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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: An equation , , is considered, where and are linear operators in a Hilbert space , , is a non-linear operator, and is a constant. Under some assumptions on and sufficient conditions are given for the solution to exist globally, i.e., for all , to be globally bounded, and to tend to zero at a specified rate as . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the nonlinear stability of traveling wave fronts for nonlocal delayed reaction–diffusion equation. We prove that these traveling wave fronts are exponentially stable to perturbation in some exponentially weighted spaces, when the initial perturbation around the traveling wave fronts decays exponentially as , but the initial perturbation can be arbitrary large in other locations. The time decay rate is also obtained by weighted energy estimates. [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]

Abstract: In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known. [Copyright &y& Elsevier]

Abstract: In this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Some well-known results are improved and generalized. [Copyright &y& Elsevier]

Abstract: In this paper, we study the behavior of solutions of second order delay differential equation where , , , are real numbers, τ is positive real number. A basic theorem on the behavior of solutions is established. As a consequence of this theorem, a stability criterion is obtained. [Copyright &y& Elsevier]

Abstract: In this paper we are concerned with the exponential asymptotic stability of the solution of a class of differential equations with state dependent delays. Our approach is based on the Crandall–Liggett approximation and the properties of semigroups. [Copyright &y& Elsevier]

Abstract: In this paper we stochastically perturb the delay Lotka–Volterra model into the stochastic delay differential equation (SDDE) The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially be divided into two categories: [(i)] If the delay Lotka–Volterra model already has some nice properties, e.g., nonexplosion, persistence, and asymptotic stability, then the SDDE will preserve these nice properties provided the noise is sufficiently small. [(ii)] When the delay Lotka–Volterra model does not have some desired properties, e.g., nonexplosion and boundedness, the noise might make the SDDE achieve these desired properties. [Copyright &y& Elsevier]

In this paper we study stability and asymptotic behaviour of a real two-dimensional system x′(t)=A(t)x(t)+B(t)x(t−r)+h(t,x(t),x(t−r)), where r>0 is a constant delay, A, B and h being the matrix functions and the vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex-valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov–Krasovskii functional. [Copyright &y& Elsevier]