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In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]

This paper examines a class of two-species reaction-diffusion-advection competition models with two time delays. A system of DDE equations was derived, both theoretically and numerically, using the Galerkin technique method. A condition is defined that helps to find the existence of Hopf bifurcation points. Full diagrams of the Hopf bifurcation points and areas of stability are investigated in detail. Furthermore, we discuss three different sources of delay on bifurcation maps, and what impacts of all these cases of delays on others free rates on the regions of the Hopf bifurcation in this model. We find two different stability regions when the delay time is positive (τ > 0), while the no-delay case (τ = 0) has only one stable region. Moreover, the effect of delays and diffusion rates on all free others parameters in this model have been considered, which can significantly impact upon the stability regions in both population concentrations. It is also found that, as diffusion increases, the time delay increases. However, as the delay maturation is increased, the Hopf points for both proliferation of the population and advection rates are decreased and it causes raises to the region of instability. In addition, bifurcation diagrams are drawn to display chosen instances of the periodic oscillation and two dimensional phase portraits for both concentrations have been plotted to corroborate all analytical outputs that investigated in the theoretical part. • To study the effect of diffusion with two different delay terms on the two-species advection competition system. • To drive theoretical outcomes of the DDE obtainable utilizing the Galerkin technique tool. • To find, theoretically, a condition that assists to obtain the existence of Hopf bifurcation points. • To construct the bifurcation diagram for the DDE and DPDE systems via numerical simulation to confirm theoretical results. [ABSTRACT FROM AUTHOR]

In the recent paper entitled “Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system” in [Appl. Math. Comput. 291 (2016) 323–339] by Amin Zarei and Saeed Tavakoli, they proposed the following new four-dimensional (4-D) quadratic autonomous hyper-chaotic system: x 1 ˙ = a ( x 2 − x 1 ) , x 2 ˙ = b x 1 − x 2 + e x 4 − x 1 x 3 , x 3 ˙ = − c x 3 + x 1 x 2 + x 1 2 , x 4 ˙ = − d x 2 , which generates double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Combining theoretical analysis and numerical simulations, they investigated some dynamical properties of that system like Lyapunov exponent spectrum, bifurcation diagram, phase portrait, Hopf bifurcation, etc. In particular, they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled “Ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems” [Int. J. Bifurc. Chaos, 21(09) (2011), 2679–2694] by P. Wang et al. However, by means of detailed theoretical analysis, we show that both the conclusion itself and the derivation of its proof in [Appl. Math. Comput. 291 (2016) 323–339] are erroneous. Furthermore, we point out that the method adopted for studying the ultimate bound of that system is not applicable at all. Therefore, the ultimate bound estimation of that system needs further studying in future work. [ABSTRACT FROM AUTHOR]

• The three-strategy game model with continuous distributed time delay is analyzed. • The two-community three-strategy games with two delays is discussed. • The existence of Hopf bifurcation in replicator equation is proved. • The rock-paper-scissors game is simulated as an example. In this paper, we combine evolutionary game with dynamics, and discuss two kinds of game models with time delay. First, based on continuous distributed (kernel function) stochastic delay, we give the asymptotic stability condition of general three-strategy game model. Second, we study the replicator dynamics of two-community three-strategy game model with two fixed time delays (i.e. τ 1 within the community and τ 2 between different communities). Based on Hopf bifurcation theory, by calculating the critical conditions for the existence of bifurcation, the influence of bifurcation on the stability of equilibrium points is analyzed. Finally, we take rock-paper-scissors game as an example to verify the correctness of the theoretical results, when the delay τ meets certain conditions or doesn't exceed the critical value, the stability of the system will not change. [ABSTRACT FROM AUTHOR]

