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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.

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.

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 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.

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.

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.