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Abstract This paper considers Hopf bifurcation of complex network with two independent delays. By analyzing the eigenvalue equations, the local stability of the system is studied. Taking delay as parameter, the change of system stability with time is studied and the emergence of inherent bifurcation is given. By changing the value of the delay, the bifurcation of a given system can be controlled. Numerical simulation results confirm the validity of the results found. [ABSTRACT FROM AUTHOR]

Highlights • A time delay model of the complex interaction between Tumor, CD8+T cells and T - helper cells. • The model undergoes Hopf bifurcation in which time delay plays an important role to destabilize the system. • Extensive numerical simulations are performed to validate our analytical findings. Abstract In this paper, we consider a three-dimensional nonlinear delay differential system (tumor cells, cytotoxic-T lymphocytes, T-helper cells) with single interaction delay. We perform linear stability of the equilibria and the existence of Hopf bifurcation in which the discrete time delay is used as a bifurcation parameter. We estimate the length of delay to preserve the stability of period-1 limit cycle. We also investigate the direction, period, and the stability of bifurcated periodic solutions by applying normal form method and center manifold theory. We observe that the discrete time delay plays an important role in stability switching. Numerical simulations are presented to illustrate the rich dynamical behavior of the model with different values for the time delay τ including the existence of periodic oscillations, which demonstrate the phenomena of long-term tumor relapse. [ABSTRACT FROM AUTHOR]

Based on biological meaning, a kind of diffusive intraguild predation (IGP: resource, IG prey and IG predator) model with delay effect is investigated in this paper. The model has Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type II functional response between IG prey and IG predator. We first give sufficient conditions on the stability of possible nonnegative constant steady-state solutions for the proposed model, which give us a complete picture of the global dynamics. Then we investigate Hopf bifurcation near the unique positive constant steady-state solution by taking delay as bifurcation parameter and derive the Hopf bifurcation threshold. It is shown that the delay can induce three types of bistability (node-node bistability, node-cycle bistability and cycle-cycle bistability), periodic oscillations and irregular oscillations triggering spatiotemporal chaos in the diffusive IGP model. Numerical simulations are performed to illustrate our theoretical results and suggest that delay can even trigger the emergence of self-organised spatiotemporal patterns, which evolve from spiral patterns to irregular spatial patterns via spatiotemporal Hopf bifurcation. In addition, the impact of diffusion on the model’s dynamics under certain time delay are also explored. [ABSTRACT FROM AUTHOR]

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, 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]

This paper deals with a three-dimensional plant-herbivore-predator model that incorporates explicitly the plant toxicity in plant-herbivore interactions. The existence and stability conditions of all the feasible equilibria are established. Our results indicate that plant toxicity may play a key role in the dynamical behavior of the system. By adding another plant species with a different toxicity level to this system, we derive threshold conditions on the invasion of the second plant species. The analysis indicates that several parameters may be critical to determine successful invasion. Numerical simulations are also provided to reinforce the theoretical conclusions. [ABSTRACT FROM AUTHOR]

• Propose a general single species animal movement model with distributed delays in diffusion and reaction. • Investigate the impact of spatiotemporal delays on stability of memory-based model. • Obtain conditions for the occurrence of Hopf and steady state bifurcations. • Explore dynamics due to the interaction of Hopf and steady state bifurcations. In this paper, we investigate the spatiotemporal dynamics of a single-species model with spatiotemporal delays characterizing spatial memory and maturation. Through stability and bifurcation analysis, we find that the spatial memory-based diffusion coefficient, the spatiotemporal diffusive delay and spatiotemporal reaction delay have important effects on the dynamics of the model and their combined impact can cause the destabilization of the positive constant steady state and give rise to steady state and Hopf bifurcations. Taking the coefficient of spatial memory diffusion as the bifurcation parameter, the critical values of steady state and Hopf bifurcations are rigorously determined. Furthermore, we apply the theoretical results to a modified diffusive logistic model with predation and obtain spatially inhomogeneous steady states and spatially homogeneous and inhomogeneous periodic solutions via numerical simulations. [ABSTRACT FROM AUTHOR]