• Incorporating time delay has a considerable impact on the dynamics of prey– predator interactions with fading memory and Allee effect. • Different values of time delays in competition and cooperation can lead to or eliminate chaos. • Predator density depends not only on the current density of prey but also on the recent past density of prey. • The stability of the system switches from stable (unstable) to unstable (stable) through Hopf bifurcation in the presence of time delays. This paper deals with a model of population dynamics comprising Allee effect and weakening memory with constant time delays. Since predator density depends on the prey density in current time and in past, a two-component model of prey-predator interactions is complemented with a third differential equation for the influence of recent past. The role of constant time delays incorporated in the functional form of Allee effect (referred as delays in competition and cooperation) is investigated analytically and numerically. Steady states of the model are obtained and the local stability analysis around the coexisting state is calculated in the presence of both delays. The critical threshold for time delays, above which the stability of the system switches from stable (unstable) to unstable (stable), is computed for various cases. Analytical findings of this paper are supported with numerical simulations, where time evolution as well as numerical bifurcation diagrams are presented. The results of this paper demonstrate that the influence of past on the prey-predator density in the present of time delay may have a considerable effect upon the system behaviour and can give important insights into underlying biological mechanism. [ABSTRACT FROM AUTHOR]

Highlights • A prey-predator model incorporating fear factor is developed. • A globally stable theorem of coexistence equilibrium is established. • The existences of Hopf bifurcation and limit cycle are shown. • The fear effect can not only reduce the population density of predator, but also stabilize the system by excluding the existence of periodic solutions. Abstract In this paper, we investigate the influence of anti-predator behaviour due to the fear of predators with a Holling-type-II prey-predator model incorporating a prey refuge. We first provide the existence and stability of equilibria of the model. Next, we give the existence of Hopf bifurcation and limit cycle. In addition, we study the impact of the fear effect on the model analytically and numerically, and find that the fear effect can not only reduce the population density of predator at the positive equilibrium, but also stabilize the system by excluding the existence of periodic solutions. Moreover, we also find that prey refuge has great impact on the persistence of the predator. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we investigate the dynamics of the S 3 − component of a J − trajectory in R × S 3. We obtain several geometric properties for the projection curve on S 2 (1 2) via the Hopf map and we generate some examples with Mathematica. We also prove that the curves in S 3 are geodesics on the Hopf tubes over the projection curve on the 2 − sphere if and only if they are Legendre curves. [ABSTRACT FROM AUTHOR]

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]

The aim of this paper is to investigate the effect of awareness coverage and delay in controlling infectious diseases. We formulate an SIS model considering individuals’ behavioral changes due to the influences of media coverage and divide the susceptible class into two subclasses: aware susceptible and unaware susceptible. Other model variables are infected human and media campaign. It is assumed that the rate of becoming aware (unaware), from unaware to aware susceptible human (from aware to unaware susceptible human), is a function of media campaign. A time delay is considered for the time that is taken by an unaware (aware) susceptible individual to become aware (unaware). An additional time delay is considered due to the time lag needed in organising awareness campaigns. The model exhibits two equilibria: the disease-free equilibrium and the endemic equilibrium. The disease-free equilibrium is stable if the basic reproduction number is smaller than unity and the endemic equilibrium exhibits a Hopf-bifurcation, in both delayed and non-delayed system, whenever it exists. Analytical and numerical results prove the significance of awareness and delay on the prevalence of infectious diseases. [ABSTRACT FROM AUTHOR]

In this paper we investigate the dynamical properties of a two prey-one predator system with quadratic self interaction represented by a three-dimensional system of differential equations by using tools of computer algebra. We first investigate the stability of the singular points. We show that the trajectories of the solutions approach to stable singular points under given conditions by numerical simulation. Then, we determine the conditions for the existence of the invariant algebraic surfaces of the system and we give the invariant algebraic surfaces to study the flow on the algebraic invariants which is a useful approach to check if Hopf bifurcation exists. [ABSTRACT FROM AUTHOR]

• The high-order fractional complex-valued BAM neural networks are proposed and their sufficient stability conditions are given. • Taking time delay as a bifurcation parameter, the general Hopf bifurcation conditions of fractional complex-valued BAM neural networks are obtained. • The relationship between fractional order and critical bifurcation point is also discussed. In this paper dynamical behaviors of a class of high–order fractional complex–valued bidirectional associative memory neural networks with multiple time delays are investigated. Firstly, they are reduced to real–valued systems by separating the real and imaginary parts. Then, stability criteria of fractional complex–valued bidirectional associative memory neural networks without delay are obtained. Concerning the delay case, the time delay is set as a bifurcation parameter and the condition of Hopf bifurcation is given by analyzing roots of characteristic equations. Finally, two numerical examples are presented and illustrate that Hopf bifurcation does happen when time delay exceeds the critical value. [ABSTRACT FROM AUTHOR]

