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This paper is concerned with the existence results for generalized transition waves of space periodic and time heterogeneous lattice Fisher-KPP equations. By constructing appropriate subsolutions and supersolutions, we show that there is a critical wave speed such that a transition wave solution exists as soon as the least mean of wave speed is above this critical speed. Moreover, the critical speed we construct is proved to be minimal in some particular cases, such as space-time periodic or space independent.

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.

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution \begin{document}${\bf u} = {\bf 0}$\end{document} of such a system is unstable and the system has a stable space-time periodic positive solution \begin{document}${\bf u^*}(t,x)$\end{document} . We first show that in any direction \begin{document}$ξ∈ \mathbb{S}^{N-1}$\end{document} , such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed \begin{document}$c^{*}(ξ)$\end{document} in the direction of \begin{document}$ξ$\end{document} . Next, we show that for any \begin{document}$c>c^{*}(ξ)$\end{document} , there are space-time periodic traveling wave solutions of the form \begin{document}${\bf{u}}(t,x) = {\bf{Φ}}(x-ctξ,t,ctξ)$\end{document} connecting \begin{document}${\bf u^*}$\end{document} and \begin{document}${\bf 0}$\end{document} , and propagating in the direction of \begin{document}$ξ$\end{document} with speed \begin{document}$c$\end{document} , where \begin{document}$Φ(x,t,y)$\end{document} is periodic in \begin{document}$t$\end{document} and \begin{document}$y$\end{document} , and there is no such solution for \begin{document}$c . We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity.

This paper is concerned with the spreading speeds and transition waves of the following two species competition system in time heterogeneous media, u t = Δ u + u ( a 1 ( t ) − b 1 ( t ) u − c 1 ( t ) v ) , v t = Δ v + v ( a 2 ( t ) − b 2 ( t ) u − c 2 ( t ) v ) , x ∈ R N , t ∈ R , where a i ( ⋅ ) , b i ( ⋅ ) , c i ( ⋅ ) ( i = 1 , 2 ) depend in a general way in t ∈ R . The notion of transition waves for such a system is first introduced in this paper. We first establish the lower bounds of spreading speed intervals and generalized spreading speed intervals. It then shows that, under certain conditions, there exists a pair of general transition waves ( u ( t , x ) , v ( t , x ) ) connecting two semitrivial solutions of this system for a given class of speed c ( t ) with c > c ∗ and for any c c ∗ , there is no such transition waves.

The current paper is devoted to the study of the spreading speeds of two species competition diffusion-advection systems in time almost periodic and space periodic media. We first show that there is a finite spreading speed interval for such diffusion-advection systems. The principal Lyapunov exponent and the principal Floquent bundle theory have been applied to study the spreading speed of time almost periodic and space periodic systems. Under some sufficient conditions, we prove that the spreading speed interval of such systems in any direction is a singleton in the partially spatially homogeneous case and the general case, respectively.

The current paper is devoted to the study of the stability of space–time periodic traveling wave solutions and positive space–time periodic entire solutions of nonlocal dispersal cooperative systems in space–time periodic habitats. We first show the existence, uniqueness and stability of positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ for such nonlocal dispersal cooperative system. The existence of space–time periodic traveling wave solution connecting $${\mathbf {0}}$$ and positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ has been established by Bao, Shen and Shen (Commun. Pure Appl. Anal. 18: 361–396, 2019). In this paper, by using comparison principle and a weight function, we further show that the space–time periodic traveling wave solution for nonlocal dispersal cooperative system is asymptotically stable, as long as the initial value is uniformly bounded in a weighted space.

The current paper is devoted to the study of spreading speeds and linear determinacy of diffusive cooperative/competitive systems of the form u t ( t , x ) = ( A u ) ( t , x ) + F ( t , u ( t , x ) ) , x ∈ H , where H = R N or Z N , A is a random or nonlocal dispersal operator in the cases H = R N , and is a discrete dispersal operator in the case H = Z N , F ( t , u ) is cooperative or competitive in u and is recurrent in t, which includes F ( t , u ) is periodic and almost periodic in t as special cases. First, the notion of spreading speed intervals for diffusive cooperative systems is introduced via the natural features of spreading speeds and some basic spreading properties are established. Next, some principal Lyapunov exponent and principal Floquent bundle theory for linear cooperative systems of ordinary differential equations is developed, which plays an important role in the study of spreading speeds of diffusive cooperative/competitive systems with time dependence and is also of independent interest. In terms of the principal Lyapunov exponent and principal Floquent bundle theory, some upper and lower bounds for the spreading speed intervals for diffusive cooperative systems are then established. Under certain conditions, it is proved that a diffusive cooperative system has a single spreading speed and is linearly determinant. Finally, by transforming a diffusive competitive system to a diffusive cooperative system and applying established results for diffusive cooperative systems, upper and lower bounds for the spreading speed intervals for diffusive competitive systems are obtained and linearly determinacy for such systems is discussed.

