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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].

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 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.

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.

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.

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.

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.

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 is the second part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc., in press, to positive random dynamical systems on finite-dimensional ordered Banach spaces. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type-K strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations., 31 pages; some clarifications have been made, and minor typos corrected. Published in the Journal of Mathematical Analysis and Applications

This is the first of a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. It focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming parts applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations., Comment: 37 pages; some typos and minor errors have been corrected. Published in Transactions of the American Mathematical Society. arXiv admin note: text overlap with arXiv:1209.3381

The current paper is concerned with the spectral theory, in particular, the principal eigenvalue theory, of nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a smooth random dispersal operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we first establish criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic type boundary conditions. It is shown that a time periodic nonlocal dispersal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance is sufficiently small, or the time average of the underlying media satisfies some vanishing condition with respect to the space variable at a maximum point or is nearly globally homogeneous with respect to the space variable. Next we obtain lower bounds of the principal spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding time averaged problems. Finally we discuss the applications of the established principal eigenvalue theory to time periodic Fisher or KPP type equations with nonlocal dispersal and prove that such equations are of monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique time periodic positive solution which is globally asymptotically stable.

Random dispersal describes the movement of organisms between adjacent spatial locations. However, the movement of some organisms such as seeds of plants can occur between non-adjacent spatial locations and is thus non-local. We propose to study a mixed dispersal strategy, which is a combination of random dispersal and non-local dispersal. More specifically, we assume that a fraction of individuals in the population adopt random dispersal, while the remaining fraction assumes non-local dispersal. We investigate how such mixed dispersal affects the invasion of a single species and also how mixed dispersal strategy will evolve in spatially heterogeneous but temporally constant environment.

This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors’ earlier work [41] for following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend the results in [41] to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

Coexistence and extinction for two species Volterra-Lotka competition systems with nonlocal dispersal are investigated in this paper. Sufficient conditions in terms of diffusion, reproduction, self-limitation, and competition rates are established for existence, uniqueness, and stability of coexistence states as well as for the extinction of one species. The focus is on environments with hostile surroundings. In this case, our results correspond to those for random dispersal under Dirichlet boundary conditions. Similar results hold for environments with non-flux boundary and for periodic environments, which correspond to those for random dispersal under Neumann boundary conditions and periodic boundary conditions, respectively.

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space l ρ p ( p ⩾ 1 ) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space l ρ 2 for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l 2 space of infinite sequences.

We study the structural stability of global attractors ( $${\mathcal{A}}$$ -stability) for two-species competition diffusion systems with Morse-Smale structure. Such systems generate semiflows on positive cones of certain infinite-dimensional Banach spaces (e.g., fractional order spaces). Our main result states that a two species competition diffusion system with Morse-Smale structure is structurally $${\mathcal{A}}$$ -stable, which implies that the set of nonlinearities for which the system possesses Morse-Smale structure is open in an appropriate space under the topology of C 2-convergence on compacta. Moreover, we provide a sufficient condition under which a system has Morse-Smale structure and provide some examples which satisfy the sufficient condition.

The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established.

The current paper deals with spatial spreading and front propagating dynamics for spatially discrete KPP (Kolmogorov, Petrovsky and Paskunov) models in time recurrent environments, which include time periodic and almost periodic environments as special cases. The notions of spreading speed interval, generalized propagating speed interval, and traveling wave solutions are first introduced, which are proper modifications of those introduced for spatially continuous KPP models in time almost periodic environments. Among others, it is then shown that the spreading speed interval in a given direction is the minimal generalized propagating speed interval in that direction. Some important upper and lower bounds for the spreading and generalized propagating speed intervals are provided. When the environment is unique ergodic and the so called linear determinacy condition is satisfied, it is shown that the spreading speed interval in any direction is a singleton (called the spreading speed), which equals the classical spreading speed if the environment is actually periodic. Moreover, in such a case, a variational principle for the spreading speed is established and it is shown that there is a front of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction.

The current paper is devoted to the study of spatial spreading and front propagating dynamics of KPP models in inhomogeneous media, particularly, in time almost periodic and space periodic media. While spatial spreading and front propagating dynamics of KPP models in time independent or periodic media has been widely studied, there is little study on such dynamics when the media is nonperiodically inhomogeneous. This paper develops some theoretical foundation for the study of the speeds of spread and propagation for KPP models in time almost periodic and space periodic media. It introduces a notion of spreading speed intervals for such models for the first time, which extends the classical concept of the spreading speeds for time independent or periodic KPP models to time almost periodic models and can be used for more general time dependent models. It also introduces a notion of generalized propagating speed intervals of front solutions to time almost periodic and space periodic KPP models for the first ...

Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent., Comment: 43 pages; published in Discrete and Continuous Dynamical Systems Series B

We investigate a dynamical order induced by coupling and/or noise in systems of coupled oscillators. The dynamical order is referred to a one-dimensional topological structure of the global attractor of the system in the context of random skew-product flows. We show that if the coupling is sufficiently strong, then the system exhibits one dimensional dynamics regardless of the strength of noise. If the coupling is weak, then it is shown numerically that the system also exhibits one dimensional dynamics provided the noise is sufficiently strong. We also show that for any coupling and any noise, the system has a unique rotation number and hence all the oscillators tend to oscillate with the same frequency eventually (frequency locking).