• In this paper, we study the dynamics of a delay differential model for three species preypredator system (of two preys and one predator), with Monod-Haldane and Holling type IIfunctional responses, and cooperation between the two-teams of prays against predation. • Two discrete time-delays are incorporated to justify the reaction time of predator with each prey. • The permanence of such system is proved. Local and global stabilities of interior steady states are discussed. • Hopf bifurcation analysis in terms of time-delay parameters is investigated, and threshold parameters τ 1 * and τ 2 * are obtained. • Sensitivity analysis, which estimates how the model predictions can vary to small changes in the parameters of the model, is also studied. Some numerical simulations are provided to show the effectiveness of the theoretical results. In this paper, we study the dynamics of a delay differential model of predator-prey system involving teams of two-prey and one-predator, with Monod-Haldane and Holling type II functional responses, and a cooperation between the two-teams of preys against predation. We assume that the preys grow logistically and the rate of change of the predator relies on the growth, death and intra-species competition for the predators. Two discrete time-delays are incorporated to justify the reaction time of predator with each prey. The permanence of such system is proved. Local and global stabilities of interior steady states are discussed. Hopf bifurcation analysis in terms of time-delay parameters is investigated, and threshold parameters τ 1 * and τ 2 * are obtained. Sensitivity analysis that measures the impact of small changes in the model parameters into the model predictions is also investigated. Some numerical simulations are provided to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model for exploited fish populations is considered, where the prey and the predator both show schooling behavior. Due to this coordinated behavior, predator-prey interaction occurs only at the outer edge of the schools formed by the populations. Positivity and boundedness of the model are discussed. Analysis of the equilibria is presented. A criterion for Hopf bifurcation is obtained. The optimal harvest policy is also discussed using Pontryagin’s maximum principle, where the effort is used as the control parameter. 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, we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton, where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered. Stochastic fluctuations are incorporated into the proposed system in form of Gaussian white noises to depict stochastic environmental factors in plankton system. For deterministic system without double time delays, existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting, and state feedback controllers are designed to eliminate singularity induced bifurcation. For deterministic system with double time delays, positivity and uniform persistence of solutions are studied, and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated. For stochastic system without double time delays, stochastic stability and existence of stochastic Hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory. For stochastic system with double time delays, existence and uniqueness of global positive solution are investigated, and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate Lyapunov functions. Numerical simulations are carried out to validate theoretical analysis. [ABSTRACT FROM AUTHOR]

• To demonstrate the Galerkin method's validity, via the accuracy of its predictions using DPDE models. • To structure a process for applying the theoretical framework to find theoretical results. • To show the impacts of diffusion and immune response delay on the model, through bifurcation diagrams. • To determine the steady-state solutions, Hopf points, and the stable and unstable zones. • To illustrate comparisons of the DDE and DPDE approaches through a selection of 2-D and 3-D solution maps. This paper studies a diffusive viral infection system with delayed immune response in the one-domain system. A system of DDE equations was explored, both analytically and numerically, using the Galerkin method. A condition that helps to find Hopf bifurcation points is determined. Full maps of the Hopf bifurcation points as well regions of stability are constructed and considered in detail. It is shown that the time delay of cytotoxic T lymphocyte (CTL) response and the diffusion parameter can significantly impact upon the stability regions. Furthermore, the influences of the other free values have been examined for their effects on stability. It is found that, as diffusion increases, the CTL response delay increases, and also as the CTL response delay is increased, the Hopf points for both generation rate and activate rate are decreased, whereas the Hopf points for the infection and death rates increased. Moreover, an increase diffusion results in an increase in the Hopf points for growth rate and activation rate, while the Hopf bifurcations are decreased for the death rate of infected cells. Bifurcation diagrams are plotted to show selected examples of limit cycle behavior (periodic oscillation), and 3-D solutions for the three concentrations in the model have been plotted to corroborate all analytical results from the theoretical section. [ABSTRACT FROM AUTHOR]

