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This paper is devoted to the study of a new delayed eco-epidemiological model with infection-age structure and Holling type II functional response. Firstly, the disease transmission rate function among the predator population is treated as the piecewise function concerning the incubation period τ 2 of the epidemic disease and the model is rewritten as an abstract nondensely defined Cauchy problem. Besides, the prerequisite which guarantees the presence of the coexistence equilibrium is achieved. Secondly, via utilizing the theory of integrated semigroup and the Hopf bifurcation theorem for semilinear equations with nondense domain, it is found that the model exhibits a Hopf bifurcation near the coexistence equilibrium, which suggests that this model has a nontrivial periodic solution that bifurcates from the coexistence equilibrium as the bifurcation parameter τ crosses the bifurcation critical value τ 0 . That is, there is a continuous periodic oscillation phenomenon. Finally, some numerical simulations are shown to support and extend the analytical results and visualize the interesting phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, the asymptotical behavior of a chemostat model for E. coli and the virulent phage T4 is analyzed. The basic reproduction number R0 is proved to be a threshold which determines the outcome of the virulent phage T4. If R0 < 1, the virus dies out; if R0 > 1, the virus persists. Sufficient conditions for the Hopf bifurcation are also established. The theoretical results show that increasing the input of nutrient will result in an increase in the equilibrium population density of the virulent bacteriophage T4, but will have no effect on the equilibrium population density of E. coli. The results also show that increasing the input of nutrient or increasing the average lytic time for the infected E. coli can destabilize the interaction between E. coli and T4. [ABSTRACT FROM AUTHOR]

In this paper, the stability and evolutionary trend of Hopf bifurcations are investigated thoroughly for the double-input SEPIC DC–DC converter with one-cycle control (OCC), which is widely used in hybrid power systems. By means of cycle-by-cycle simulations, the evolutionary trend of bifurcation behaviors is identified. The inherent mechanisms of two types of bidirectional Hopf bifurcations are uncovered based on the proposed averaged model. Moreover, stability boundaries in the parameter space are derived with the help of eigenvalue analysis to optimize the circuit design. Furthermore, some important indices with respect to coupling effects are evaluated to assess the coupling effects in the system. Finally, PSpice circuit experiments are performed to verify the bifurcation analysis. [ABSTRACT FROM AUTHOR]

This paper takes the reaction–diffusion approach to deal with the quiescent females phase, so as to describe the dynamics of invasion of aedes aegypti mosquitoes, which are divided into three subpopulations: eggs, pupae and female. We mainly investigate whether the time of quiescence (delay) in the females phase can induce Hopf bifurcation. By means of analyzing the eigenvalue spectrum, we show that the persistent positive equilibrium is asymptotically stable in the absence of time delay, but loses its stability via Hopf bifurcation when time delay crosses some critical value. Using normal form and center manifold theory, we investigate the stability of the bifurcating branches of periodic solutions and the direction of the Hopf bifurcation. Numerical simulations are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the influence of anti-predator behavior in prey due to the fear of predators with a Beddington–DeAngelis prey–predator model analytically and numerically. We give the existence and stability of equilibria of the model, and provide the existence of Hopf bifurcation. In addition, we investigate the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also prevent the occurrence of limit cycle oscillation and increase the stability of the system. [ABSTRACT FROM AUTHOR]

