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In this paper, following previous works of ours, we deal with a mathematical model of hepatitis B virus (HBV) infection. We assume spatial diffusion of free HBV particles, logistic growth for both healthy and infected hepatocytes, and use the standard incidence function for viral infection. Moreover, one time delay is introduced to account for actual virus production. Another time delay is used to account for virus maturation. The existence, uniqueness, positivity and boundedness of solutions are established. Analyzing the model qualitatively and using a Lyapunov functional, we establish the existence of a threshold $ \mathcal {T}_0 $ T 0 such that, if the basic reproduction number $ \mathcal {R}_0 $ R 0 verifies $ \mathcal {R}_0\leq \mathcal {T}_0{ \lt }1 $ R 0 ≤ T 0 < 1 , the infection-free equilibrium is globally asymptotically stable. When $ \mathcal {R}_0 $ R 0 is greater than one, we discuss the local asymptotic stability of the unique endemic equilibrium and the occurrence of a Hopf bifurcation. Also, when $ \mathcal {R}_0 $ R 0 is greater than one, the system is uniformly persistent, which means that the HBV infection is endemic. Finally, we carry out some relevant numerical simulations to clarify and interpret the theoretical results. [ABSTRACT FROM AUTHOR]

In this era of information technology, information leakage and file corruption due to computer virus intrusion have been serious issues. How to detect and prevent the spread of the computer virus is the major challenge we are facing now. To target this problem, a class of virus propagation models with hybrid control scheme are formulated to investigate the dynamic evolution and prevention from a spatial-temporal perspective in this paper. Diffusion-induced Turing instability is detected in response to the computer virus propagation. The introduction of hybrid control scheme can effective suppress Turing instability and turn the propagation system back to a stable state. And then, the time delay is selected as the bifurcation parameter. If the time delay exceeds the bifurcation threshold, the propagation will be destabilised and a Hopf bifurcation will occur. The hybrid control tactic can not only regulate the occurrence of Hopf bifurcation well, but also optimise the properties of bifurcating period solutions. In the end, the correctness and validity of the theoretical results are verified via numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the dynamic properties of a class of three-dimensional SIR models. First, we use the complete discriminant theory of polynomials to obtain the parameter conditions for the topological types of each fixed point. Second, by employing the centre manifold theorem and bifurcation theory, we prove that the system may undergo transcritical, flip and Neimark–Sacker bifurcations at fixed points. Finally, we simulated all the codimension-1 bifurcations. We found that the system also has codimension-2 bifurcations, i.e. 1:2, 1:3 and 1:4 strong resonances. [ABSTRACT FROM AUTHOR]

In this paper, we classify the phase portraits of an SIR epidemic dynamics model. Depending on the values of the parameters, this model can exhibit seven different phase portraits. In particular, from a biological point of view we prove that the unique attractors of this model are one or two equilibrium points depending on the values of the parameters and from the phase portraits follow the basins of attraction of these equilibria. [ABSTRACT FROM AUTHOR]

A reaction-diffusion-ODE system of stoichiometric producer-grazer type is considered in this paper. Since the system has nonsmooth nonlinearity, it is shown that the system has new dynamics different from the smooth case. We construct various types of discontinuous steady states and investigate their asymptotic behavior. In addition, the steady-state bifurcation near a constant solution is studied by treating the diffusion coefficient as a bifurcation parameter and the existence of Hopf bifurcation is derived. Our results cover the case where the number of positive equilibria of the kinetic system (i.e. without diffusion) changes from one to three in the spatial interval. Finally, some numerical simulations are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In a photovoltaic permanent magnet synchronous motor pumping system (PVPMSM), both photovoltaic modules and switching-mode converters present nonlinear and time-variant characteristics, which result in a difficult control problem. This paper presents an in-depth analysis of the operating point dynamics. A potential trade-off exists between maximum power and stability of the system operating point. Analysis of parametric singularities based on bifurcation theory permits to optimize the maximum power-stability trade-off. Tracing the bifurcation structures related to the parameter set (T, G), we can determine the maximum power point(MPP) in midway and far enough from the bifurcation points from these structures. An algorithm aiming to determine the MPP within the range of stability through power bifurcation analysis is proposed in this paper. Our methodology was validated with the simulation that was carried out on MATCONT/MATLAB, and the results are demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction-diffusion model incorporating dynamic variables for nutrient, phytoplankton, and zooplankton. Moreover, we account for the impact of time delay in the growth of phytoplankton following nutrient uptake. Our theoretical analysis reveals that the time delay can trigger the emergence of persistent oscillations in the model via a Hopf bifurcation. We also analytically track the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Our simulation results demonstrate stability switches occurring for the positive equilibrium with an increasing time lag. Furthermore, the model exhibits homogeneous periodic-2 and 3 solutions, as well as chaotic behaviour. These findings highlight that the presence of time delay in the phytoplankton growth can bring forth dynamical complexity to the nutrient-plankton system of aquatic habitats. [ABSTRACT FROM AUTHOR]