This paper investigates an issue of bifurcation control for a novel incommensurate fractional-order predator–prey system with time delay. Firstly, the associated characteristic equation is analyzed by taking time delay as the bifurcation parameter, and the conditions of creation for Hopf bifurcation are established. It is demonstrated that time delay can heavily effect the dynamics of the proposed system and each order has a major influence on the creation of bifurcation simultaneously. Then, a linear delayed feedback controller is introduced to successfully control the Hopf bifurcation for such system. It is shown that the control effort is markedly influenced by feedback gain. It is further found that the onset of the bifurcation can be delayed as feedback gain decreases. Finally, two illustrative examples are exploited to verify the validity of the obtained newly results. [ABSTRACT FROM AUTHOR]

Complex-valued neural networks (CVNNs) with integer-order have attracted much attention, and which have been well discussed. Fractional complex-valued neural networks (FCVNNs) are more suitable to describe the dynamical properties of neural networks, but have rarely been studied. It is the first time that the stability and bifurcation of a class of delayed FCVNN is investigated in this paper. The activation function can be expressed by separating into its real and imaginary parts. By using time delay as the bifurcation parameter, the dynamical behaviors that including local asymptotical stability and Hopf bifurcation are discussed, the conditions of emergence of bifurcation are obtained. Furthermore, it reveals that the onset of the bifurcation point can be delayed as the order increases. Finally, an illustrative example is provided to verify the correctness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Based on Lorenz system, a new four-dimensional quadratic autonomous hyper-chaotic attractor is presented in this paper. It can generate double-wing chaotic and hyper-chaotic attractors with only one equilibrium point. Several properties of the new system are investigated using Lyapunov exponents spectrum, bifurcation diagram and phase portraits. Using the center manifold and normal form theories, the local dynamics, the stability and Hopf bifurcation at the equilibrium point are analyzed. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lagrangian function and local maximizer point. By properly choosing P and Q matrices, an estimation of the ultimate bound region, as a function of the Lagrange coefficient, is obtained using local maximizer point and reduced Hessian matrix. To demonstrate the evolution of the bifurcation and to show the ultimate bound region, numerical simulations are provided. [ABSTRACT FROM AUTHOR]

• This paper presents a novel non-smooth rumor spreading model. • Complex dynamics are studied by using the theory of partial differential equations. • Nonlinear control function is adopted in modeling and we have proved its effectiveness. To reflect the government and media refutation to rumor propagation, we have established a reaction-diffusion rumor propagation model by considering a non-smooth control function. Firstly, we obtain the rumor propagation threshold according to the next generation matrix theory, prove the existence and uniqueness of solution and discuss the existence of the equilibrium points. Secondly, we discuss the stability of the equilibrium points with the impact of the spatial diffusion. Thirdly, the conditions for spatially homogeneous and discontinuous Hopf bifurcation are presented. Finally, several numerical simulations are given to show the possible impact, and the factors affecting rumor propagation are theoretically analyzed, which proves the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

• This paper is concerned with the detailed dynamics analysis of the Lengyel-Epstein system with a discrete delay. • Under the assumption that the positive equilibrium of the model is locally asymptotically stable in the absence of delay, the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail. • The phenomenon that the equilibrium becomes ultimately unstable after passing through multiple stability switches and Hopf bifurcations at some certain critical values of delay is found. • By means of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formulae determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions are obtained. • To verify our theoretical conclusions, some numerical simulations for specific examples are also included at the end of this article. This paper examines the dynamical analysis of the Lengyel-Epstein system with a discrete delay in detail. Under the assumption that the unique positive equilibrium of the model is locally asymptotically stable in the absence of the delay, the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail. It is found that under suitable conditions on the other parameters, the delay doesn't affect the stability of the equilibrium, namely, the equilibrium is absolutely stable while under the other conditions on the other parameters, the equilibrium will become ultimately unstable after passing through multiple stability switches and Hopf bifurcations at some certain critical values of delay. Particularly, by means of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formulae determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions are obtained. To verify our theoretical conclusions, some numerical simulations for specific examples are also included at the end of this article. [ABSTRACT FROM AUTHOR]

