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Abstract: It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the stability analysis of a delay difference system of the form , , where a, b and c are real numbers and k is a positive integer. We establish some exact conditions for the zero solution of the system to be asymptotically stable. [Copyright &y& Elsevier]

Abstract: Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function. [Copyright &y& Elsevier]