This paper is concerned with the time periodic traveling front solutions of pyramidal shapes for time periodic Lotka–Volterra competition–diffusion system in R N with N ≥ 3 . It should be mentioned that the existence, uniqueness and stability of two-dimensional time periodic traveling curved fronts of time periodic Lotka–Volterra competition–diffusion system in R 2 have been studied yet. By the supersolution and subsolution technique coupled with comparison principle, in this paper we construct the existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for time periodic Lotka–Volterra competition–diffusion system in R N .

This paper is concerned with the existence, monotonicity, asymptotic behavior and uniqueness of traveling wave solutions for a three-species competitive–cooperative system with nonlocal dispersal and bistable dynamics. By considering a related truncated problem, we first establish the existence and strict monotonicity of traveling waves by means of a limiting argument and a comparative lemma. Then the asymptotic behavior of traveling waves is investigated by using Ikehara’s lemma and bilateral Laplace transform. Finally, we obtain the uniqueness of wave speed and traveling wave by sliding method.

The current paper is devoted to the study of the existence and stability of generalized transition waves of the following time-dependent reaction-diffusion cooperative system \begin{document}$ \begin{equation*} \frac{{\partial} \mathbf{u}}{{\partial} t}(t,x) = \Delta \mathbf{u}(t,x)+\mathbf{F}(t,\mathbf{u}(t,x)),\quad (x,t)\in {\mathbb{R}}^{N}\times {\mathbb{R}},\, \mathbf{u}\in \Bbb{R}^{K},\, K>1. \end{equation*} $\end{document} Here \begin{document}$ \mathbf{F}(t,\mathbf{u}(t,x)) $\end{document} depends on \begin{document}$ t\in\Bbb{R} $\end{document} in a general way. Recently, the spreading speeds and linear determinacy of the above time-dependent system have been studied by Bao et al. [J. Differential Equations 265 (2018) 3048-3091]. In this paper, using the principal Lyapunov exponent and principal Floquent bundle theory of linear cooperative systems, we prove the existence of generalized transition waves in any given direction with speed greater than the spreading speed by constructing appropriate subsolutions and supersolutions. When the initial value is uniformly bounded with respect to a weighted maximum norm, we further show that all solutions converge to the generalized transition wave solutions exponentially in time.

This paper is concerned with space periodic traveling wave solutions of the following Lotka-Volterra competition system with nonlocal dispersal and space periodic dependence, {partial derivative u(1)/partial derivative t = integral R-N kappa(y - x)u(1)(t, y)dy - u(1)(t, x) + u(1)(a(1)(x) - b(1)(x)u(1) - c(1)(x)u(1)), x is an element of R-N partial derivative u(2)/partial derivative t = integral R-N kappa(y - x)u(2)(t, y)dy - u(1)(t, x) + u(2)(a(2)(x) - b(2)(x)u(2) - c(2)(x)u(2)), x is an element of R-N. Under suitable assumptions, the system admits two semitrivial space periodic equilibria (u*(1) (x), 0) and (0, u*(2)(x)), where (u*(1)(x), 0) is linearly and globally stable and (0, u*(2)(x)) is linearly unstable with respect to space periodic perturbations. By sub- and supersolution techniques and comparison principals, we show that, for any given xi is an element of SN-1, there exists a continuous periodic traveling wave solution of the form (u(1)(t, x), u(2)(t, x)) = (Phi(1)(x - ct xi, ct xi), 4), Phi(2)(x - ct xi, ct xi)) connecting (u*(1) (.), 0) and (0, u*(2)(.)) and propagating in the direction of xi. with speed c > c*(xi), where c*(xi) is the spreading speed of the system in the direction of. Moreover, for c c* (xi), we also prove the asymptotic stability and uniqueness of traveling wave solution using squeezing techniques. (C) 2016 Elsevier Inc. All rights reserved.