An intraguild predator–prey model including prey refuge and hunting cooperation is investigated in this paper. First, for the corresponding ordinary differential equation model, the existence and stability of all equilibria are given, and the existence of Hopf bifurcation, direction and stability of bifurcating periodic solutions are investigated. Then, for partial differential equation model, the diffusion-driven Turing instability is obtained. What is more, the existence and non-existence of the non-constant positive steady state of the reaction–diffusion model are established by the Leray–Schauder degree theory and some priori estimates. Next, some numerical simulations are performed to support analytical results. The results showed that prey refuge can change the stability of model and even have a stabilizing effect on this model, meanwhile the hunting cooperation can make such model without diffusion unstable, but make such model with diffusion stable. Lastly, a brief conclusion is concluded in the last section. [ABSTRACT FROM AUTHOR]

This paper is concerned with a prey–predator system involving herd behaviour and strong Allee effect. First, the local stability analysis of all the possible equilibrium points are investigated. Second, the existence of Hopf bifurcation around the unique positive equilibrium are carried out. Finally, this paper is ended with a numerical simulation to understand the dynamics of the system. As it turns out that the 'Allee effect constant' plays a very important role in biological invasion mechanism, such as controlling and preventing biological invasion or invading successfully. [ABSTRACT FROM AUTHOR]

Tracing the rapid progress of communication network, the control of dynamic evolution of network has become a central issue. There are a lot of tipping phenomena in network congestion systems. Therefore, tipping control principally centres on traditional control policies, and some advanced control approaches need to be supplemented. In this paper, a fractional-order proportional-integral-derivative (PID) controller is introduced to a network congestion model to ponder corresponding bifurcation-induced tipping regulation. First, a fractional-order congestion model with fractional-order PID controller is constructed. Then the onset of the tipping induced by Hopf bifurcation of the uncontrolled model is studied. By contrast, the tipping point can be delayed under the controller for the controlled model. Some conditions under which Hopf bifurcation occurs are given. The stable and unstable ranges of control parameters for the controlled model are also deduced. At last, some simulated examples are given to verify the theoretical results and demonstrate the superiority of the controller in tipping regulation. Moreover, the bidirectional effects of the controller are displayed by manipulating the control parameters. [ABSTRACT FROM AUTHOR]

The search for the Hopf bifurcation point in power systems can be made as the load is increased by small steps, and a power flow is solved for every load until a pair of purely imaginary eigenvectors has been found (zero real part). However, a search through successive power flows is time-consuming and does not guarantee that the Hopf bifurcation point will be found. This paper proposes a methodology and strategies that mitigate these problems. To increase the possibility of finding the Hopf bifurcation point, a continuation power flow is used to stablish search paths as the loading is incremented according to a fraction of the maximum loading point of the power system. It is only at specific converged continuation power flow that the dynamic state Jacobian matrix is constructed and has its eigenvalues computed. To facilitate the understanding and application of the proposal, it is used the MATPOWER (version 7.0) computer program and two available power systems. [ABSTRACT FROM AUTHOR]

Nonlinear magneto-electric response characteristics of giant magnetostrictive-piezoelectric composite sensors in harmonic and stochastic magnetic fields are studied in this paper. Van der Pol nonlinear differential items are introduced to explain the hysteresis phenomena of both giant magnetostrictive material and piezoelectric ceramics. The nonlinear dynamic model of giant magnetostrictive-piezoelectric composite sensors in harmonic and stochastic magnetic fields is developed. The expression of dynamic response of the system is obtained, and the bifurcation characteristics of the system are analyzed. The results of this paper are helpful for optimal design and improvement of giant magnetostrictive-piezoelectric composite sensors. [ABSTRACT FROM PUBLISHER]

