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This note is to fix some gaps in the proofs of some of the results in our paper [M.A. Onyido, W. Shen, J. Differ. Equ. 295 (2021) 1–38].

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics,$$ \begin{cases} u_t=\Delta u-\chi_1\nabla\cdot (\frac{u}{w}\nabla w)+u(a_1-b_1u-c_1v),\quad &x\in\Omega\cr v_t=\Delta v-\chi_2\nabla\cdot (\frac{v}{w}\nabla w)+v(a_2-b_2v-c_2u),\quad &x\in\Omega\cr 0=\Delta w-\mu w +\nu u+ \lambda v,\quad &x\in\Omega\cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0,\quad &x\in\partial\Omega, \end{cases} $$ where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, and $\chi_i,a_i,b_i,c_i$ ($i=1,2$) and $\mu,\nu,\lambda$ are positive constants. Among others, we prove that for any given nonnegative initial data $u_0,v_0\in C^0(\bar\Omega)$ with $u_0+v_0\not \equiv 0$, (0.1) has a unique globally defined classical solution $(u(t,x;u_0,v_0),v(t,x;u_0,v_0),w(t,x;u_0,v_0))$ with $u(0,x;u_0,v_0)=u_0(x)$ and $v(0,x;u_0,v_0)=v_0(x)$ in any space dimensional setting. Moreover, we prove that there is $\chi^*(\mu,\chi_1,\chi_2)>0$ satisfying \begin{align*} &\chi^*(\mu,\chi_1,\chi_2)= \begin{cases} \frac{\mu \chi^2}{4} &{\rm if}\,\, 0\chi^*(\mu,\chi_1,\chi_2) $$ implies $$\limsup_{t\rightarrow\infty} \|u(t,\cdot;u_0,v_0)+v(t,\cdot;u_0,v_0)\|_\infty\leq M^*\,{\rm and}\,\liminf_{t\rightarrow\infty}\inf_{x\in\Omega}(u(t,x,u_0,v_0)+v(t,x;u_0,v_0))\ge m^*$$ for some positive constants $M^*,m^*$ independent of $u_0,v_0$, the latter is referred to as combined pointwise persistence.

The current paper is concerned with the forced waves of Keller–Segel chemoattraction systems in shifting environments of the form, 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=u_{xx}-\chi (uv_x)_x +u(r(x-ct)-bu),\quad x\in \mathbb {R}\\ 0=v_{xx}- \nu v+\mu u,\quad x\in \mathbb {R}, \end{array}\right. } \end{aligned}$$ where $$\chi $$ , b, $$\nu $$ , and $$\mu $$ are positive constants, $$c\in \mathbb {R}$$ , the resource function r(x) is globally Holder continuous, bounded, $$r^*=\sup _{x\in {\mathbb R}}r(x)>0$$ , $$r(\pm \infty ):=\lim _{x\rightarrow \pm \infty }r(x)$$ exist, and either $$r(-\infty )2\chi \mu $$ . In the case that $$r(-\infty )\frac{\chi \mu r^*}{2\sqrt{\nu }(b-\chi \mu )}- 2\sqrt{\frac{r^*(b-2\chi \mu )}{b-\chi \mu }}$$ . In the case that $$r(\pm \infty )0$$ , where $$\lambda _\infty $$ is the generalized principal eigenvalue of the operator $$u(\cdot )\mapsto u_{xx}(\cdot )+cu_{x}(\cdot )+r(\cdot )u(\cdot )$$ on $${\mathbb R}$$ in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.