In this paper, we have analyzed a tri-trophic food chain model consisting of phytoplankton, zooplankton and fish population in an aquatic environment. Here, the pelagic water column is divided into two layers namely, the upper layer and the lower layer. The zooplankton population makes a diel vertical migration (DVM) from lower portion to upper portion and vice-versa to trade-off between food source and fear from predator (Fish). Here, mathematical model has been developed and analyzed in a rigorous way. Apart from routine calculations like boundedness and positivity of the solution, local stability of the equilibrium points, we performed Hopf bifurcation analysis of the interior equilibrium point of our model system in a systematic way. It is observed that the migratory behavior of zooplankton plays a crucial role in the dynamics of the model system. Both the upward and downward migration rates of DVM leads the system into Hopf bifurcation. The upward migration rate of zooplankton deteriorates the stable coexistence of all the species in the system, whereas the downward migration rate enhance the stability of the system. Further, we analyze the non-autonomous version of the system to capture seasonal effect of environmental variations. We have shown that under certain parametric restrictions periodic coexistence of all the species of our system is possible. Finally, extensive numerical simulation has been performed to support our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we construct a mathematical model to investigate the interaction between the tumor cells, the immune cells and the helper T cells (HTCs). We perform mathematical analysis to reveal the stability of the equilibria of the model. In our model, the HTCs are stimulated by the identification of the presence of tumor antigens. Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells, which leads to the elimination of the bistable behavior of the tumor-immune system, i.e. the equilibrium corresponding to the high steady state of tumor cells is destabilized. Choosing immune intensity c as bifurcation parameter, there exists saddle-node bifurcation. Besides, there exists a critical value c ∗ , at which a Hopf bifurcation occurs. The stability and direction of Hopf bifurcation are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive predator–prey model with Monod–Haldane functional response. We study the Turing instability and Hopf bifurcation of the coexisting equilibriums. We investigate the Turing–Hopf bifurcation through some key bifurcation parameters. In addition, we obtain a normal form for the Turing–Hopf bifurcation. Finally, we show numerical simulations to illustrate the theoretical results. For parameters around the critical value of the Turing–Hopf bifurcation, we demonstrate that the predator–prey model exhibits complex spatiotemporal dynamics, including spatially homogeneous periodic solutions, spatially inhomogeneous periodic solutions, and spatially inhomogeneous steady-state solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter k measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when k is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as k is not large. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of a delayed reaction–diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive predator–prey model is considered in which the predator and prey populations both exhibit schooling behavior. The system's spatial dynamics are captured via a suitable threshold parameter, and a sequence of spatiotemporal patterns such as hexagons, stripes and a mixture of the two are observed. Specifically, the linear stability analysis is applied to obtain the conditions for Hopf bifurcation and Turing instability. Then, employing the multiple-scale analysis, the amplitude equations near the critical point of Turing bifurcation are derived, through which the selection and stability of pattern formations are investigated. The theoretical results are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis. [ABSTRACT FROM AUTHOR]

To understand the complex dynamics of Kingni-Jafari system with hidden attractors, the first objective of this paper is to study the global dynamics, and give a complete description of the dynamics of Kingni-Jafari system at infinity by using the Poincaré compactification of a polynomial vector field in . The second objective of this paper is to prove the existence of periodic solutions in the Kingni-Jafari system by classical Hopf bifurcation and degenerate Hopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution that bifurcates from a zero-Hopf equilibrium was derived in the Kingni-Jafari system. The theoretical analysis and simulations demonstrate the rich dynamics of the system. [ABSTRACT FROM AUTHOR]

In this paper, a system of neural networks in neutral form with time delay is investigated. Further, by introducing delay as a bifurcation parameter, it is found that Hopf bifurcation occurs when is across some critical values. The direction of the Hopf bifurcations and the stability are determined by using normal form method and center manifold theory. Next, the global existence of periodic solution is established by using a global Hopf bifurcation result. Finally, an example is given to support the theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper is concerned with bifurcation and stability in an autoimmune model, which was established to study an important phenomenon - blips arising from such models. This model has two equilibrium solutions, disease-free equilibrium and disease equilibrium. The positivity of the solutions of the model and the global stability of the disease-free equilibrium have been proved. In this paper, we particularly focus on Hopf bifurcation which occurs from the disease equilibrium. We present a detailed study on the use of center manifold theory and normal form theory, and derive the normal form associated with Hopf bifurcation, from which the approximate amplitude of the bifurcating limit cycles and their stability conditions are obtained. Particular attention is also paid to the bifurcation of multiple limit cycles arising from generalized Hopf bifurcation, which may yield bistable phenomenon involving equilibrium and oscillating motion. This result may explain some complex dynamical behavior in real biological systems. Numerical simulations are compared with the analytical predictions to show a very good agreement. [ABSTRACT FROM AUTHOR]

This paper demonstrates that the Chua Corsage Memristor, when connected in series with an inductor and a battery, oscillates about a locally-active operating point located on the memristor's DC - curve. On the operating point, a small-signal equivalent circuit is derived via a Taylor series expansion. The small-signal admittance is derived from the small-signal equivalent circuit and the value of inductance is determined at a frequency where the real part of the admittance of the small-signal equivalent circuit of Chua Corsage Memristor is zero. Oscillation of the circuit is analyzed via an in-depth application of the theory of Local Activity, Edge of Chaos and the Hopf-bifurcation. [ABSTRACT FROM AUTHOR]