• A model for HTLV-I infection that includes uninfected & infected CD4+T cells and HTLV-I-specific CD8+T cells. • To better understand the dynamics of HTLV-I infection, we incorporate a CTL immune response delay. • Applying Lyapunov functional procedures to determine global stability by the reproduction numbers R0 and R1. • The model with delay experiences a destabilization of the infected steady state leading to Hopf bifurcation and periodic solutions. • We estimate the length of delay to preserve the stability of bifurcating limit cycle using Nyquist criteria. This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R 0 and R 1 , basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E 0 is globally stable if R 0 ≤ 1 , and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E 1 is globally stable if R 1 ≤ 1 < R 0 , which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E 2 is globally asymptotic stable if R 1 > 1 , which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R 0 and R 1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h , removal rate of virus-specific CTLs d 4 , spontaneous infected CD4+T cell activation d 2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored. [ABSTRACT FROM AUTHOR]

In this paper, a systematical analysis of a predator–prey system based on metaphysiological approach is proposed, which considers changes in the aggregate biomass density rather than changes in the demographics. Bifurcation behaviors including transcritical and Hopf bifurcation are observed in this type of system, and bubble phenomena can be seized by setting appropriate values for parameters. The obtained results provide some new insights in population dynamics, which play important roles in maintaining population diversity. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a neutral neural network model with two delays is investigated. The condition to ensure the stability of the zero solution of the system is decided by choosing τ 1 and τ 2 as parameters, respectively. Then the Hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by Hassard and Kazarinoff. Global existence of periodic solution is studied by using the global Hopf bifurcation theory. Finally, some numerical simulations are carried out to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a three species dynamical system is explored. The system consisting of two logistically growing competing species and the third species acts as a predator as well as host. It is predating over second species with Holling type II functional response, while first species is benefited from the third species. In addition, the prey species move into a refuge to avoid high predation. The essential mathematical features of the proposed model are studied in terms of boundedness, persistence, local stability and bifurcation. The existence of transcritical bifurcations have been established about two axial points. It has been observed that survival of all three species may be possible due to commensalism. Numerical simulations have been performed to show the Hopf bifurcation about interior equilibrium point. The existence of period-2 solution is observed. Further, the bifurcations of codimension-2 have also been investigated. [ABSTRACT FROM AUTHOR]

This paper proposes and investigates a delayed model for the dynamics and control of a mosquito population which is subject to an integrated strategy that includes pesticide release, the use of mechanical controls and the use of the sterile insect technique (SIT). The existence of positive equilibria is characterized in terms of two threshold quantities, being observed that the “richer” equilibrium (with more mosquitoes in the aquatic phase) has better chances to be stable, while a longer duration of the aquatic phase has the potential to destabilize both equilibria. It is also found that the stability of the trivial equilibrium appears to be mostly determined by the value of the maturation rate from the aquatic phase to the adult phase. A nonstandard finite difference (NSFD) scheme is devised to preserve the positivity of the approximating solutions and to keep consistency with the continuous model. The resulting discrete model is transformed into a delay-free model by using the method of augmented states, a necessary condition for the existence of optimal controls then determined. The particularities of different control regimes under varying environmental temperature are investigated by means of numerical simulations. It is observed that a combination of all three controls has the highest impact upon the size of the aquatic population. At higher environmental temperatures, the oviposition rate is seen to possess the most prominent influence upon the outcome of the control measures. [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]

In this paper, a delayed biological economic system which considers a plankton system with harvest effort on phytoplankton is proposed. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the plankton dynamics, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. Then by numerical simulations the impact of delay, diffusion and economic interest on plankton system are explored. It is found that delay can cause system into chaos and can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, diffusion and economic profit can also affect the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]

In this paper, a BAM neural network with multiple time delays and diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation under two different cases are established, respectively. By using the normal form theory and the center manifold reduction of partial functional differential equations (PFDEs), explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we firstly present the general framework of calculation of normal forms of non-resonance and weak resonance double Hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to Faria and Magalh a ˜ es. Then, the dynamical behavior of van der Pol–Duffing oscillator with delayed position and velocity feedback is considered. Specifically, the dynamical classification near the double Hopf bifurcation point is investigated by analyzing the obtained normal form. Finally, the numerical simulations support the theoretical results and present some interesting phenomena. [ABSTRACT FROM AUTHOR]

