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In this paper, we propose a novel fractional-order two-gene regulatory network model with delays, which can describe the memory and hereditary properties of genetic regulatory networks more suitably. It is the first time that the dynamics of the stability and Hopf bifurcation are investigated for the delayed fractional-order model of two-gene regulatory network. The total delay is chosen as the bifurcation parameter of the network, and the sufficient conditions of the stability and Hopf bifurcation are achieved through analyzing its characteristic equation. It is found that the delayed fractional-order genetic network can generate a Hopf bifurcation when the total delay passes through some critical values, which can be determined exactly by dealing with the characteristic equation of the network. Finally, the validity of our theoretical analysis is illustrated by carrying out the numerical simulation for the example, and some desirable dynamical behaviors of the case are obtained by choosing the appropriate fractional order. It is discovered that the onset of the Hopf bifurcation increases distinctly when the fractional order deceases. Therefore, the stability domain of the network is inversely proportion to the fractional order of the network. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we proposed and analyzed a prey predator system with stage-structured for the predator population. A time delay is incorporated due to the gestation for the matured predator. All the possible non-negative equilibria are obtained and their local as well as global behavior are studied. Choosing delay as a bifurcation parameter, the existence of the Hopf bifurcation of the system has been investigated. Moreover, we use the normal form method and the center manifold theorem to examine the direction of the Hopf bifurcation and the nature of the bifurcating periodic solution. Some numerical simulations are given to support the analytical results. Some interesting conclusions are obtained from our analysis and it is given at the end of the paper. [Copyright &y& Elsevier]

A kind of stage-structured predator–prey model with Beddington–DeAngelis functional response incorporating a prey refuge is investigated in this paper. By analyzing the corresponding characteristic equations, the local stability of the equilibria is investigated. Moreover, Hopf bifurcations occur at the positive equilibrium as the delay τ crosses some critical values. Further, the influence of prey refuge on densities of predator species and prey species is investigated. Numerical simulations are carried out to illustrate our main results. [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, a delayed ratio-dependent predator–prey model with Holling type III functional response and stage structure for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By utilizing normal form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations supporting the theoretical analysis are given. [ABSTRACT FROM AUTHOR]

We propose a predator–prey ecoepidemic model with parasitic infection in the prey. We assume infection time delay as the time of transmission of disease from susceptible to infectious prey. We examine the effects of supplying additional food to predator in the proposed model. The essential theoretical properties of the model such as local and global stability and in addition bifurcation analysis is done. The parameter thresholds at which the system admits a Hopf bifurcation are investigated in presence of additional food with non-zero time lag. The conditions for permanence of the system are also determined in this paper. Theoretical analysis results are verified through numerical simulations. By supplying additional food we can control predator population in the model. Most important observation is that we can control parasitic infection of prey species by supplying additional food to predator. Eliminating the most infectious individuals from the prey population, predator quarantine the infected prey and prevent the spreading of disease. [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: Based on the stretch-twist-fold flow, a generalized stretch-twist-fold flow is introduced. By choosing an appropriate bifurcation parameter, Hopf bifurcations occur in this system when the bifurcation parameter exceeds a critical value. The formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are presented. In addition, the paper also investigates the bifurcations of the heteroclinic orbits for this system. The existence and its associated existing regions are given for two heteroclinic orbits, respectively. Finally, some numerical simulations for justifying the theoretical analysis are presented. [Copyright &y& Elsevier]

Abstract: This paper is concerned with spatially homogeneous and inhomogeneous autocatalysis models with arbitrary order. For the spatially homogeneous model, the existence and stability of Hopf bifurcation surrounding the interior equilibrium are considered. For the model subject to zero-flux boundary conditions, Turing instability of the interior equilibrium is firstly discussed and Turing instability region regarding the diffusion coefficients is established. Then by the center manifold theory and normal form method, the existence of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained, which is much more difficult and complicated than dealing with the spatially homogeneous case. Finally, to verify our analytical results, some examples of numerical simulations are also included. [Copyright &y& Elsevier]

Abstract: In this paper, we consider several three-dimension discrete neural network models representing the models containing symmetry with time-delayed connections. The existence of symmetric property is investigated. We examine the relation between the structure of a discrete network and the spatio-temporally symmetric periodic dynamics it can support. We illustrate the existence of spatio-temporally periodic solutions through a direct construction of symmetric groups, and show that these solutions can be stable in some region of parameter space. We give computer simulations to support the theoretical predictions. [Copyright &y& Elsevier]

Abstract: In this paper we consider an unidirectional ring of n neurons with distributed delays. The effects of the delay, the coupling strengths and the network size on the stability of the system are investigated. Taking the average delay as a bifurcation parameter, we find two critical values depending on the network size and the coupling strengths, at which the system undergoes Hopf bifurcations. By using the method of multiple scales, we can show that these Hopf bifurcating periodic solutions are asymptotically stable. Finally, the theoretical results are illustrated by numerical simulations for the neural networks with three or four neurons. [Copyright &y& Elsevier]