The generation of action potentials (oscillations) in biological systems is a complex, yet poorly understood nonlinear dynamical phenomenon involving ions. This paper reveals that the time-varying calcium ion and the time-varying potassium ion, which are essential for generating action potentials in Barnacle giant muscle fibers are in fact generic memristors in the perspective of electrical circuit theory. We will show that these two ions exhibit all the fingerprints of memristors from the equations of the Morris-Lecar model of the Barnacle giant muscle fibers. This paper also gives a textbook reference to understand the difference between memristor and nonlinear resistor via analysis of the potassium ion-channel memristor and calcium ion-channel nonlinear resistor. We will also present a comprehensive in-depth analysis of the generation of action potentials (oscillations) in memristive Morris-Lecar model using small-signal circuit model and the Hopf bifurcation theorem. [ABSTRACT FROM AUTHOR]

This paper deals with a diffusive predator–prey model with Bazykin functional response. The parameter regions for the stability and instability of the unique constant steady state are derived. The Turing (diffusion-driven) instability which induces spatial inhomogeneous patterns, the existence of time-periodic orbits which produce temporal inhomogeneous patterns, the existence and nonexistence of nonconstant steady state positive solutions are proved. Numerical simulations are presented to verify and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an eco-epidemiological model with time delay representing the gestation period of the predator is investigated. In the model, it is assumed that the predator population suffers a transmissible disease and the infected predators may recover from the disease and become susceptible again. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free and coexistence equilibria are established, respectively. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the coexistence equilibrium, the disease-free equilibrium and the predator-extinct equilibrium of the system, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we incorporate immune systems into an HIV model, which considers both logistic target-cell proliferation and viral cell-to-cell transmission. We study the dynamics of this model including the existence and stability of equilibria. Based on the existence of equilibria, we focus on the backward bifurcation and forward bifurcation. Considering the stability of equilibria, Hopf bifurcation is discussed by identifying the basic reproduction number R 0 as bifurcation parameter. The direction and stability of Hopf bifurcation are investigated by computing the first Lyapunov exponent. Specially, the effects of immune response on the basic reproduction number R 0 and viral dynamics are addressed by deriving the sensitivity analysis. As a result, we find that the removal rate of infected cells by cytotoxic T lymphocytes (CTLs), γ 1 , is the predominant factor of R 0 . However, we conclude from numerical results that it is unfeasible to decrease R 0 by increasing the value of γ 1 constantly. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions. These dynamics are investigated by the proposed model to point out the importance and complexity of immune responses in fighting HIV replication. [ABSTRACT FROM AUTHOR]

Very recently, Zhang et al. considered an epidemic model on adaptive networks [Zhang et al., 2019], in which Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation are studied. Degenerate Hopf bifurcation is investigated via simulation and a numerical example is given to show the existence of two limit cycles. However, whether the codimension of the Hopf bifurcation is two is still open. In this paper, we will rigorously prove that the codimension of the Hopf bifurcation is two. That is, the maximal two limit cycles can bifurcate from the Hopf critical point. Moreover, the conditions for the existence of two limit cycles are derived. [ABSTRACT FROM AUTHOR]

In this paper, we study the local bifurcations of an enzyme reaction system with positive parameters α , β , γ and integer n ≥ 2. This system is orbitally equivalent to a polynomial differential system of order n + 2. Although not all coordinates of its equilibria can be calculated because of the high order polynomial, parameter conditions for the existence of each equilibrium are given. Moreover, qualitative properties of each equilibrium are determined. It shows that various bifurcations, including saddle-node bifurcation, Bogdanov–Takens bifurcation and Hopf bifurcation, may occur in this system as the parameters are varied. With Lyapunov quantities, the maximum order of the weak focus in this system is proved to be at most two by resultant elimination and real-root isolation. Furthermore, parameter conditions of its exact order are obtained. Finally, numerical simulations demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Recently, experimental studies show that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we introduce temporal fractional derivatives into a six-neuron bidirectional associative memory (BAM) neural network with leakage delay. By selecting two different bifurcation parameters and analyzing corresponding characteristic equations, it is verified that the delayed fractional neural network generates Hopf bifurcation when the bifurcation parameters pass through some critical values. In order to measure how much is the impact of leakage delay on Hopf bifurcation, sensitivity analysis methods, such as scatter plots and partial rank correlation coefficients (PRCCs), are introduced to assess the sensitivity of bifurcation amplitudes to leakage delay. Numerical examples are carried out to illustrate the theoretical results and help us gain an insight into the effect of leakage delay. [ABSTRACT FROM AUTHOR]

