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Sea lice infection is one of the major threats in the marine fishery, especially for farmed salmon. In this paper, we propose a mathematical model for the growth of sea lice with a three-stage structure: non-infectious immature, infectious immature and adults where the level of non-infectious immature development depends on the size of the adult population. We first describe the nonlinear dynamics by a system of partial differential equations, then, by mathematical techniques and an appropriate change of variables transform it into a system of delay differential equations with constant delay. We address the system threshold dynamics in the established model with respect to the adult reproduction number R s , including the global stability of the trivial steady state when R s < 1 , persistence and global attractivity of the unique positive steady state when R s > 1 . Numerical simulations are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, effects of the synaptic delay of signal transmissions on the pattern formation of nonlinear waves in a bidirectional ring of neural oscillators is studied. Firstly, the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. Meanwhile, using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of Lie groups, we discuss the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Finally, Hopf bifurcation directions and corresponding stabilities of bifurcating periodic orbits are derived by using the normal form approach and the center manifold theory. These theoretical results are significant to complement experimental and numerical observations made in living neuronal systems and artificial neural networks, in order to better understand the mechanisms underlying the system’s dynamics. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a differential–algebraic biological economic system with time delay where the model with Holling type II functional response incorporates a constant prey refuge and prey harvesting. By considering time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the differential–algebraic biological economic system based on the new normal form approach of the differential–algebraic system and the normal form approach and the center manifold theory. Finally, numerical simulations illustrate the effectiveness of our results. [Copyright &y& Elsevier]

Abstract: This paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions. [Copyright &y& Elsevier]

Abstract: A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper. [ABSTRACT FROM AUTHOR]

Abstract: The objective of this paper is to systematically study the boundedness, persistence and stability of the nonlinear malaria epidemic model with latent periods. First, we consider the simplified model with the approximation , when is small enough so that the function does not vary too rapidly over the time interval [], and study the stability of the trivial and the positive equilibrium points. Second, when the latent periods are equal (and not small enough), we will investigate the stability of the positive equilibrium point and prove the existence of Hopf Bifurcations and discuss the stability independent of the delays. Third, in the case when the latent periods are different, we will employ the Lyapunov functional method to establish some sufficient conditions for the local asymptotic stability of the positive equilibrium point. [Copyright &y& Elsevier]

Abstract: By introducing discrete time delay into the model for producing 1,3-propanediol by microbial continuous fermentation, we consider the stability and Hopf bifurcation of the delay differential system. Through numerical simulations, we get the rule of branch value changing with parameter and draw the pictures of periodic solutions and phase diagrams with specified parameters. The effect of time delay suggests that the system can qualitatively describe oscillatory phenomena occurring in the experiment. [Copyright &y& Elsevier]

Abstract: In this paper, shunting inhibitory cellular neural networks with continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, some new sufficient conditions for the existence and exponential stability of the almost periodic solutions are established. [Copyright &y& Elsevier]

Abstract: In this article, we study the primitive equations (PEs) of the ocean with delays. We prove the existence and uniqueness of their strong solution when the external force contains some delays. We also discuss the asymptotic behaviour of their weak solutions and the stability of their stationary solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a simple discrete two-neuron network model with three delays. The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms. We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution. Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark–Sacker bifurcations. The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem. Our results are a very important generalization to the previous works in this field. [Copyright &y& Elsevier]

Abstract: The dynamics of a physiological control systems described by a first-order nonlinear delay differential equations are investigated. we proved that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838]. [Copyright &y& Elsevier]

Abstract: We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay and show that the distributed delay can destabilize the entire system. In particular, it is shown that a Hopf bifurcation can occur. [Copyright &y& Elsevier]

In this paper, we consider the complex dynamics of a novel mathematical model for a feedback control system of the gut microbiota, which was proposed by Dong, et al. The main work of the present paper is to study the effect of antibiotics injection on the gut microbiota through some dynamical methods, such as double Hopf bifurcation analysis and so on. We first use DDE-BIFTOOL to find the non-resonant double Hopf bifurcation points of the system, and draw the bifurcation diagram with two bifurcation parameters, τ 1 and τ 2 , i.e., respective measurement delays. Then we study small perturbations of two delay differential equations at these double Hopf bifurcation points, and the method of multiple scales is employed to obtain two common complex amplitude equations. By analyzing the amplitude equations, we can derive the classification and unfolding of these double Hopf bifurcation points. Finally, we verify the results by numerical simulations. We find more complicated dynamic behaviors of the system via analytical method. For example, there exists stable equilibrium, stable periodic solution or even the co-existing stable periodic solutions in respective region. And the numerical simulations are consistent with the analytic results, meanwhile it implies that the MMS is effective and accurate. All complex dynamical phenomena found in the present paper can be very helpful for the researchers to understand the mechanism of the system of gut microbiota. And it is also very significant to microbiology and engineering control. It reveals that the measurement delays can induce the complicated dynamics in this system and to the ends of excellent performance, we should take the proper values of these delays. • Find complicated behaviors in delayed gut microbiota system esp. bi-stability. • Measurement delays can induce complicated dynamics and careful to take values. • Complicated behaviors be very helpful to understand mechanism of gut mocrobiota. [ABSTRACT FROM AUTHOR]