In recent years, it has been drawn a lot of attention to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. However, for several important chemotaxis models, only in the space two dimensional setting, it has been shown that logistic kinetics is sufficient to enforce the global existence of classical solutions (see [15] and [34]). The current paper is to give a confirmed answer to the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \begin{equation} \begin{cases} u_t=\Delta u-\chi\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u),\quad x\in \Omega\cr 0=\Delta v-\mu v+\nu u,\quad x\in \Omega \quad \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad x\in\partial\Omega, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$, $\chi$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, and $\mu,\nu$ are positive constants. We prove that, for any given nonnegative initial data $0\not\equiv u_0\in C^0(\bar \Omega)$, (0.1) has a unique globally defined classical solution $(u(t,x;u_0),v(t,x;u_0))$ with $u(0,x;u_0)=u_0(x)$, which shows that, in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $\chi$. In addition, the solutions are shown to be uniformly bounded under the conditions $a_{\inf}>\frac{\mu \chi^2}{4}$ when $\chi \leq 2$ and $a_{\inf}>\mu (\chi-1)$ when $\chi>2$.

The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space and time dependent logistic source, 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu=\Delta u -\chi \nabla \cdot (u\nabla v)+u(a(x,t)-b(x,t)u),&{}\quad x\in {\mathbb R}^N,\\ 0=\Delta v-\lambda v+\mu u ,&{}\quad x\in {\mathbb R}^N, \end{array}\right. } \end{aligned}$$ where $$N\ge 1$$ is a positive integer, $$\chi , \lambda $$ and $$\mu $$ are positive constants, and the functions a(x, t) and b(x, t) are positive and bounded. In the first of the series (Salako and Shen in Math Models Methods Appl Sci 28(11):2237–2273, 2018), we studied the phenomena of pointwise and uniform persistence for solutions with strictly positive initial data, and the asymptotic spreading for solutions with compactly supported or front like initial data. In the second of the series (Salako and Shen in J Math Anal Appl 464(1):883–910, 2018), we investigate the existence, uniqueness and stability of strictly positive entire solutions of (0.1). In particular, in the case of space homogeneous logistic source (i.e. $$a(x,t)\equiv a(t)$$ and $$b(x,t)\equiv b(t)$$ ), we proved in Salako and Shen (J Math Anal Appl 464(1):883–910, 2018) that the unique spatially homogeneous strictly positive entire solution $$(u^*(t),v^*(t))$$ of (0.1) is uniformly and exponentially stable with respect to strictly positive perturbations when $$00$$ with $$\chi \mu \big (1+\frac{\sup _{t\in {\mathbb R}}a(t)}{\inf _{t\in {\mathbb R}}a(t)}\big ) {c}^*_{\chi }$$ and every unit vector $$\xi $$ , (0.1) has a transition front solution of the form $$(u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))$$ satisfying that $$C'(t)=\frac{a(t)+\kappa ^2}{\kappa }$$ for some positive number $$\kappa $$ , $$\liminf _{t-s\rightarrow \infty }\frac{C(t)-C(s)}{t-s}=\underline{c}$$ , and $$\begin{aligned} \lim _{x\rightarrow -\infty }\sup _{t\in {\mathbb R}}|U(x,t)-u^*(t)|=0 \quad \text {and}\quad \lim _{x\rightarrow \infty }\sup _{t\in {\mathbb R}}|\frac{U(x,t)}{e^{-\kappa x}}-1|=0. \end{aligned}$$ Furthermore, we prove that there is no transition front solution $$(u(x,t),v(x,t))=(U(x\cdot \xi -C(t),t),V(x\cdot \xi -C(t),t))$$ of (0.1) connecting (0, 0) and $$(u^*(t),v^*(t))$$ with least mean speed less than $$2\sqrt{\underline{a}}$$ , where $$\underline{a}=\liminf _{t-s\rightarrow \infty }\frac{1}{t-s}\int _{s}^{t}a(\tau )d\tau $$ .