Hopf bifurcation is one of the important dynamical behaviors. It could often cause some phenomena, such as quasiperiodicity and intermittency. Consequently, chaos will happen due to such dynamical behaviors. Since chaos appears in the Lorenz-like system, to understand the dynamics of such system, Hopf bifurcation will be explored in this paper. First, the stability of equilibrium points is presented. Then Hopf bifurcation of the Lorenz-like system is investigated. By applying the normal form theory, the conditions guaranteeing the Hopf bifurcation are derived. Further, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also presented. Finally, numerical simulations are given to verify the theoretical analysis. It is found that Hopf bifurcation could happen when conditions are satisfied. The stable bifurcating periodic orbit is displayed. Chaos will also happen when parameter further increases. [ABSTRACT FROM AUTHOR]

In this paper, we aim to investigate the dynamics of a gene regulatory network which is a time-delayed version of the model proposed by Elowitz and Leibler [Nature 403 (2000) 335–338] . Based on the normal form theory and center-manifold reduction, Hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated. We also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network. It shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation. More precisely, (i) the amplitude increases with small time delay, while the change of amplitude is not sensitive to relatively large time delay. However, the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter, where amplitude always increases quickly and linearly with the transcriptional rate; (ii) the period of oscillation increases as the time delay increases, but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value, which implies that the robustness of period to the transcriptional rate variations occurs. Our numerical simulations also support the theoretical conclusions, namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks. [ABSTRACT FROM AUTHOR]

In this paper, we investigate theoretically and numerically a 2-D spatio-temporal dynamics of a predator-prey mathematical model which incorporates the Holling type II and a modified Leslie–Gower functional response and logistic growth of the prey. This system is modeled by a reaction diffusion equations defined on a disc domain { ( x , y ) ∈ R 2 / x 2 + y 2 < R 2 } with Dirichlet initial conditions and Neumann boundary conditions. We study the local and global stability of the positive equilibrium point. We show that the diffusion can induce instability of the uniform equilibrium point which is stable with respect to a constant perturbation as shown by Turing in 1950s and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation in a spatial domain. We perform the computations and generalize, on a circular domain, the results presented in Camara and Aziz-Alaoui [6]. [ABSTRACT FROM AUTHOR]

In this paper we study a model of HCV with saturation and delay, we stablish the local and global stability of system also we stablish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. We present a sensitivity analysis for the basic reproductive number. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed diffusive predator–prey model with herd behavior. Firstly, by choosing the appropriate bifurcation parameter, the stability of the positive equilibria and the existence of Hopf bifurcations, induced by diffusion and delay respectively, are investigated by analyzing the corresponding characteristic equation. Then, applying the normal form theory and the center manifold argument for partial functional differential equations, the formula determining the properties of the Hopf bifurcation are obtained. Furthermore, the instability of the Hopf bifurcation leads to the emergence of spatial patterns. Finally, some numerical simulations are also carried out to illustrate and expand the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a tumor–immune system (T–IS) interaction model with two discrete delays, namely the immune activation delay for effector cells (ECs) and activation delay for helper T cells (HTCs). By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Our results exhibit that both delays do not affect the stability of tumor-free equilibrium. However, they are able to destabilize the immune-control equilibrium and cause periodic solutions. We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors. The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium. Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium. [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 study the nonlinear dynamics of a modified van der Pol oscillator. More precisely, we study the local codimension one, two and three bifurcations which occur in the four parameter family of differential equations that models an extension of the classical van der Pol circuit with cubic nonlinearity. Aiming to contribute to the understand of the complex dynamics of this system we present analytical and numerical studies of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. [ABSTRACT FROM AUTHOR]

This paper deals with a robust stage structured predator–prey model with Beddington–DeAngelis-type functional response. The proposed mathematical model consists of a system of three nonlinear ordinary differential equations to stimulate the interactions between prey population, juvenile predators and adult predator population. The positivity, boundedness and the conditions for uniform persistence have been derived. The dynamical behavior of the system both analytically and numerically investigated from the point of view of local stability, persistence and global stability. The global stability of the system has been derived by using the theory of competitive systems, stability of periodic orbits and compound matrices for the interior equilibrium point. Depending on the conversion rate of the prey population to juvenile predator, the model exhibits Hopf-bifurcation. The model admit periodic solutions which is produced from the stage structure of the predator populations. Numerical simulations have been accomplished to validate our analytical findings. [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]

