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Consider the class of planar systems of first-order rational difference equations 1´ where , and the parameters are nonnegative and such that both terms in the right-hand side of (1′) are nonlinear. In this paper, we prove the following discretized Poincaré-Bendixson theorem for the class of systems (1′). If the map associated to system (1′) is bounded, then the following statements are true: In particular, system (1′) cannot exhibit chaos when its associated map is bounded. Moreover, we show that if both equilibrium curves of (1′) are reducible conics and the map associated to system (1′) is unbounded, then every solution converges to one of up to infinitely many equilibria or to or . MSC:39A05, 39A11.

This paper studies the global dynamics of a general diffusive hepatitis B virus (HBV) infection model. The model includes both enveloped viruses and DNA containing capsids. Two immune responses are recruited to attack the virus and infected hepatocytes. These are the cytotoxic T-lymphocytes (CTL) which kill the infected liver cells, and B cells which send antibodies to attack the virus. The non-negativity and boundedness of the solutions are discussed. The existence of spatially homogeneous equilibrium points is examined. The global stability of all possible equilibrium points is proved by choosing suitable Lyapunov functionals. Some numerical simulations are performed to enhance the theoretical results and present the behavior of solutions in space and time.

In this paper, we propose and study a two-species stage structured amensalism model with a cover for the first species. By developing a new analysis technique or, more precisely, by combining the differential inequality theory and the Lyapunov function method, we obtain sufficient conditions ensuring the global attractivity of positive and boundary equilibria, respectively. Our study shows that the final density of the first species is an increasing function of the partial cover, and if the stage structured species is globally asymptotically stable, then there exists a threshold such that if the cover is greater than this threshold, the species can still exist in the long run, whereas if the cover is too small, then the first species is driven to extinction.

In this paper, we investigate the asymptotic behavior for a kind of resource competition model with environmental noises. Considering the impact of white noise on birth rate and death rate separately, we first prove the existence of a positive solution, and then a sufficient condition to maintain permanence and extinction is obtained by using a proper Lyapunov functional, stochastic comparison theorem, strong law of large numbers for martingales, and several important inequalities. Furthermore, the stochastic final boundedness and path estimation are studied. Finally, the fact that the intensity of white noise has a very important influence on the permanence and extinction of the system’s solution is illustrated by some numerical examples.

An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method is proposed in this paper. The parallel algorithm consists of a parallel solver for linear tridiagonal equations and parallel vector arithmetic operations. For the parallel solver, in order to solve the linear tridiagonal equations efficiently, a new tridiagonal reduced system is developed with an elimination method. The experimental results show that the parallel algorithm is in good agreement with the analytic solution. The parallel implementation with 16 parallel processes on two eight-core Intel Xeon E5-2670 CPUs is 14.55 times faster than the serial one on single Xeon E5-2670 core.

In this paper, the authors investigate the iterated exponent of convergence of zeros of ( ), where f is a solution of some second-order linear differential equation, is an entire function satisfying or ( ). We obtain some results which improve and generalize some previous results in (Chen in Acta Math. Sci. Ser. A 20(3):425-432, 2000; Chen and Shon in Chin. Ann. Math. Ser. A 27(4):431-442, 2006; Tu et al. in Electron. J. Qual. Theory Differ. Equ. 23:1-17, 2011) and provide us with a method to investigate the iterated exponent of convergence of zeros of ( ). MSC:34A20, 30D35.