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first of the series, we investigated the dynamical behaviors of logistic type chemotaxis models on the half line R + , which are formally corresponding limit systems of the free boundary problems. In the second of the series, we establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

In the current series of two papers, we study the long time behavior of nonnegative solutions to the following random Fisher–KPP equation, 1 $$\begin{aligned} u_t =u_{xx}+a(\theta _t\omega )u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$ where $$\omega \in \Omega $$ , $$(\Omega , {\mathcal {F}},{\mathbb {P}})$$ is a given probability space, $$\theta _t$$ is an ergodic metric dynamical system on $$\Omega $$ , and $$a(\omega )>0$$ for every $$\omega \in \Omega $$ . We also study the long time behavior of nonnegative solutions to the following nonautonomous Fisher–KPP equation, 2 $$\begin{aligned} u_t=u_{xx}+a_0(t)u(1-u),\quad x\in {{\mathbb {R}}}, \end{aligned}$$ where $$a_0(t)$$ is a positive locally Holder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $$a(\omega )$$ , we show that the constant solution $$u=1$$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $$[2\sqrt{{\underline{a}}}, 2\sqrt{{\bar{a}}}]$$ , where $${\underline{a}}$$ and $${\bar{a}}$$ are the least and the greatest means of $$a(\cdot )$$ , respectively, and hence the spreading speed interval is linearly determinate. It is shown that the solution of (1) with a nonnegative initial function which is bounded away from 0 for $$x\ll -1$$ and is 0 for $$x\gg 1$$ propagates at the speed $$2\sqrt{{\hat{a}}}$$ , where $${\hat{a}}$$ is the mean of $$a(\cdot )$$ . Under some assumption on $$a_0(\cdot )$$ , we also show that the constant solution $$u=1$$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $$a(\omega )$$ and $$a_0(t)$$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher–KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).

To date, most of electrocaloric devices reported can be simplified as a multilayer structure in which thermal source and sink are different materials at two ends. The thermal conduction in the multilayer structure is the key for the performance of the devices. In this paper, the analytical solutions for the thermal conduction in a multilayer structure with four layers are introduced. The middle two layers are electrocaloric materials. The analytical solutions are also simplified for a hot/cold plate with two sides being different media—a typical case for thermal treatment of materials. The analytical solutions include series with infinite terms. It is proved that these series are convergent so the sum of a series can be calculated using the first N terms. The equation for calculating the N is introduced. Based on the case study, it is found that the N is usually a small number, mostly less than 40 and rarely more than 100. The issues related to the application of the analytical solutions for the simulation of real electrocaloric devices are discussed, which includes the usage of multilayer ceramic capacitor, influence of electrodes, and characterization of thin film.

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line \begin{document}$ \mathbb{R}^+ $\end{document} , which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

The present paper is devoted to the investigation of population dynamics under climate change. The evolution of species is modelled by a reaction-diffusion equation in a spatio-temporally heterogeneous environment described by a climate envelope that shifts with a time-dependent speed function. For a general almost-periodic speed function, we establish the persistence criterion in terms of the sign of the approximate top Lyapunov exponent and, in the case of persistence, prove the existence of a unique forced wave solution that dominates the population profile of species in the long run. In the setting for studying the effects of fluctuations in the shifting speed or location of the climate envelope, we show by means of matched asymptotic expansions and numerical simulations that the approximate top Lyapunov exponent is a decreasing function with respect to the amplitude of fluctuations, yielding that fluctuations in the shifting speed or location have negative impacts on the persistence of species, and moreover, the larger the fluctuation is, the more adverse the effect is on the species. In addition, we assert that large fluctuations can always drive a species to extinction. Our numerical results also show that a persistent species under climate change is invulnerable to mild fluctuations, and becomes vulnerable when fluctuations are so large that the species is endangered. Finally, we show that fluctuations of amplitude less than or equal to the speed difference between the shifting speed and the critical speed are too weak to endanger a persistent species.