In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington–DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

The diffusive Holling–Tanner predator–prey model with no-flux boundary conditions and nonlocal prey competition is considered in this paper. We show the existence of spatially nonhomogeneous periodic solutions, which is induced by nonlocal prey competition. In particular, the constant positive steady state may lose the stability through Hopf bifurcation when the given parameter passes through some critical values, and the bifurcating periodic solutions near such values could be spatially nonhomogeneous and orbitally asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper, the Turing pattern formation in a diffusive toxin producing phytoplankton–zooplankton model with three-dimensional patch is investigated. Firstly, the conditions for the existence of Hopf bifurcation and stationary pattern are obtained based on the linear stability analysis. Then, the amplitude equations are derived by using nonlinear analysis and standard multiple-time-scale analysis. This study shows that there exists a pattern transition from spot pattern to stripe pattern with the decrease of the amount of toxicity, and the death rate of zooplankton also has a significant effect on the pattern structure. The obtained results suggest that the toxicity of phytoplankton and the death rate of zooplankton both play vital roles in the spatial distribution of population density. [ABSTRACT FROM AUTHOR]

In this paper, a model with time delay describing biodegradation of Microcystins (MCs) is investigated. Firstly, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Furthermore, an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the applications of the results. [ABSTRACT FROM AUTHOR]

In this paper, the well-known Vallis model for El Niño is analyzed for the parameter condition P ≠ 0. The conditions for the stability of the equilibrium points are derived. The condition for Hopf bifurcation occurring in the system for P = 0 and P ≠ 0 are investigated. The multistability feature of the Vallis model when P ≠ 0 is explained with forward and backward continuation bifurcation plots and with the coexisting attractors. The creation of period doubling followed by their annihilation via inverse period-doubling bifurcation known as antimonotonicity occurrence in the Vallis model for P ≠ 0 is presented for the first time in the literature. [ABSTRACT FROM AUTHOR]

This paper considers a general virus dynamics model with cell-mediated immune response and direct cell-to-cell infection modes. The model incorporates two types of intracellular distributed time delays and a discrete delay in the CTL immune response. Under certain conditions, the model exhibits a global threshold dynamics with respect to two parameters: the basic reproduction number and the reproduction number of immune response. We use suitable Lyapunov functionals and apply Lasalle's invariance principle to establish the global asymptotic stability of the two boundary equilibria. We also perform a bifurcation analysis for the positive equilibrium to show that the time delays may lead to sustained oscillations. To determine the direction of the Hopf bifurcation and the stability of the periodic solutions, the method of multiple time scales is applied. Finally, we carry out numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of hysteresis current-controlled quadratic buck-boost converter is investigated in detail. The system model is derived based on the sliding mode approach and also in its dimensionless form for algebraic brevity. The stability of the system is disclosed with the aid of the movement of eigenvalues. Onset of Hopf bifurcation is identified when the complex conjugate eigenvalue pair crosses the imaginary axis of the complex plane. The stability boundary is drawn to benefit the power electronics engineer for a stable and reliable design. The computer simulation of the switched model is performed using MATLAB/Simulink software to uncover the sequential occurrence of nonlinear behavior exhibited due to Hopf bifurcation for variation in control parameters and the input voltage. The phase portrait disclosing the subtle periodicity is plotted at different operating points to elicit that the stable period-1 attractor bifurcates to the quasi-periodic orbit and finally to a limit cycle. The precise dynamic of the phase portrait is also captured using the Poincare section. Experimental outputs are presented for confirming the low-frequency bifurcation scenario witnessed in the simulated and analytical results. [ABSTRACT FROM AUTHOR]