This paper is concerned with a predator–prey model with prey-taxis and linear prey harvesting under the homogeneous Neumann boundary condition. The stability of the unique positive constant solution of the predator–prey model without prey-taxis is derived. Also, the emergence of Hopf bifurcation is concluded by choosing the proper Hopf bifurcation parameters. By the center manifold theorem and normal form, we compute the direction of Hopf bifurcation and the stability of the bifurcating solution. Moreover, the stationary pattern with prey-taxis is investigated. The conclusions show that prey harvesting and prey-taxis can enrich the dynamics. [ABSTRACT FROM AUTHOR]

The diffusive Schnakenberg model with gene expression time delay is considered. In this paper, the stability, diffusion-driven instability, and time delay-induced Hopf bifurcation have been investigated. By linear stability analysis, we find the parameter areas where the unique positive equilibrium is stable and Turing instability can occur for a certain relationship of diffusion rates. Then we obtain a series of critical values for the time delay at which the spatially homogeneous and inhomogeneous periodic solutions may emerge. Based on the explicit formula determining the properties of the Hopf bifurcation, we employ numerical simulations for parameters both in the stable region and Turing instability region. The numerical simulations show that delay can destabilize the stability of the positive equilibrium solution and eventually induce spatially homogeneous and inhomogeneous periodic solutions. Furthermore, the spatiotemporal patterns in the two spaces dimension from the Turing instability regime provide an indication of the wealth of patterns that the delayed system can exhibit. [ABSTRACT FROM AUTHOR]

Based on the theory of non-linear dynamic systems, the influence of the perturbation of the wheel rotation speed on the quasi-steady curved motions of a two-axle railway bogie system with a realistic wheel/rail contact relation is investigated in this paper. Since the wheel/rail contact relation is non-linear, it is tabulated in a wheel/rail contact table. The bifurcation diagram of the bogie system is constructed with gradually increasing and decreasing speed in the speed range . A supercritical Hopf bifurcation, where the stable non-trivial stationary solution loses its stability, is found in both models with and without perturbation of the wheel rotation speed. In the model without perturbation of the wheel rotation speed, the first chaotic motions develop at the speed where the wheel flange contact starts. A period-doubling cascade of the bogie system through pitchfork bifurcations, which explains the transition from periodic solutions to chaotic motions of the bogie system, is found. Several jumps happen at higher speeds because of the coexistence of multiple attracting solutions, which should be avoided. A comprehensive investigation of the hysteresis phenomena is made. However, in the model with a perturbation of the wheel rotation speed, no chaotic motions are found. [ABSTRACT FROM AUTHOR]

In this paper, an improved heterogeneous continuum model is presented accounting for multiple optimal velocity functions with probabilities, in which self-stabilizing effect and velocity uncertainty are considered simultaneously. The linear stability criterion of this new model is obtained by applying linear stability theory. By means of nonlinear analysis method, the Korteweg-de Vries-Burgers (KdV-Burgers) equation is deduced to discuss the evolution process of traffic flow density wave. Through bifurcation analysis, the conditions and stability of the Hopf bifurcation for the new model are derived in detail. Numerical simulations are performed to study the influences of above mentioned factors on the stability of traffic flow. Fuel consumption and emissions are also explored in this paper. It is notable that when the Hopf bifurcation point is selected as the initial point of density evolution, significant periodic oscillations will occur in the traffic flow. The results of numerical simulations coincide well with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R 0 and show that if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, whereas if R 0 > 1 , then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fear effect predator–prey model with mutual interference or group defense. For the model with mutual interference, we show the interior equilibrium is globally stable, and the mutual interference can stabilize the predator–prey system. For the model with group defense, we discuss the singular dynamics around the origin and the occurrence of Hopf bifurcation, and find that there is a separatrix curve near the origin such that the orbits above which tend to the origin and the orbits below which tend to limit cycle or the interior equilibrium. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the influence of the waning immunity on the dynamics of Tilapia Lake Virus (TiLV) transmission in wild and farmed tilapia within freshwater. We formulate a model for which susceptible individuals can contract the disease in two ways: (i) direct mode caused by contact with infected individuals; (ii) indirect mode due to the presence of pathogenic agents in the water. We obtain an age-structured model which combines both age since infection and age since recovery. We derive an explicit formula for the reproductive number R 0 and show that the disease-free equilibrium is locally asymptotically stable when, R 0 < 1. We discuss on the form of the waning immunity parameter and show numerically that a Hopf bifurcation may occur for suitable immunity parameter values, which means that there is a periodic solution around the endemic equilibrium when, R 0 > 1. [ABSTRACT FROM AUTHOR]