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on \begin{document}$ \mathbb{R}^{N} $\end{document}, \begin{document}$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\end{equation} $\end{document} where \begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document} are positive constants and \begin{document}$ N $\end{document} is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption \begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}, the global existence of a unique classical solution \begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document} of (1) with \begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document} and \begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document} for every nonnegative, bounded, and uniformly continuous function \begin{document}$ u_0(x) $\end{document}, and every nonnegative, bounded, uniformly continuous, and differentiable function \begin{document}$ v_0(x) $\end{document}. Next, under the same assumption \begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function \begin{document}$ u_0 $\end{document} is bounded below by a positive constant independent of \begin{document}$ (u_0, v_0) $\end{document} when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function \begin{document}$ u_0 $\end{document}. We show that there is \begin{document}$ K = K(a,\lambda,N)>\frac{N}{4} $\end{document} such that if \begin{document}$ b>K \chi\mu $\end{document} and \begin{document}$ \lambda\geq \frac{a}{2} $\end{document}, then for every strictly positive initial function \begin{document}$ u_0(\cdot) $\end{document}, it holds that \begin{document}$ \lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0. $\end{document}

This series of two papers is devoted to the study of the principal spectral theory of nonlocal dispersal operators with almost periodic dependence and the study of the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence. In this first part of the series, we investigate the principal spectral theory of nonlocal dispersal operators from two aspects: top Lyapunov exponents and generalized principal eigenvalues. Among others, we provide various characterizations of the top Lyapunov exponents and generalized principal eigenvalues, establish the relations between them, and study the effect of time and space variations on them. In the second part of the series, we will study the asymptotic dynamics of nonlinear nonlocal dispersal equations with almost periodic dependence applying the principal spectral theory to be developed in this part.

We study the long-time behavior of the skew-product semiflow generated by scalar reaction-diffusion equation on the circle with almost periodic forcing: u t = u x x + f ( t , u , u x ) , t > 0 , x ∈ S 1 = R / 2 π Z , where f ( t , u , u x ) is uniformly almost-periodic in t. Almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic. Contrary to the time-periodic cases (for which any ω-limit set Ω can be embedded into a periodically forced circle flow), we show that, for almost-periodic forcing, the problem that whether Ω can be embedded into an almost-periodically forced circle flow is strongly related to the dimension of the center space V c ( Ω ) associated with Ω. On the one hand, if dim V c ( Ω ) ≤ 1 then Ω is either spatially-inhomogeneous or spatially-homogeneous; and moreover, any spatially-inhomogeneous Ω can be embedded into an almost periodically-forced circle flow. On the other hand, when dim V c ( Ω ) > 1 , it is shown that the above embedding property cannot hold anymore. These reveal that for such system there are essential differences between time periodic forcing and non-periodic forcing.

This paper is concerned with the asymptotic dynamics of two species competition systems of the form \begin{document}$ \begin{equation*} \begin{cases} u_t(t,x) = \mathcal{A} u+u(a_1(t,x)-b_1(t,x)u-c_1(t,x)v),\quad x\in {\mathbb{R}} \cr v_t(t,x) = \mathcal{A} v+ v(a_2(t,x)-b_2(t,x)u-c_2(t,x) v),\quad x\in {\mathbb{R}} \end{cases} \end{equation*} $\end{document} where \begin{document}$ (\mathcal{A}u)(t,x) = u_{xx}(t,x) $\end{document} , or \begin{document}$ (\mathcal{A}u)(t,x) = \int_{ {\mathbb{R}} }\kappa(y-x)u(t,y)dy-u(t,x) $\end{document} ( \begin{document}$ \kappa(\cdot) $\end{document} is a smooth non-negative convolution kernel supported on an interval centered at the origin), \begin{document}$ a_i(t+T,x) = a_i(t,x) $\end{document} , \begin{document}$ b_i(t+T,x) = b_i(t,x) $\end{document} , \begin{document}$ c_i(t+T,x) = c_i(t,x) $\end{document} , and \begin{document}$ a_i $\end{document} , \begin{document}$ b_i $\end{document} , and \begin{document}$ c_i $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) are spatially homogeneous when \begin{document}$ |x|\gg 1 $\end{document} , that is, \begin{document}$ a_i(t,x) = a_i^0(t) $\end{document} , \begin{document}$ b_i(t,x) = b_i^0(t) $\end{document} , \begin{document}$ c_i(t,x) = c_i^0(t) $\end{document} for some \begin{document}$ a_i^0(t) $\end{document} , \begin{document}$ b_i^0(t) $\end{document} , \begin{document}$ c_i^0(t) $\end{document} , and \begin{document}$ |x|\gg 1 $\end{document} . Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. In particular, we study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds. We also study a relevant problem, that is, the continuity of the spreading speeds of time periodic two species competition systems with respect to time periodic perturbations, and prove that the spread speeds of such systems are lower semicontinuous with respect to time periodic perturbations.