In this paper, the KdV–Burgers–Kuramoto chaotic system with distributed delay feedback is studied. The local stability of equilibrium points of this system is analyzed and the conditions under which Hopf bifurcation occurs are obtained by choosing the mean time delay as a bifurcation parameter. The direction and stability of bifurcating periodic solutions are derived by means of the normal form theory and the center manifold theorem. Numerical simulations are also illustrated which are in agreement with our theoretical results. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional complex Ginzburg-Landau equation which is a generic modulation equation describing the nonlinear evolution of patterns in fluid dynamics. The existence of a Hopf bifurcation from the basic solution was proved by Park [Bifurcation and stability of the generalized complex Ginzburg-Landau equation, Pure Appl. Anal. 7(5) (2008) 1237-1253]. We prove in this paper that the solution bifurcates to traveling waves which have constant amplitudes. We also prove that there exist kink-profile traveling waves which have variable amplitudes. The structure of the traveling waves is examined and it is proved by means of the center manifold reduction method and some perturbation arguments, that the variable amplitude traveling waves are quasi-periodic and they connect two constant amplitude traveling waves. [ABSTRACT FROM AUTHOR]

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of food chain predator-prey model. This model is given by a reaction-diffusion system defined on a circular spatial domain, which includes three-state variables namely, prey and intermediate predator and top predator and incorporates the Holling type II and a modified Leslie-Gower functional response. The aim of this paper is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The local and global stabilities of the positive steady-state solution and the conditions that enable the occurrence of Hopf bifurcation and Turing instability in the circular spatial domain are proved. In the end, we carry out numerical simulations to illustrate how biological processes can affect spatiotemporal pattern formation in a disc spatial domain and different types of spatial patterns with respect to different time steps and diffusion coefficients are obtained. [ABSTRACT FROM AUTHOR]

This paper considers the time-delayed feedback control for Maglev system with two discrete time delays. We determine constraints on the feedback time delays which ensure the stability of the Maglev system. An algorithm is developed for drawing a two-parametric bifurcation diagram with respect to two delays τ1 and τ2. Direction and stability of periodic solutions are also determined using the normal form method and center manifold theory by Hassard. The complex dynamical behavior of the Maglev system near the domain of stability is confirmed by exhaustive numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, analytical solutions of periodic evolutions of the Brusselator with a harmonic diffusion are obtained through the generalized harmonic balance method. The stability and bifurcation of the periodic evolutions are determined. The bifurcation tree of period-1 to period-8 evolutions of the Brusselator is presented through frequency-amplitude characteristics. To illustrate the accuracy of the analytical periodic evolutions of the Brusselator, numerical simulations of the stable period-1 to period-8 evolutions are completed. The harmonic amplitude spectrums are presented for the accuracy of the analytical periodic evolution, and each harmonics contribution on the specific periodic evolution can be achieved. This study gives a better understanding of periodic evolutions to chaos in the slowly varying Brusselator system, and the bifurcation tree of period-1 evolution to chaos are clearly demonstrated, which can help one understand a route of periodic evolution to chaos in chemical reaction oscillators. From this study, the generalized harmonic balance method is a good method for slowly varying systems, and such a method provides very accurate solutions of periodic motions in such nonlinear systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with a delayed predator–prey model with nonconstant death rate and constant-rate prey harvesting. We mainly study the impact of the time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively. By choosing time delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the time delay passes some critical values. In addition, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, some numerical simulations are carried out to depict our theoretical results. [ABSTRACT FROM AUTHOR]

A scalar population model with delayed growth rate of Allee effect type is considered in this paper. The stability of equilibria and associated supercritical Hopf bifurcations are analyzed. The basins of attraction of the two locally stable equilibria are characterized in terms of parameter values. In particular, when the time delay is large, the basin of attraction of the persistence equilibrium and limit cycle shrinks to a single point, so a global extinction of population occurs as a combined result of Allee effect and time delay. [ABSTRACT FROM AUTHOR]

Recently there are several works studying the finance model ẋ = z + x (y − a) , ẏ = 1 − b y − x 2 , ż = − x − c z , where a , b and c are positive parameters. The first objective of this paper is to show that this model exhibits one small-amplitude periodic solution emerging from a Hopf bifurcation at the equilibrium point (0 , 1 / b , 0) and in the second one we show that this system does not have invariant algebraic surfaces for any value of the parameters. [ABSTRACT FROM AUTHOR]