• We introduced three-dimensional fractional diffusion into the reaction-diffusion system. • It is maybe the first time to discuss three-dimensional pattern dynamics with fractional cross-diffusion. • The increase of the cross-fractional diffusion coefficient causes the bistable state of the three-dimensional pattern. • The fractional-order α affects the stability of the 3D pattern. • Proper protection of prey refuge is beneficial to the stability of the ecosystem. In this paper, we study the pattern dynamics in a spatial fractional predator-prey model with cross fractional diffusion, herd behavior and prey refuge. In this model, herd behavior exists in the population of predators and the prey. The spatial dynamics of the system are obtained through appropriate threshold parameters, and a series of three-dimensional patterns are observed, such as tubes, planar lamellae and spherical droplets. Specifically, linear stability analysis is applied to obtain the conditions of Hopf bifurcation and Turing instability. Then, by utilizing the central manifold reduction theory analysis, the amplitude equation near the critical point of Turing bifurcation is deduced to study the selection and stability of pattern formation. The theoretical results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

• New fractional-order six-neuron bi-directional associative memory (BAM) neural networks involving multiple delays are investigated. • Existence, uniqueness and boundedness of solution for the involved neural networks are studied. • The stability and the existence of Hopf bifurcation of the involved neural networks are analyzed. • Up to now, there are few papers that deal with fractional-order high-dimensional BAM neural networks. This study mainly explores fractional-order six-neuron bi-directional associative memory (BAM) neural networks involving multi-delays. Taking advantage of contraction mapping principle, we prove that the solution of the addressed BAM neural networks exists and is unique. Utilizing a acceptable function, we confirm that the solution of the addressed BAM neural networks is bounded. By applying a suitable variable substitution, new fractional order six-neuron BAM neural networks involving mult-delays are converted to a class of fractional order six-neuron BAM neural networks with single delay. Using the stability criterion and bifurcation theory of fractional order differential dynamical systems, we carry out a detailed discussion on the stability and the onset of Hopf bifurcation of the established BAM neural networks. The study shows that the time delay is an important factor which affects the stability behavior and Hopf bifurcation of the involved neural networks. Numerical simulation plots are presented to illustrate our derived key conclusions. The derived analytical findings of the study play a vital role in optimizing and designing neural networks. [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]

• Comparative analysis on a three-disk dynamo system with and without time delay are performed. • Regarding the local Hopf bifurcation, three different unfolding structures of delay-induced double Hopf bifurcations are presented. • For the global transitions and phenomena, the coexistence of periodic or quasi-periodic attractors and chaotic motion are observed. The disk dynamo plays an important role in studying the geodynamo and much research works have been devoted to the understanding of dynamo dynamics. This paper further investigates an extended disk dynamo system having three coupled conducting disks and incorporates the interaction-induced time delay in the dynamic governing equations. By carrying out a comparative analysis, the dynamic behaviors of the coupled three-disk dynamo system with and without time delay are studied to explore novel and complex nonlinear dynamic phenomena in the coupled delayed dynamo system. It is found that the double Hopf bifurcations can be induced in the time-delayed dynamo system. Three different topological structures of the unfolding are obtained under different time delays. Accordingly, it is shown that the novel dynamic behaviors, including quasi-periodic torus, three-dimensional torus and the coexistence of multiple attractors, can appear in the time-delayed dynamo system. Furthermore, by performing the continuation analysis on the periodic orbit generated from the Hopf bifurcation of equilibrium, some new coexistence patterns, e.g., the coexistence of periodic orbits and chaos, the coexistence of quasi-periodic orbits and chaos, are observed in the dynamo system with time delay. Based on the obtained results, it is believed that the inclusion of time delay in the modelling of the three-disk dynamo system is necessary and meaningful for developing an in-depth understanding of dynamo dynamics. Finally, the results of theoretical analyses are verified by the numerical simulations. [ABSTRACT FROM AUTHOR]

• The Lotka-Volterra competition model is reconstructed with maturation delays. • Under the identical delays, the steady state loses stability if the delays are larger than some critical value. • The stability switching curve is analytically constructed and numerically verified in the case of dint delays. • The unstable system can generate various dynamics including Hopf cycles, multistability and chaos. This paper reexamines the Lotka-Volterra competition model with two delays. The steady state is shown to be locally asymptotically stable without delay. If the two delays are identical, then the model becomes a one-delay system. The critical value of the delay is determined when stability might be lost. If the delays are different, then the stability switching curves are analytically defined and numerically verified. It is demonstrated that the unstable two-delay system may exhibit periodic behavior, multistability, quasi period-doubling cascade and even complicated dynamics depending on model parameters. [ABSTRACT FROM AUTHOR]