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.

The current series of three papers is concerned with the asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space- and time-dependent logistic source: [Formula: see text] where [Formula: see text] is a positive integer, [Formula: see text] and [Formula: see text] are positive constants, and the functions [Formula: see text] and [Formula: see text] are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading in [Formula: see text]. To this end, under some explicit condition on the parameters, we first show that [Formula: see text] has a unique nonnegative time global classical solution [Formula: see text] with [Formula: see text] for every [Formula: see text] and every nonnegative bounded and uniformly continuous initial function [Formula: see text]. Next we show the pointwise persistence phenomena of the solutions in the sense that, for any solution [Formula: see text] of [Formula: see text] with strictly positive initial function [Formula: see text], there are [Formula: see text] such that [Formula: see text] and show the uniform persistence phenomena of solutions in the sense that there are [Formula: see text] such that for any strictly positive initial function [Formula: see text], there is [Formula: see text] such that [Formula: see text] We then discuss the spreading properties of solutions to [Formula: see text] with compactly supported initial function and prove that there are positive constants [Formula: see text] such that for every [Formula: see text] and every nonnegative initial function [Formula: see text] with nonempty compact support, we have that [Formula: see text] and [Formula: see text] We also discuss the spreading properties of solutions to [Formula: see text] with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions of [Formula: see text] and the existence of transition fronts of [Formula: see text], respectively.

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.

The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N \tau v_t=\Delta v-v+u, \quad x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ represents the population density of a mobile species and $v(x,t)$ represents the population density of a chemoattractant, and $\chi$ represents the chemotaxis sensitivity. We prove that for every $\tau >0$, there is $0

In this paper, we consider two species chemotaxis systems with Lotka–Volterra competition reaction terms. Under appropriate conditions on the parameters in such a system, we establish the existence of traveling wave solutions of the system connecting two spatially homogeneous equilibrium solutions with wave speed greater than some critical number c ∗ . We also show the non-existence of such traveling waves with speed less than some critical number c 0 ∗ , which is independent of the chemotaxis. Moreover, under suitable hypotheses on the coefficients of the reaction terms, we obtain explicit range for the chemotaxis sensitivity coefficients ensuring c ∗ = c 0 ∗ , which implies that the minimum wave speed exists and is not affected by the chemoattractant.

In the current paper, we consider the following parabolic–elliptic semilinear Keller–Segel model on RNRN, {ut=∇⋅(∇u−χu∇v)+au−bu2,x∈RN,t>0,0=(Δ−I)v+u,x∈RN,t>0, where χ>0,a≥0,b>0 are constant real numbers and N is a positive integer. We first prove the local existence and uniqueness of classical solutions (u(x,t;u0),v(x,t;u0))(u(x,t;u0),v(x,t;u0)) with u(x,0;u0)=u0(x)u(x,0;u0)=u0(x) for various initial functions u0(x)u0(x). Next, under some conditions on the constants a,b,χa,b,χ and the dimension N , we prove the global existence and boundedness of classical solutions (u(x,t;u0),v(x,t;u0))(u(x,t;u0),v(x,t;u0)) for given initial functions u0(x)u0(x). Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants a,b,χa,b,χ and the dimension N , we show that for every strictly positive initial function u0(⋅)u0(⋅), limt→∞⁡supx∈RN⁡[|u(x,t;u0)−ab|+|v(x,t;u0)−ab|]=0, and that for every nonnegative initial function u0(⋅)u0(⋅) with non-empty and compact support supp(u0)supp(u0), there are 0cup⁎(u0).