In this paper, we derive a delayed epidemic model to describe the characterization of cytotoxic T lymphocyte (CTL)-mediated immune response against virus infection. The stability of equilibria and the existence of Hopf bifurcation are analysed. Theoretical results reveal that if the basic reproductive number is greater than 1, the positive equilibrium may lose its stability and the bifurcated periodic solution occurs when time delay is taken as the bifurcation parameter. Furthermore, we investigate an optimal control problem according to the delayed model based on the available therapy for hepatitis B infection. With the aim of minimizing the infected cells and viral load with consideration for the treatment costs, the optimal solution is discussed analytically. For the case when periodic solution occurs, numerical simulations are performed to suggest the optimal therapeutic strategy and compare the model-predicted consequences. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper aims to analyse stability and Hopf bifurcation of the HIV-1 model with immune delay under the functional response of the Holling II type. The global stability analysis has been considered by Lyapunov–LaSalle theorem. And stability and the sufficient condition for the existence of Hopf Bifurcation of the infected equilibrium of the HIV-1 model with immune response are also studied. Some numerical simulations verify the above results. Finally, we propose a novel three dimension system to the future study. [ABSTRACT FROM AUTHOR]

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey model that takes into account the interactions between three species of fish, zooplankton, and phytoplankton has been developed. The logistic growth of phytoplankton has been considered in the absence of zooplankton. It is assumed that zooplankton eats phytoplankton. It is considered that the growth rate of zooplankton decreases due to the fear of fish. The release of toxins by phytoplankton and fish consumption are two further factors that are assumed to decrease the growth rate of zooplankton. The harvesting of fish has also been considered in this study. Local and global stability of the system around the equilibrium points has been investigated. The existence condition of oscillatory behaviour of the system has been studied through Hopf bifurcation. It is found that the increasing rate of minimum cost of fear η may stabilize the system. It is obtained that the rate of increase of release of toxin substance ρ by phytoplankton may be responsible for the stability of the system. It is also found that the system may become unstable for the increase of level of fear ξ of zooplankton. Some numerical simulation results are presented to test the theoretical findings. [ABSTRACT FROM AUTHOR]

Recently, by studying some field experimental data and research articles it has been observed that due to predation, predators not only affect prey by direct killing, but they also induce the predation fear in prey which reduces its reproduction rate. The grouping behaviour of animals is one of the most important and interesting phenomena in the field of population dynamics. As a result of prey-predator forming groups, many authors have considered different functional responses in their proposed models. In reality, the amount of biomass converted into the predator due to the prey consumption is not instantaneous. To incorporate the gestation delay in this paper we have proposed and analysed a delayed prey-predator model with square root functional response and fear effect in the prey. We have discussed the qualitative analysis of solutions of the considered model in presence and absence of delay. Moreover, the transcritical bifurcation, Hopf bifurcation with it's direction and the impact of the fear effect on both populations are studied here. Finally, some numerical simulations through graphical illustrations are carried out to support the analytical results.. i [ABSTRACT FROM AUTHOR]

This paper is designed based on the combined bioeconomic harvesting of Holling-Tanner prey-predator competition model with Beddington-DeAngelis functional response with two different delays. The situation and existence of the steady state points for proposed model are discussed. The conditions for local stability of the interior steady state points together with the existence of Hopf bifurcation at the interior steady state point are addressed. The situations for stability and the direction of bifurcating periodic solutions are obtained by using center manifold theory. Also total revenue is calculated using bionomic equilibrium, and it is maximized by using Pontryagin's maximum principle. Some numerical simulations together with pictorial representations are placed in the paper for showing the effects of various parameters in the considered model. Conclusions with the future study are described at last. [ABSTRACT FROM AUTHOR]

