document} with \begin{document}$ N\geq 3 $\end{document} . In this paper, we continue to study the uniqueness and asymptotic stability of such time-periodic pyramidal traveling front in the three-dimensional whole space. For any given admissible pyramid, we show that the time periodic pyramidal traveling front is uniquely determined and it is asymptotically stable under the condition that given perturbations decay at infinity. Moreover, the time periodic pyramidal traveling front is uniquely determined as a combination of two-dimensional periodic V-form waves on the edges of the pyramid.

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium \begin{document}$ (0, 0) $\end{document} to some unknown positive steady state for wave speed \begin{document}$ c>c^* = \max\left\{2, 2\sqrt{dr}\right\} $\end{document} and there is no such traveling wave solutions for \begin{document}$ c , where \begin{document}$ d $\end{document} and \begin{document}$ r $\end{document} respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., \begin{document}$u_{t}=J*u-u+f(u)$\end{document} . Here the kernel \begin{document}$J$\end{document} is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds \begin{document}$c^{*}$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} , where \begin{document}$c^*$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel \begin{document}$J$\end{document} is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

In this paper, we study the asymptotic stability of traveling wave fronts to the Allen-Cahn equation with a fractional Laplacian. The main tools that we used are super- and subsolutions and squeezing methods.

This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.

We study the asymptotic spreading properties and periodic traveling wave solutions of a time periodic and diffusive SI epidemic model with demographic structure (follows the logistic growth). Since the comparison principle is not applicable to the full system, we analyze the asymptotic spreading phenomena for susceptible class and infectious class by comparing with respective relevant periodic equations with KPP-type. By applying fixed point theorem to a truncated problem on a finite interval, combining with limit idea, the existence of periodic traveling wave solutions are derived. The results show that the minimal wave speed exactly equals to the spreading speed of infectious class when susceptible class is abundant.

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 \begin{document} $\mathcal{R}_{0}$ \end{document} and establish the threshold-type results on the global dynamics in terms of \begin{document} $\mathcal{R}_{0}$ \end{document} . Some general qualitative properties of \begin{document} $\mathcal{R}_{0}$ \end{document} are presented, then the paper is devoted to studying how the advection and diffusion of the infected individuals affect the reproduction number \begin{document} $\mathcal{R}_{0}$ \end{document} for the special case that \begin{document} $γ(x,t)-β(x,t) = V(x,t)$ \end{document} is monotone with respect to spatial variable \begin{document} $x$ \end{document} . Our results suggest that if \begin{document} $V_{x}(x,t)≥0,\not\equiv0$ \end{document} and \begin{document} $V(x, t)$ \end{document} changes sign about \begin{document} $x$ \end{document} , the advection is beneficial to eliminate the disease, whereas if \begin{document} $V_{x}(x,t)≤0,\not\equiv0$ \end{document} and \begin{document} $V(x, t)$ \end{document} changes sign about \begin{document} $x$ \end{document} , the advection is bad for the elimination of disease.

This paper is concerned with the global stability of V-shaped traveling fronts in reaction-diffusion equations with combustion and degenerate monostable nonlinearity. The existence of such curved fronts has been recently proved by [ 39 ]. In this paper, by constructing new subsolutions, we show the asymptotic stability of V-shaped traveling fronts.

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 \begin{document} $\mathcal{R}_0$ \end{document} for the model system, which gives the threshold dynamics in the sense that the disease will die out if \begin{document} $\mathcal{R}_0＜1$ \end{document} and the disease will be uniformly persistent if \begin{document} $\mathcal{R}_0>1.$ \end{document} Furthermore, it is shown that there is at least one positive steady state when \begin{document} $\mathcal{R}_0>1.$ \end{document} Finally, in terms of general birth function for adult individuals, through introducing two numbers \begin{document} $\check{\mathcal{R}}_0$ \end{document} and \begin{document} $\hat{\mathcal{R}}_0$ \end{document} , we establish sufficient conditions for the persistence and global extinction of the disease, respectively.

This paper is concerned with the following two-species Lotka-Volterra competition-diffusion system in the three-dimensional spatial space {partial derivative/partial derivative tu(1) (x, t) = Delta u(1) (x, t) + u(1) (x, t) [1 - u(1) (x, t) - k(1)u(2) (x, t)] , {partial derivative/partial derivative tu(2) (x, t) = d Delta u(2) (x, t) + ru(2) (x, t) [1 - u(2) (x, t) - k(2)u(1) (x, t)] , where x is an element of R-3 and t > 0. For the bistable case, namely k(1) , k(2) > 1, it is well known that the system admits a one-dimensional monotone traveling front Phi(x + ct) = (Phi(1) (x + ct), Phi(2) (x + ct)) connecting two stable equilibria E-u = (1, 0) and E-v = (0, 1), where c is an element of R is the unique wave speed. Recently, two-dimensional V-shaped fronts and high-dimensional pyramidal traveling fronts have been studied under the assumption c > 0. In this paper it is shown that for any s > c > 0, the system admits axisymmetric traveling fronts psi(x', x(3) + st) = Phi(1)(x ', x(3) + st), Phi(2)(x ', x(3) + st) in R-3 connecting E-u = (1, 0) and E-v = (0, 1), where x' is an element of R-2. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the x 3-axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When s tends to c, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in R-3. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level set is discussed.