The present paper deals with a prey–predator model with prey refuge in proportion to both species, and the independent harvesting of each species. Our study shows that using refuge as control, it can break the limit cycle of the system and reach the required state of equilibrium level. We have established the optimal harvesting policy. The boundedness, feasibility of interior equilibria and bionomic equilibrium have been determined. The main observation is that the coefficient of refuge plays an important role in regulating the dynamics of the present system. Moreover, the variation of the coefficient of refuge changes the system from stable to unstable and vice-versa. Some numerical illustrations are given in order to support our analytical and theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a delayed mathematical model was developed based on experimental data to understand how the time delays required for transcription and translation in Mdm2 gene expression affect the kinetic behavior of the p53-Mdm2 network. Taking the time delays as the main research parameters, the stability of the system at the positive equilibrium was studied by using the theoretical method of delay differential equation. We found that such delays can induce oscillations by undergoing a supercritical Hopf bifurcation. Then, we used the normal form theory and the center manifold reduction to study the direction and stability of the bifurcation in detail. Furthermore, we also studied the effects of the length of time delays and the model parameters by numerical simulations. We found that time delays in Mdm2 synthesis are required for p53 oscillations and the length of such delays can determine the amplitude and period of the oscillations. In addition, the model parameters can also change the stability of the system. These results illustrate that the repair process after DNA damage can be regulated by varying time delays and the model parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey system with double Allee effects and time delay is considered. On the one hand, by using time delay as bifurcation parameter, the stability of the interior equilibrium point is studied. It is shown that under certain assumptions the equilibrium point of the delay model is locally asymptotically stable for all delay values, otherwise, there is a critical value, such that the equilibrium point is locally stable when time delay is less than the critical value, and Hopf bifurcation occurs when the delay crosses the critical value. Numerical simulations are carried out to validate the analytical results. On the other hand, we investigate the following four submodels respectively: the time-delayed system without Allee effect, the time-delayed system with Allee effect on predator, the time-delayed system with Allee effect on prey and the time-delayed system with double Allee effects. Compared with the situation without Allee effects, Allee effect on predator causes a decline of the equilibrium state of the predator, the equilibrium state of the prey and critical time delay are almost unaltered; Allee effect on prey and Allee effect on prey and predator all lead to the descent of the equilibrium state of the predator and prey, inversely, and critical time delay has an obvious rise. A comparison of the four cases indicates that Allee effect can alter the magnitudes of predator density and prey density at future time, the time that predator and prey cost to keeping steady state is also changed. [ABSTRACT FROM AUTHOR]

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means - both "locally" and "globally" and both "finitely" and "infinitely" - to comprehensively explore a given system. [ABSTRACT FROM AUTHOR]

Dynamic models have been successfully used for the prediction of long-term evolutionary behavior of the growth of tumor cells. In this paper, based on previous research, time delay in the process of the growth of tumor cells is further considered and a class of dynamic models with two-time delays and general functional response function is proposed. This model describes the effect of tumor-induced vascular endothelial growth factor (VEGF) on angiogenesis. First, the model is reduced to a two-dimensional one by means of the quasi-steady-state approximation method. This model has forward and backward bifurcations of the equilibria. Then, local stability of all equilibria, global stability of the tumor cell-free equilibrium and the vascular endothelial cell-free equilibrium, and the persistence of this model are studied in detail. It is found that time delay of tumor-induced vascular endothelial cells proliferation does not change local stability of the equilibria, while time delay associated with the growth of tumor cells can lead to the existence of Hopf bifurcation periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed and analyzed a mathematical model for the eco-epidemiological system with disease in prey population, placing particular emphasis on the effects of an incubation time delay and the influence of weak Allee effect on the growth of the predator population. We investigate basic properties of the proposed model including positivity, boundedness, uniform persistence and the stability of the biologically feasible equilibrium points. Using the time delay as bifurcation parameter, we explore the stability of the interior equilibrium point and observe that Hopf bifurcation can occur when the incubation time delay crosses some threshold value. We have identified the stability regions associated with the extinction of populations, stability of the steady states, and high-periodic oscillations in disease transmission. Analytical findings are supported by numerical illustrations that demonstrate the behavior of the proposed model in different dynamical regimes. Stability criteria for the biologically feasible equilibrium points were obtained and validated with numerical simulations. We found most sensitive system parameters using the PRCC sensitivity analysis. We observe that our model simulation exhibits high-periodic oscillations due to the increasing value of time delay parameter τ and the disease transmission rate γ x . [ABSTRACT FROM AUTHOR]