The present paper is devoted to the investigation of various properties of transition fronts in one-dimensional nonlocal equations in heterogeneous media of ignition type, whose existence has been established by the authors of the present paper in a previous work. It is first shown that transition fronts are continuously differentiable in space with uniformly bounded and uniformly Lipschitz continuous space partial derivative. This is the first time that space regularity of transition fronts in nonlocal equations is ever studied. It is then shown that transition fronts are uniformly steep. Finally, asymptotic stability, in the sense of exponentially attracting front-like initial data, of transition fronts is studied.

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle $S^1$, \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z}, \end{equation*} where $f$ is independent of $t$ or $T$-periodic in $t$. Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some $\omega$-limit set). More precisely, in the autonomous case, it is proved that any non-wandering point is either a fixed point or generates a rotating wave on the circle. In the periodic case, it is proved that any non-wandering point is a periodic point or generates a rotating wave on a torus. In particular, if $f(t,u,-u_x)=f(t,u,u_x)$, then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case., Comment: 35pages

In this paper, we consider the asymptotic dynamics of non-autonomous fractional reaction-diffusion equations of the form \[u_{t}+(-\Delta)^{s}u+f(u)=g(t)\] complemented with the Dirichlet boundary condition on a bounded domain. First, we obtain some higher-attraction results for pullback attractors, that is, without any additional $t$-differentiability assumption on the forcing term $g$, for any space dimension $N$ and any growth power $p\geq2$ of $f$, the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback attractor can indeed attract every $L^{2}(\Omega)$-bounded set in the $L^{2+\delta}(\Omega)$-norm for every $\delta\in [0,\infty)$ as well as in the $W_{0}^{s,2}(\Omega)$-norm. Then, we construct a family of Borel probability measures $\{\mu_{t}\}_{t\in\mathbb{R}}$, whose supports satisfy the higher-attraction results. Finally, we investigate the relationship between such the Borel probability measures and time-dependent statistical solutions for this fractional Laplacian equation.

The current paper is concerned with the persistence and spreading speeds of the following Keller-Segel chemoattraction system in shifting environments, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-\chi(uv_x)_x +u(r(x-ct)-bu),\quad x\in\R\cr 0=v_{xx}- \nu v+\mu u,\quad x\in\R, \end{cases} \end{equation} where $\chi$, $b$, $\nu$, and $\mu$ are positive constants, { $c\in\R$ }, $r(x)$ is H\"older continuous, bounded, $r^*=\sup_{x\in\R}r(x)>0$, $r(\pm \infty):=\lim_{x\to \pm\infty}r(x)$ exist, and $r(x)$ satisfies either $r(-\infty)\chi\mu$ and $b\ge \big(1+\frac{1}{2}\frac{(\sqrt{r^*}-\sqrt{\nu})_+}{(\sqrt{r^*}+\sqrt{\nu})}\big)\chi\mu$. In the case that $r(-\infty)c^*:=2\sqrt{r^*}$, then the species becomes extinct in the habitat. If the moving speed $ -c^*\leq cc^*$, then the species will become extinct in the habitat. If $\lambda_{\infty}$, defined to be the generalized principle eigenvalue of the operator $u\to u_{xx}+cu_{x}+r(x)u$, is negative and the degradation rate $\nu$ of the chemo-attractant is grater than or equal to some number $\nu^*$, then the species will also become extinct in the habitat. If $\lambda_{\infty}>0$, then the species will persist surrounding the good habitat.

