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Dynamical behaviours of a tritrophic food chain model with strong Allee effect in the prey are studied in this paper. Positivity and boundedness of the system are discussed. Some global results on extinction of the species are derived. Stability analysis of the equilibrium points is presented. The effect of discrete time-delay is studied, where the delay may be regarded as the gestation period of the superpredator. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically. [ABSTRACT FROM AUTHOR]

In this paper, an SEIR epidemiological model with saturation incidence and a time delay describing the latent period of the disease is investigated, where it is assumed that the susceptible population is subject to logistic growth in the absence of the disease. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By means of Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if the basic reproduction number is greater than unity, sufficient conditions are obtained for the global stability of the endemic equilibrium. Numerical simulations are carried out to illustrate some theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the effect of distributed time delays on dynamics of a mathematical model of gene expression. Both the weak and strong delay kernels are discussed. Sufficient conditions for the local stability of the unique equilibrium are obtained. Taking the average delay as a bifurcation parameter, we investigate the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the method of multiple time scales. Finally, numerical simulation is carried out to illustrate our theoretical results. It shows both subcritical and supercritical Hopf bifurcations can happen. [ABSTRACT FROM AUTHOR]

Abstract: The ratio-dependent theory is favored by researchers since it is more suitable for describing the relationship between predator and its prey. In this paper, a ratio-dependent predator–prey system with Holling type II functional response, two time delays and stage structure for the predator is investigated. Firstly, by choosing the two time delays as the bifurcation parameter, the sufficient conditions for the local stability and the existence of Hopf bifurcation with respect to both delays are established. Furthermore, based on the normal form method and center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Finally, numerical simulations are given to verify the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel–Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic equation. For the Hopf bifurcation analysis, the delay parameter is chosen as a bifurcation parameter. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. (1981) [7]. Furthermore, the direction of the bifurcation, the stability and the period of periodic solutions are given. Finally, the theoretical results are supported by some numerical simulations. [Copyright &y& Elsevier]

Abstract: A generalized model of the two-neuron network with mixed delays is studied. The main purpose of this paper is to explore the linear stability of the trivial solution and Hopf bifurcation of a two-neuron network with continuous and discrete delays. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solutions are also studied. Finally, the numerical simulations are given to illustrate the theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze the stability and Hopf bifurcation of the biological economic system based on the new normal form and the Hopf bifurcation theorem. The basic model we consider is owed to a ratio-dependent predator–prey system with harvesting, compared with other researches on dynamics of predator–prey population, this system is described by differential-algebraic equations due to economic factor. Here μ as bifurcation parameter, it is found that periodic solutions arise from stable stationary states when the parameter μ increases close to a certain limit. Finally, numerical simulations illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]