In this paper, we study time-delayed reaction–diffusion systems with advection subject to Lotka–Volterra competition dynamics over one-dimensional domains. These systems model the population dynamics of two groups of competing species, with one dispersing randomly and the other a combination of random and biased dispersal (to avoid competition). We show that time-delay(s) in the interspecific competition mechanism can induce instability of the homogeneous equilibrium to the reaction–advection–diffusion systems, and further promote the appearance of time-oscillating spatially inhomogeneous distributions of the species. Our results indicate that these time-delayed systems (both single and double time-delays) can be used to model the well-observed time-periodic distributions of interacting species in natural fields, compared to the systems without time-delay(s). [ABSTRACT FROM AUTHOR]

In this paper, a delayed human immunodeficiency virus (HIV) model with apoptosis of cells has been studied. Both immunological and intracellular delay have been incorporated to make the model more relevant. Firstly, the model has been investigated using local stability analysis. Next, the global stability analysis of steady states has been performed. The stability switch criteria taking the delay as the bifurcating parameter, leading to Hopf bifurcation has been studied. The transition of the system from order to chaos has been explored, and the analytical results have been verified by numerical simulations. The results thus can be used to describe the extensive dynamics exhibited by the model introduced in this article. The effects of apoptosis on viral load has been studied in the model numerically. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington-DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results. [ABSTRACT FROM AUTHOR]

This paper deals with reduction of oscillations of a continuous spinning shaft using proportional-derivative time delay controller. The stable and unstable zones of the system are obtained using the Routh-Hurwitz criteria. For small values of proportional coefficients, the Hopf bifurcation occurs in the high values of time delays while in the higher values of proportional coefficients instability occurs in the lower values. Also, in small values of derivative coefficients the Hopf bifurcation happens in the lower values of time delays, while in the higher values of derivative coefficients mentioned bifurcation occurs in the higher values. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model with immunity age is investigated, where the constant treatment rate and the loss of the acquired immunity are incorporated. The well-posedness of the model is verified by changing it into an abstract non-densely defined Cauchy problem, and the conditions for the existence of disease-free equilibrium and the endemic equilibria are found. The theoretic analysis showed that the disease-free equilibrium is globally asymptotically stable as the basic reproduction number is less than unity, and the numerical simulation illustrated that it is asymptotically stable as the number is greater than unity. Combining numerical simulations, the instability and the local stability of different endemic equilibrium, and the existence of saddle-node bifurcation, and Hopf bifurcation are analyzed. Again, we think it is possible that the Bogdanov–Takens bifurcation may occur for the model under some conditions. Both non-periodic and periodic behaviors are shown when the disease persists in population, where the duration that the recovered individual stays in the recovery class plays an important role in the spread of the disease. [ABSTRACT FROM AUTHOR]

Mode transition plays an important role in the operation of hybrid electric vehicles. The clutch is a key actuator for mode transition, but friction resulting from its use makes it difficult to achieve seamless mode transition. Moreover, as is usual in an electromechanical coupling system, the hybrid powertrain may exhibit complex dynamic behaviours as operating parameters change during mode transition. This paper considered changes in the clutch friction coefficient, and established a mathematical model of the hybrid powertrain in a parallel-series hybrid electric vehicle (PSHEV) during mode transition. The Routh-Hurwitz criterion and bifurcation theory are introduced to analyse the influence of speed difference on the dynamic characteristics of the hybrid powertrain, and the corresponding instability threshold were given. Furthermore, an optimised adaptive control methodology was proposed to control the dynamic behaviours, and tests on PSHEV were carried out. The results show that the variations of speed difference may lead to the occurrence of Fold and Hopf bifurcations, and that once the speed difference reaches the instability threshold, it may cause the system to enter a chaotic state and exhibit random oscillation. The proposed control methodology can effectively expand the security domain and suppress the oscillation to improve smoothness of mode transition. [ABSTRACT FROM AUTHOR]