Rumors may cause serious harm to social productivity and life. Studying the dynamics of rumor propagation can help provide corresponding countermeasures to control communication effectively. Based on the classic S I infectious disease model, this paper studies a rumor spreading model with a time delay caused by communication delay between rumor communicators and nonsmooth threshold propagation function in reality. In this model, we solve non-negative equilibrium points firstly. By analyzing the characteristic equations of smooth and nonsmooth rumor systems, we establish local stability at non-negative equilibrium points and find the existence of Hopf bifurcation from two aspects of diffusion coefficient and delay, respectively. In addition, considering the discontinuity of rumor system, we separate it into two systems and get the condition of Hopf bifurcation when the system is discontinuous. We further verify the feasibility of the theoretical results by simulation results. [ABSTRACT FROM AUTHOR]

In this paper, dynamic instability of functionally graded carbon nanotubes (CNTs)-reinforced composite joined conical-cylindrical shell in supersonic flow is analyzed numerically. The higher-order shear deformation theory is applied to describe the stress–strain state of thin-walled structure. The assumed-mode method is used to derive the finite degrees-of-freedom dynamical system, which describes the structure motions. The structure motions are expanded by using the eigenmodes, which are obtained by the Rayleigh–Ritz method. The trial functions, which satisfy the continuity conditions at the cylindrical-cone junction, are used to obtain the eigenmodes. The properties of free vibrations of thin-walled structure are analyzed numerically. The dynamic instability of the joined conical-cylindrical shell in supersonic flow is analyzed using the characteristic exponents. As follows from the numerical study, the dynamic instability is arisen due to the Hopf bifurcation. The dependences of the supersonic flow critical pressure on the Mach number and the type of CNTs distribution are analyzed numerically. [ABSTRACT FROM AUTHOR]

For a vector field with Hopf bifurcation at the origin, when periodic excitation is introduced in which the frequency of the excitation is far less than the natural frequency, bursting oscillations, characterized by the alternations between the large-amplitude and the small-amplitude oscillations, can often be observed. The paper tries to answer the common question of how many types of bursting attractors may appear in such a vector field. Based on the normal form of Hopf bifurcation up to the third order, by regarding the whole exciting term as a slow-varying bifurcation parameter, all possible equilibrium branches of the generalized autonomous system are derived, the bifurcation sets of which divide the parameter space into regions associated with different types of dynamical behaviors. It is found that, there exist totally four possible combinations of the regions that the trajectory of the full system may visit as the slow-varying parameter varies. Accordingly, four qualitatively different behaviors, i.e. periodic symmetric oscillations, periodic symmetric mixed-mode oscillations, fold/Hopf/Hopf/fold and fold-Hopf/fold-Hopf bursting attractors, are presented, the mechanism of which is obtained by overlapping the transformed phase portrait and the equilibrium branches in the generalized autonomous system. Furthermore, the dynamics of the generalized autonomous system in different regions in the parameter space may influence the structures of the bursting oscillations when the trajectory enters the corresponding regions, while the bifurcations at the boundaries of the regions may lead to changes between the quiescent states and spiking states in the bursting attractors. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Leslie–Gower predator–prey model with strong Allee effect on prey and fear effect on predator. We discuss the existence and local stability of equilibria by making full use of qualitative analytical theory. It is shown that the above system exhibits at most two positive equilibria and it can undergo a series of bifurcation phenomena. We indicate that the dynamical behavior of the model is closely related to the fear effect on predator. In detail, when the fear effect parameter p = p ∗ , the system will undergo degenerate Hopf bifurcation. There exist two limit cycles (the inner is stable and the outer is unstable). However, when p = p ∗ , the system will undergo degenerate Bogdanov–Takens bifurcation. Also, by numerical simulation, we conclude that the stronger the fear effect, the bigger the density of prey species. The above shows that fear effect on predator is beneficial to the persistence of the prey species. Our results can be seen as a complement to previous works [González-Olivares et al., 2011; Pal & Mandal, 2014]. [ABSTRACT FROM AUTHOR]