The current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains. We first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models. Next, under the same conditions on the coefficients, we show that pointwise { persistence} occurs, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough. It should be pointed out that in \cite{TaoWin15}, the authors established the persistence of mass for globally defined positive solutions, which indicates that any extinction phenomenon, if occurring at all, necessarily must be spatially local { in }nature, whereas the population as a whole always persists. The pointwise persistence proved in the current paper implies that not only the population as a whole persists, but also it persists at any location eventually. It also implies the existence of strictly positive entire solutions.

In this work, we extend the classical real-valued framework to deal with complex-valued dissipative dynamical systems. With our new complex-valued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data \begin{document}$u_{0}$\end{document} , we construct a unique complex invariant measure \begin{document}$\mu$\end{document} on a metric space which is acted by a continuous semigroup \begin{document}$\{S(t)\}_{t\geq 0}$\end{document} possessing a global attractor \begin{document}$\mathcal{A}$\end{document} . Moreover, it is shown that the support of \begin{document}$\mu$\end{document} is not only contained in global attractor \begin{document}$\mathcal{A}$\end{document} but also in \begin{document}$\omega(u_{0})$\end{document} . Next, the structure of the measure \begin{document}$\mu$\end{document} is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex Ginzburg-Landau equation and a nonlinear Schrodinger equation and construct complex invariant measures for these two complex-valued equations.

The current paper is devoted to the investigation of wave propagation phenomenon in reaction-diffusion equations with ignition type nonlinearity in time heterogeneous and random media. It is proven that such equations in time heterogeneous media admit transition fronts or generalized traveling wave solutions with time dependent profiles and that such equations in time random media admit generalized traveling wave solutions with random profiles. Important properties of generalized traveling wave solutions, including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts, are also obtained.

The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system, \begin{document}$\label{IntroEq0-2}\begin{cases}u_{t}=Δ{u}-χ\nabla·(u\nabla{v})+u(1-u),{x}∈\mathbb{R}^N,\\{0}=Δ{v}-v+u,{x}∈\mathbb{R}^N,\end{cases}$ \end{document} where $u(x, t)$ represents the population density of a mobile species and $v(x, t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We first give a detailed study in the case $N=1$. In this case, it has been shown in an earlier work by the authors of the current paper that, when $0 \begin{document}$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [|u(x,t;{u_0}) - 1| + |v(x,t;{u_0}) - 1|] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} 0 and \begin{document}$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [u(x,t;{u_0}) + v(x,t;{u_0})] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} c > c_ + ^*(\chi ).$ \end{document} We also show that if $0 \begin{document}$\mathop {\lim }\limits_{\chi \to 0} {c^*}(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ + ^*(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ - ^*(\chi ) = 2.$ \end{document} We then consider the extensions of the results in the case $N=1$ to the case $N \ge 2$.

In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.

The structure of the ω -limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion equation u t = u x x + f ( t , u , u x ) , t > 0 , x ∈ S 1 = R / 2 π Z , where f is uniformly almost periodic in t and satisfies f ( t , u , u x ) = f ( t , u , − u x ) . We show that any ω -limit set Ω contains at most two minimal sets. Moreover, any hyperbolic ω -limit set Ω is a spatially-homogeneous 1-cover of hull H ( f ) . When dim ⁡ V c ( Ω ) = 1 ( V c ( Ω ) is the center space associated with Ω), it is proved that either Ω is a spatially-homogeneous, or Ω is a spatially-inhomogeneous 1-cover of H ( f ) .

In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In the first part of the series, we showed that a spreading–vanishing dichotomy occurs for such free boundary problems (see [16] ). In this second part of the series, we investigate the spreading speeds of such free boundary problems in the case that the spreading occurs. We first prove the existence of a unique time almost periodic semi-wave solution associated to such a free boundary problem. Using the semi-wave solution, we then prove that the free boundary problem has a unique spreading speed.

The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile of the unique generalized traveling wave is of the same recurrence as the media. In particular, if the media is time almost periodic, then so is the wave profile of the unique generalized traveling wave.

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.