In this paper, a computer virus propagation model with two delays and infectivity in latent period is investigated. First, the conditions which guarantee the local stability of the positive equilibrium and the existence of the Hopf bifurcation are derived by choosing the different combination of the two delays as the bifurcation parameter. Moreover, some specific properties for determining the stability and direction of the Hopf bifurcation are obtained by employing the normal form theory and the centre manifold theorem. Finally, a numerical simulation is carried out to verify the correctness of our obtained theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper investigates Turing instability and Hopf bifurcation for a diffusive plankton system with time delay and cell size. To determine the effects of diffusion and cell size on the dynamics of the system, we first study the system without time delay, where the conditions of stability of coexisting equilibrium and Turing instability are obtained through Routh-Hurwitz criterion. Then we give the existence of Hopf bifurcation using time delay as bifurcation parameter by analyzing the distribution of eigenvalues, and derive the property of Hopf bifurcation by applying the centre manifold theory. Finally, numerical simulation shows that different cell size increases the variety of dynamics for diffusive plankton system. [ABSTRACT FROM AUTHOR]

In this paper, we revisit a Nicholson's blowflies model with proportional delay, the stability and bifurcation of whose discrete version have not been studied. By using the method of semidiscretization, we dig deeply out the stability and Neimark–Sacker bifurcation of its discrete model. Especially, some results for the existence and stability of Neimark–Sacker bifurcation are derived by using the centre manifold theorem and bifurcation theory. Numerical simulations are also formulated to verify the existence of Neimark–Sacker bifurcation derived. [ABSTRACT FROM AUTHOR]

In this paper, a delayed diffusive predator–prey model with schooling behaviour and Allee effect is investigated. The existence and local stability of equilibria of model without time delay and diffusion are given. Regarding the conversion rate as bifurcation parameter, Hopf bifurcation of diffusive system without time delay is obtained. In addition, the local stability of the coexistent equilibrium and existence of Hopf bifurcation of system with time delay are discussed. Moreover, the properties of Hopf bifurcation are studied based on the centre manifold and normal form theory for partial functional differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a stage-structured predator–prey model with mutual interference, in which includes two discrete delays. By theoretical analysis, we establish the stability of the unique positive equilibrium and the existence of Hopf bifurcation when the maturation delay for predators is used as the bifurcation parameter. Our results exhibit that the maturation delay for preys does not affect the stability of the positive equilibrium. However, the maturation delay for predator is able to destabilize the positive equilibrium and causes periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and display the differential impacts of two type delays and mutual interference. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a discrete-time Lotka–Volterra predator–prey model that can undergo flip bifurcations and Hopf bifurcations by using the centre manifold theorem and bifurcation theory. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits verify the theoretical analysis, which exhibits complex dynamical behaviours such as period-7, 13, 14, 21, 27, 32 orbits, a cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. Furthermore, the feedback control method is used to stabilize the chaotic orbits at an unstable fixed point. This result further reveals richer dynamics of the discrete model compared with the continuous model. It is found out that the appropriately carrying capacity can be helpful to stabilize the model. Instead, the higher or lower carrying capacity may destabilize the system producing more complex dynamical behaviours. [ABSTRACT FROM AUTHOR]

In order to study whether haemoglobin (Hb) can replace peroxidase and has good catalytic properties. The key to exploring the characteristics of Hb peroxidase is to establish a suitable kinetic model, which is studied in this paper. First, according to the Hb catalytic reaction, a nonlinear system is established and improved. It is proved that the established system is in line with the practical significance. The stability of the original system is judged by analysing the stability of the simplified system. Then, considering the effect of time delay on Hb catalytic reaction, a nonlinear time-delay catalytic reaction system is obtained. For convenient application, the system is linearized using Taylor's formula, and the dynamic characteristics of Hopf bifurcation are analysed. The response diagrams of three system are plotted by setting perturbation parameters, and their variations are observed to analyse the differences among them. The results show that the nonlinear time-delay system can better describe the characteristics of the catalytic reaction. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed two-species predator-prey model with general functional response under the homogeneous Neumann boundary condition. We discuss the stability of the trivial and semi-trivial solutions and obtain the spatially nonhomogeneous bifurcation solutions stemming from the semi-trivial trivial solutions (θ a , 0) and (0 , θ b). Besides, the stability and some results of Hopf bifurcation at the spatially nonhomogeneous bifurcation steady-state solutions are investigated by analyzing the distribution of the eigenvalues. The method we applied here is mainly based on spectral analysis, comparison principle, Lyapunov-Schmidt reduction, and bifurcation theory. [ABSTRACT FROM AUTHOR]