This paper is concerned with traveling curved fronts in reaction diffusion equations with degenerate monostable and combustion nonlinearities. For a given admissible pyramidal in three-dimensional spaces, the existence of a pyramidal traveling front has been proved by Wang and Bu [ 30 ] recently. By constructing new supersolutions and developing the arguments of Taniguchi [ 25 ] for the Allen-Cahn equation, in this paper we first characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces, and then establish the uniqueness and asymptotic stability of such three-dimensional pyramidal traveling fronts under the condition that given perturbations decay at infinity.

This paper is concerned with propagation phenomena for an epidemic model describing the circulation of a disease within two populations or two subgroups in periodic media, where the susceptible individuals are assumed to be motionless. The spatial dynamics for the cooperative system obtained by a classical transformation are investigated, including spatially periodic steady state, spreading speeds and pulsating travelling fronts. It is proved that the minimal wave speed is linearly determined and given by a variational formula involving linear eigenvalue problem. Further, we prove that the existence and non-existence of travelling wave solutions of the model are entirely determined by the basic reproduction ratio \begin{document}$ \mathcal{R}_{0} $\end{document} . As an application, we prove that if the localized amount of infectious individuals are introduced at the beginning, then the solution of such a system has an asymptotic spreading speed in large time and that is exactly coincident with the minimal wave speed.

This paper is concerned with a nonlocal reaction-diffusion equation with the form \begin{eqnarray} \frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}+u\left\{ 1+\alpha u-\beta u^{2}-(1+\alpha-\beta)(\phi\ast u) \right\}, \quad (t,x)\in (0,\infty) \times \mathbb{R}, \end{eqnarray} where $\alpha $ and $\beta$ are positive constants, $0c^*$. At the same time, we show that there is no such traveling wave solutions for speed $cc^*$, we further show that the steady state is the unique positive equilibrium. Using the lower and upper solutions method, we also establish the existence of monotone traveling wave fronts connecting the zero equilibrium and the positive equilibrium. Finally, for a specific kernel function $\phi(x):=\frac{1}{2\sigma}e^{-\frac{|x|}{\sigma}}$ ($\sigma>0$), by numerical simulations we show that the traveling wave solutions may connects the zero equilibrium to a periodic steady state as $\sigma$ is increased. Furthermore, by the stability analysis we explain why and when a periodic steady state can appear.

This paper is to study traveling fronts of reaction-diffusion equations with space-time periodic advection and nonlinearity in $\mathbb{R}^N$ with $N\geq3$. We are interested in curved fronts satisfying some ``pyramidal" conditions at infinity. In $\Bbb{R}^3$, we first show that there is a minimal speed $c^{*}$ such that curved fronts with speed $c$ exist if and only if $c\geq c^{*}$, and then we prove that such curved fronts are decreasing in the direction of propagation. Furthermore, we give a generalization of our results in $\mathbb{R}^N$ with $N\geq4$.

In this paper, we study the nonlocal dispersal Logistic equation \begin{equation*} \begin{cases} u_t=Du+\lambda m(x)u-c(x)u^p &\text{ in }{\Omega}\times(0,+\infty),\\ u(x,0)=u_0(x)\geq0&\text{ in }{\Omega}, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\lambda>0$ and $p>1$ are constants. $Du(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy$ represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel $J$, $m\in C(\bar{\Omega})$ and may change sign in $\Omega$. The function $c$ is nonnegative and has a degeneracy in some subdomain of $\Omega$. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of $c$ on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.

In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.

This paper is concerned with the existence, uniqueness and stability of traveling curved fronts for reaction-diffusion bistable systems in two-dimensional space. By establishing the comparison theorem and constructing appropriate supersolutions and subsolutions, we prove the existence of traveling curved fronts. Furthermore, we show that the curved front is globally stable. Finally, we apply the results to three important models in biology.

Recently, we derived a lattice model for a single species with stage structure in a two-dimensional patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay (IMA J. Appl. Math., 73 (2008), 592-618.). The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. In this paper we study the asymptotic stability of traveling wavefronts of this model when the immature population is not mobile. Under the assumption that the birth function satisfies monostable condition, we prove that the traveling wavefront is exponentially stable by means of weighted energy method, when the initial perturbation around the wave is suitably small in a weighted norm. The exponential convergent rate is also obtained.