Abstract:
In this paper, the Lotka-Volterra 3-species mutualism models with diffusion and delay effects is investigated. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the unique positive steady-state solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition. Our approach to the problem is based on inequality skill and the method of the upper and lower solutions for a more general reaction—diffusion system. Finally, some numerical simulations are given to illustrate our results.

Abstract:
This paper is concerned with the following nonlinear difference equation where the initial data , , are nonnegative integers, and , B, C, and D are arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the work of inar (2004), Yang et al. (2005), and Berenhaut et al. (2007). In addition, some numerical simulations are also shown to support our analytic results.

Abstract:
Characteristics of the spatial distribution of selected dissolved heavy metals were analyzed during large scale surveys from August 12 to 25.2003 in the Bohai Sea.Dissolved Pb was the only element with average concentrations higher than the grade-one sea water quality standard of China.The spatial distribution of dissolved Pb in surface water was similar to those of Cd,Cu and As,where the isopleths generally indicated decreasing values from the bays to the central areas.Only for Hg did the high concentrations not only appear in Liaodong Bay,Bohai Bay and Laizhou Bay,but also in the Central Area,viz.not only in inshore but also in offshore areas.Vertical distributions of dissolved Pb,Cd,Cu and As were largely uniform,while that of dissolved Hg increased with depth.We infer that the input of pollutants from land was the main influencing factor for the detected distribution patterns of dissolved heavy metals,followed by the dynamics of sea water,release from bottom sediments and biochemical processes.Comparing with historical data,average concentrations of dissolved heavy metals appear to decline in recent years.

Abstract:
In this paper, the existence and stability of periodic solution in prey predator model with diffusion and distributed delay effects are investigated by using the method of upper and lower solutions and comparison principle. It is shown that under some appropriate conditions the trivial solution and semi-trivial periodic solution of the models are globally asymptotically stable, the systems have a pair of periodic quasi-solutions and the sector between the quasi-solutions is an attractor of the models with respect to every nonnegative initial function.

Abstract:
Almost periodic solution of a three-species competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained by Q. C. Lin (1999), F. D. Chen and X. X. Chen (2003), and Y. Q. Liu, S. L, Xie, and Z. D. Xie (1996). 1. Introduction In this paper, we study the following three-species competition system with grazing rates and diffusions: where , is the bounded open subset of with smooth boundary , which represent the habitat domain for three species. System (1.1) is supplement with boundary conditions and initial conditions: where denotes the outward normal derivation on , and represent the density of th species at point and the time of . Here, , , , , , and ( ) denote the diffusivity rates, competition rates, and grazing rates, respectively. They are almost periodic functions in real number field . is a Laplace operator on . System (1.1)–(1.3) describes the interaction among three species and is an important model in biomathcmatics, which has been intensively investigated, and much attention is carried to the problem [1–8]. When there is no diffusion, Jiang [1] and Lin [2] studied the existence, uniqueness, and stability on periodic solution and almost periodic solution for two-species competition system under the condition that the coefficients are the periodic function and almost periodic function, respectively; F. D. Chen and X. X. Chen [3] extended the results in [2] to n-species case. When there are no diffusion and grazing rates, Zhang and Wang [4, 5] investigated the existence of a positive periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay and the existence of multiple positive periodic solutions for a generalized delayed population model with exploited term by using the continuation theorem of coincidence degree theory; Hu and Zhang [6] established criteria for the existence of at least four positive periodic solutions for a discrete time-delayed predator-prey system with nonmonotonic functional response and harvesting by employing the continuation theorem of coincidence degree theory. When there are no grazing rates, Pao and Wang [7] proved the stability for invariable coefficient case by utilizing the method of upper and lower solutions. Liu et al. [8] showed

Abstract:
We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.

Abstract:
We study the asymptotic behavior of the solutions for the following nonlinear difference equation xn+1=∑i=1sAkixn ki/(B0+∑j=1tBljxn lj),n=0,1,…, where the initial conditions x r,x r+1,…,x1,x0 are arbitrary nonnegative real numbers, k1,…,ks,l1,…,lt are nonnegative integers, r=max{k1,…,ks,l1,…,lt}, and Ak1,…,Aks,B0,Bl1,…,Blt are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.