Narrow Results

In this study, we consider a nonlocal almost periodic reaction–diffusion–advection model to study the global dynamics of a single phytoplankton population under the assumption that nutrients are abundant and their metabolism is only affected by light intensity. First, we prove that the single phytoplankton species model is strongly monotone with respect to the order induced by cone $X$. Second, we characterize the upper Lyapunov exponent ${\lambda }^{\ast }$ for a class of almost periodic reaction–diffusion–advection equations, and provide a numerical method to compute it. On this basis, we prove that ${\lambda }^{\ast }$ is the threshold parameter for studying the global dynamic behavior of the population model. Our results show that if ${\lambda }^{\ast }<0$, phytoplankton species will become extinct, and if ${\lambda }^{\ast }>0$, phytoplankton species will be uniformly persistent. Finally, we verified the above results using numerical simulations.

In this paper the convergence behavior of delayed cellular neural networks without almost periodic coefficients are considered. Some sufficient conditions are established to ensure that all solutions of the networks converge exponentially to an almost periodic function, which are new, and also complement previously known results.

We study the asymptotic behavior of nonautonomous discrete Reaction–Diffusion systems defined on multidimensional infinite lattices. We show that the nonautonomous systems possess uniform attractors which attract all solutions uniformly with respect to the translations of external terms when time goes to infinity. These attractors are compact subsets of weighted spaces, and contain all bounded solutions of the system. The upper semicontinuity of the uniform attractors is established when an infinite-dimensional reaction–diffusion system is approached by a family of finite-dimensional systems. We also examine the limiting behavior of lattice systems with almost periodic, rapidly oscillating external terms in weighted spaces. In this case, it is proved that the uniform global attractors of nonautonomous systems converge to the global attractor of an averaged autonomous system.

This paper presents the definition and some properties of (NC⁽ⁿ⁾)-almost periodic functions, i.e. uniformly almost periodic functions in the sense of Levitan with all derivatives of order n.

We will present a criterion for left, right and both-sided invertibility of matrix Wiener-Hopf plus Hankel operators with the same Fourier symbol in the Wiener subclass of the almost periodic algebra. The criterion is based on the value of a certain mean motion constructed from a particular Hausdorff set which is bounded away from zero.