We investigate the spectral theory of the following general nonautonomous evolution equation ∂ t u ( t , x ) = A ( u ( t , ⋅ ) ) ( x ) + h ( t , x ) u ( t , x ) , x ∈ D , where D is a bounded subset of R N which can be a smooth domain or a discrete set, A is a general linear dispersal operator (for example a Laplacian operator, an integral operator with positive kernel or a cooperative discrete operator) and h ( t , x ) is a smooth function on R × D ¯ . We first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal Lyapunov exponent of a time-dependent dispersal equation is always greater than or equal to that of the time-averaged one. Several results about the principal eigenvalue of time-periodic parabolic equations are extended to general time-periodic dispersal ones. Finally, the investigation is generalized to random time-dependent dispersal equations.

The paper extends the Faber--Krahn inequality for elliptic and periodic parabolic problems to random and general nonautonomous parabolic problems. Under proper assumptions, it also provides necessary and sufficient conditions for the Faber-Krahn inequality being equality.

The purpose of this paper is to investigate uniform persistence for nonautonomous and random parabolic Kolmogorov systems via the skew-product semiflows approach. It is first shown that the uniform persistence of the skew-product semiflow associated with a nonautonomous (random) parabolic Kolmogorov system implies that of the system. Various sufficient conditions in terms of the so-called unsaturatedness and/or Lyapunov exponents for uniform persistence of the skew-product semiflows are then provided. Among others, it is shown that if the associated skew-product semiflow has a global attractor and its restriction to the boundary of the state space has a Morse decomposition which is unsaturated or whose external Lyapunov exponents are positive, then it is uniformly persistent. More specific conditions are discussed for uniform persistence in n -species, particularly 3-species, random competitive systems.

This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of B. Tang, Y. Kuang and H. Smith in SIAM J. Math. Anal., 24 (1993), 1331-1339.

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.

The current paper concerns with traveling wave solutions to time periodic two-dimensional lattice differential equations of bistable type. Employing the so-called ‘vanishing viscosity’ approach, we obtain the existence of monotone traveling wave solutions connecting the two spatially homogeneous periodic solutions. Uniqueness of the wave speed and uniqueness of continuous wave front are shown by repeatedly utilizing comparison arguments. The existence and uniqueness results of the current paper together with the stability results proved in an author's earlier paper generalize those in the time-independent case.

A two species competition model with diffusion is considered. The parameters describing the growth, interaction, and self-limitation of the species are spatially inhomogeneous and temporally almost periodic. The boundary conditions are homogeneous and of Neumann or Dirichlet type. First, a convergence theorem is derived in the single species case. Roughly speaking, it states that one of the following alternatives will occur: either every positive solution converges to a unique strictly positive almost periodic solution, every positive solution converges to the trivial solution, or every positive solution is neither bounded away from the trivial solution nor converges to it. Then appropriate conditions for uniform persistence of both species as well as for extinction of one of the species are established. Moreover, it is shown that uniform persistence implies coexistence in the sense that there is a strictly positive solution whose hull is almost automorphic. The above results generalize earlier work in the time independent and time periodic cases for both single species population models and two species competition models. The approach developed in this paper for dealing with almost periodic equations can be applied to more general nonautonomous equations, as we will indicate by briefly discussing applications where merely time recurrence is supposed.

We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed.

We study ℤd action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of ℤd-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a ℤd-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.

Skew-product semi-flow Π t : X × Y → X × Y {\Pi _t}:X \times Y \to X \times Y which is generated by \[ { u t = u x x + f ( y ⋅ t , x , u , u x ) , t > 0 , 0 > x > 1 , y ∈ Y , D or N boundary conditions \left \{ \begin {gathered} {u_t} = {u_{xx}} + f(y \cdot \,t,x,u,{u_x}),\qquad t > 0,\;0 > x > 1,\;y \in Y, \hfill \\ D\;{\text {or }}N\;{\text {boundary conditions}} \hfill \\ \end {gathered} \right . \] is considered, where X X is an appropriate subspace of H 2 ( 0 , 1 ) , ( Y , R ) {H^2}(0,1),\;(Y,\,\mathbb {R}) is a minimal flow with compact phase space. It is shown that a minimal set E ⊂ X × Y E \subset X \times Y of Π t {\Pi _t} is an almost 1 - 1 1{\text { - }}1 extension of Y Y , that is, set Y 0 = { y ∈ Y | card ⁡ ( E ⊂ P − 1 ( y ) ) = 1 } {Y_0} = \{ y \in Y|\operatorname {card} (E \subset {P^{ - 1}}(y)) = 1\} is a residual subset of Y Y , where P : X × Y → Y P:X \times Y \to Y is the natural projection. Consequently, if ( Y , R ) (Y,\mathbb {R}) is almost periodic minimal, then any minimal set E ⊂ X × Y E \subset X \times Y of Π t {\Pi _t} is an almost automorphic minimal set. It is also proved that dynamics of Π t {\Pi _t} is closed in the category of almost automorphy, that is, a minimal set E ⊂ X × Y E \subset X \times Y of Π t {\Pi _t} is almost automorphic minimal if and only if ( Y , R ) (Y,\mathbb {R}) is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.