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space $$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$ where $$\Delta $$ denotes $$\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }$$ , $$x,y\in {\mathbb {R}}$$ and $$d>0$$ is a constant, the functions $$r_i(t),a_i(t)$$ and $$b_i(t)$$ are T-periodic and Holder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution $$\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) $$ connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.

Recently, the spreading speeds and periodic traveling wave solutions for a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats have been investigated. In this paper, we continue to study the stability and convergence rate for such time and space periodic traveling waves. We show that if the initial function perturbation is uniformly bounded with respect to a weighted maximum norm, the periodic traveling wave solution is globally exponentially stable when the wave speed is greater than the spreading speed.

This paper is concerned with propagation phenomena of a general class of partially degenerate nonlocal dispersal cooperative systems in time and space periodic habitats. We first show that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions. Next, we prove that if the wave speed is greater than the spreading speed, there exists a time and space periodic traveling wave solution connecting the stable positive time and space periodic steady state and 0 . Finally, we apply these results to two species partially degenerate competition systems and a partially degenerate epidemic model with nonlocal dispersal in time and space periodic habitats.

Xin Zhang,1 Shuangsuo Dang,1 Fanpu Ji,1 Juanjuan Shi,1 Yaping Li,1 Mei Li,1 Xiaoli Jia,1 Yueqiang Wan,2 Xiongxiong Bao,3 Wenjun Wang1 1Department of Infectious Diseases, Second Affiliated Hospital of Xi’an Jiaotong University, Xi’an 710004, China; 2Department of Internal Medicine, Tuberculosis Hospital of Shaanxi Province, Xi’an 710100, China; 3School of Science, Chang’an University, Xi’an 710064, China Introduction: According to our clinical experience, cellulitis is common in summer; however, very few studies have mentioned this trend. Methods: Using Google Trends, we analyzed the monthly data of Google searches for “­cellulitis” from 31 countries on 6 continents. Results: Seasonality explained 34%–92% of the variability in search volume, with peaks occurring in summer months. Conclusion: The analyses offered new insights into the epidemiology of cellulitis on national and international scales. Clinical data are needed to validate the Internet search data. Keywords: cellulitis, Internet search, seasonality

This paper is concerned with the time periodic Lotka–Volterra competition–diffusion system { u t = d 1 ( t ) u x x + u ( r 1 ( t ) − a 1 ( t ) u − b 1 ( t ) v ) , v t = d 2 ( t ) v x x + v ( r 2 ( t ) − a 2 ( t ) u − b 2 ( t ) v ) , x ∈ R , t > 0 , where d i ( t ) , r i ( t ) , a i ( t ) , b i ( t ) ∈ C θ 2 ( R ) are T -periodic functions, 0 θ 1 , i = 1 , 2 . Under certain conditions, the system admits two stable semi-trivial periodic solutions ( p ( t ) , 0 ) and ( 0 , q ( t ) ) and a unique coexistence periodic solution ( u 0 ( t ) , v 0 ( t ) ) , which is unstable and satisfies 0 u 0 ( t ) p ( t ) and 0 v 0 ( t ) q ( t ) for t ∈ [ 0 , T ] . In this paper we prove that the system admits a time periodic traveling wave solution ( u ( x , t ) , v ( x , t ) ) = ( U ( x + c t , t ) , V ( x + c t , t ) ) connecting two periodic solutions ( p ( t ) , 0 ) and ( 0 , q ( t ) ) as x → ± ∞ , where c is the wave speed. By using a dynamical method, we show that the time periodic traveling wave solution ( U ( x + c t , t ) , V ( x + c t , t ) ) is asymptotically stable and unique modulo translation for front-like initial values.

The current paper {establishes} criteria for the existence of principal eigenvalues of time periodic cooperative linear nonlocal dispersal systems with Direchlet type, Neumann type or periodic type boundary conditions. It is shown that such a nonlocal dispersal system has a principal eigenvalue in the following cases: the nonlocal dispersal distance is sufficiently small; the spatial inhomogeneity satisfies a so called vanishing condition; or the spatial inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the principal eigenvalue of a time periodic cooperative linear nonlocal dispersal system (if it exists) is algebraically simple. A linear nonlocal dispersal system may not have a principal eigenvalue. The results established in the current paper extend those in literature for time independent or periodic nonlocal dispersal equations to time periodic cooperative nonlocal dispersal systems and {will} serve as {a} basic tool for the study of cooperative nonlinear systems with nonlocal dispersal., Comment: 15 pages