• Ecological tug-of-war between juvenile predator and prey has been discussed. • Exponential and the logistic functional responses have been analysed. • Extinction of predators as well as perfect survival of all species are illustrated. • Mathematical impact of counter-attack as well as its ecological perspective are elaborated. The dynamics of a predator-prey model with stage-structure on predator and counter-attacking behavior such that the prey can attack juvenile predator, is investigated in this paper. Adult predators may easily achieve success in overcoming prey strategies designed to protect themselves from attack but juvenile predators, which are in the learning stage of predation, may be vulnerable to counterattack by preys in self-defence due to their comparatively smaller size and eventually get killed. Our main objective is to understand how the dynamics of counter-attacking by the preys against juvenile predators help in its survival. Both exponential and logistic growth of the preys are considered in this investigation. In qualitative analysis, permanence, impermanence, and stabilities of all the equilibrium points of the system are discussed to investigate the dynamical behavior of the system. The inclusion of sub-critical and super-critical Hopf bifurcations makes the dynamics more interesting. The mathematical approaches advocated here complement and enhance the theoretical and numerical approaches applied to the system. [ABSTRACT FROM AUTHOR]

• eaction-diffusion predator-prey-parasite system is analyzed in 2D domain. • Dispersion relation (DR) is established. • Various instabilities are observed depending on the nature of the roots of DR. • Conditions for diffusion driven instability are established. • Several types of spatiotemporal patterns are observed. This paper deals with the spatial pattern formation in a diffusive predator-prey-parasite (PPP) model, where predator feeds on infected prey following type II response function and infection spreads among prey species through horizontal transmission. The study is accomplished with respect to an ecological parameter that quantifies the reproductive gain of predator and two epidemiological parameters which measure the force of infection and virulence of the disease. We show analytically that the interior equilibrium loses its stability through Hopf bifurcation in the absence of diffusion if the reproductive gain of predator crosses some threshold value. In case of diffusive system, it is shown that the interior equilibrium, which is otherwise stable, may lose its stability due to diffusion. Criteria for the occurrence of various types of instability, like Turing, Hopf-Turing and pure Hopf, associated with the PPP model are presented with illustrations. Our simulation results reveal that this diffusion-driven instability creates various spatio-temporal patterns, like spot, stripe, mixture of spots & stripes and spiral patterns, depending upon the values of ecological and diffusion parameters. Turing instability and the corresponding patterns are also observed with the variation of two epidemiological parameters. Interestingly, the epidemiological parameters that measure the infection rate and virulence of the disease show opposite patterns with their increasing values. [ABSTRACT FROM AUTHOR]

This paper investigates a three-dimensional mixing competitive system with one exponential growth rate and two rational growth rates, whose nullclines are linearly determined. In total, 33 stable nullcline classes exist. Hopf bifurcations are studied in classes 26-31. We provide examples to prove the existence of at least two limit cycles in each of the classes 27-31. [ABSTRACT FROM AUTHOR]

• Propose for the first time a network organized semiarid vegetation model. • Study the stability and direction of Hopf bifurcation at the coexistence equilibrium. • Compare different variants of modeling vegetation and water. • Find that fragmented habitats more likely induce variation in plant density. Spatiotemporal dynamics of vegetation models are traditionally investigated on a spatially continuous domain. The increasingly fragmented agricultural landscape necessitates network-organized models. In this paper, we develop a semiarid vegetation model to describe the spatiotemporal dynamics between plant and water on a network accounting for fragmented habitats which are connected by dispersal of seeds. Time delay is introduced to account for time lag in water uptake. By linear stability analysis we show that the coexistence equilibrium is asymptotically stable in the absence of time delay, but loses its stability via Hopf bifurcation when time delay is beyond a threshold. Applying the center manifold theory, we derive the explicit formulas that determine the stability and direction of the Hopf bifurcation. Numerical simulations demonstrate the emergence of spatial patterns on a network. Comparing our network-organized model to other model variants, we find that increasing landscape fragmentation is more likely to generate the variation of plant density among different habitats. [ABSTRACT FROM AUTHOR]