It is known that solutions of nonlocal dispersal evolution equations do not become smoother in space as time elapses. This lack of space regularity would cause a lot of difficulties in studying transition fronts in nonlocal equations. In the present paper, we establish some general criteria concerning space regularity of transition fronts in nonlocal dispersal evolution equations with a large class of nonlinearities, which allows the applicability of various techniques for reaction–diffusion equations to nonlocal equations, and hence serves as an initial and fundamental step for further studying various important qualitative properties of transition fronts such as stability, uniqueness and asymptotic speeds. We also prove the existence of continuously differentiable and increasing interface location functions, which give a better characterization of the propagation of transition fronts and are of great technical importance.

This paper is devoted to the study of persistence and extinction of a species modeled by nonlocal dispersal evolution equations in moving habitats with moving speed c . It is shown that the species becomes extinct if the moving speed c is larger than the so called spreading speed c ∗ , where c ∗ is determined by the maximum linearized growth rate function. If the moving speed c is smaller than c ∗ , it is shown that the persistence of the species depends on the patch size of the habitat, namely, the species persists if the patch size is greater than some number L ∗ and in this case, there is a traveling wave solution with speed c , and it becomes extinct if the patch size is smaller than L ∗ .

In this paper, we investigate the stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations with general time and space dependence. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability and uniqueness of generalized traveling waves connecting the unique strictly positive entire solution and the trivial solution zero. Applying the general stability and uniqueness theorem, we then prove the existence, stability and uniqueness of periodic traveling wave solutions of lattice Fisher-KPP equations in time and space periodic media, and the existence, stability and uniqueness of generalized traveling wave solutions of lattice Fisher-KPP equations in time heterogeneous media. The general stability result established in this paper implies that the generalized traveling waves obtained in many cases are asymptotically stable under well-fitted perturbation., Comment: 27 pages

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller–Segel chemoattraction system, 0.1 $$\begin{aligned} \left\{ \begin{aligned}&u_t=u_{xx}-\chi (uv_x)_x +u(a-bu),\quad x\in {{\mathbb {R}}},\\&0=v_{xx}- \lambda v+\mu u,\quad x\in {{\mathbb {R}}}, \ \end{aligned}\right. \end{aligned}$$ where $$\chi $$ , a, b, $$\lambda $$ , and $$\mu $$ are positive constants. Assume $$b>\chi \mu $$ . Then if in addition $$\big (1+\frac{1}{2}\frac{(\sqrt{a}-\sqrt{\lambda })_+}{(\sqrt{a}+\sqrt{\lambda })}\big )\chi \mu { \le } b$$ holds, it is proved that $$c_0^*=2\sqrt{a}$$ is the spreading speed of the solutions of (0.1) with nonnegative continuous initial function $$u_0$$ with nonempty compact support, that is, $$\begin{aligned} \limsup _{|x|\ge ct, t\rightarrow \infty }u(t,x;u_0)=0\quad \forall \, c>c_0^* \end{aligned}$$ and $$\begin{aligned} \liminf _{|x|\le ct,t\rightarrow \infty } u(t,x;u_0)>0\quad \forall \, 02\chi \mu $$ and $$\lambda \ge a$$ holds, then $$c_0^*=2\sqrt{a}$$ is the minimal speed of the traveling wave solutions of (0.1) connecting (0, 0) and $$(\frac{a}{b},\frac{\mu }{\lambda }\frac{a}{b})$$ , that is, for any $$c\ge c_0^*$$ , (0.1) has a traveling wave solution connecting (0, 0) and $$(\frac{a}{b},\frac{\mu }{\lambda }\frac{a}{b})$$ with speed c, and (0.1) has no such traveling wave solutions with speed less than $$c_0^*$$ . Note that $$c_0^*=2\sqrt{a}$$ is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, 0.2 $$\begin{aligned} u_t=u_{xx}+u(a-bu),\quad x\in {{\mathbb {R}}}. \end{aligned}$$ Hence, if $$\lambda \ge a$$ and $$b>\chi \mu $$ , or $$\lambda