An Susceptible-Vaccinated-Exposed-Infectious-Recovered computer virus model with nonlinear incidence rate and two delays is proposed and its Hopf bifurcation is investigated. The existence of Hopf bifurcation at the viral equilibrium is established by analyzing its distribution of characteristic values. Moreover, the stability and the direction of the Hopf bifurcation are obtained by using the normal form theory and centre manifold theory. Finally, a numerical simulation is performed to validate the theoretical results obtained in the paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with a delayed eco-epidemic model with a Leslie–Gower Holling type III functional response. The main results are given in terms of permanence and Hopf bifurcation. First of all, sufficient conditions for permanence of the model are established. Directly afterward, sufficient conditions for local stability and existence of Hopf bifurcation are obtained by regarding the delay as bifurcation parameter. Finally, properties of the Hopf bifurcation are investigated with the aid of the normal form theory and centre manifold theorem. Numerical simulations are carried out to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of delay differential equations that models the oncolytic virotherapy on solid tumours with the delay of viral infection in the presence of the innate immune response. We conduct qualitative and numerical analysis, and provide possible medical implications for our results. The system has four equilibrium solutions. Fixed point analysis indicates that increasing the burst size and infection rate of the viruses has positive contribution to the therapy. However, increasing the immune killing infection rate, the immune stimulation rate, or the immune killing virus rate may lead the treatment failed. The viral infection time delay induces backward Hopf bifurcations, which means that the therapy may fail before time delay increases passing through a Hopf bifurcation. The parameter analysis also shows how saddle-node and Hopf bifurcations occur as viral burst size and other parameters vary, which yields further insights into the dynamics of the virotherapy. [ABSTRACT FROM AUTHOR]

A delayed SEIR epidemic model with saturated incidence and saturated treatment function is considered in this paper. Sufficient conditions for the existence of local Hopf bifurcation are established by regarding the possible combination of the two delays as the bifurcation parameter. General formula for the direction, period and stability of the bifurcated periodic solutions are derived by using the normal form method and the centre manifold theory. Finally, some numerical simulations are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an age-structured HIV infection model with logistic growth for target cell. We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state. By linearizing the model at steady state and analysing the associated characteristic transcendental equations, we study the local asymptotic stability of the steady state. Furthermore, by using Hopf bifurcation theorem in Liu et al., we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values. Finally, we perform some numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]

A stage-structured prey–predator model with prey refuge is assumed, where predators have two stages as immature and mature. We consider here the harvesting on prey only and analyze the model using Beddington-DeAngelis functional response. Local stability as well as global stability at an interior equilibrium point are investigated in the proposed model. We choose different types of bifurcation like Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation in the formulated model. Moreover, in this paper, we check the optimality of the harvesting function with the help of optimal control. We provide the numerical simulation of the proposed model for the confirmation of analytical results. [ABSTRACT FROM AUTHOR]

The prey–predator systems exist in almost all ecological systems and the refuge stage is highly concerned with such ecological systems in which the prey goes out of reach of the attacking region of the predator. In this paper, dynamical behaviour of predator and prey species with Holling type III response function is investigated. We add refuge in prey species and also additional predators are introduced into the system. Firstly, the crisp model is formulated with these effects and the boundedness, existence, conditions of local and global stability, local bifurcation and Hopf bifurcation are discussed. The crisp model is transformed into the fuzzy model to incorporate the unpredictable behaviour and using the Root Mean Square (RMS) defuzzification method, the fuzzy model is reduced to the defuzzified model in order to avoid the complexity of the analysis of the solution. Further, numerical simulations are performed with the help of MATLAB, MATCONT package and we examine solution curves and phase portraits in two different cases, in the presence and in the absence of additional predators along with the Hopf bifurcation scenarios. Finally, the sensitivities of the controlling parameters are plotted graphically and presented their significances in controlling the infection of the prey. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a mathematical model for an inflammatory response to bacterial infection of homogeneous tissues. Specifically, we provide a detailed analysis of the Lauffenburger-Kennedy bacterial infection model and show that the model exhibits three possible equilibria corresponding to a bacteria-free and two endemic compromised steady states. Asymptotic results of the steady states along with the existences of saddle-node connection Hopf bifurcations are shown under certain conditions of the parameters. Within the biological ranges of the parameter values, we observe that the system can exhibit both forward and backward bifurcation. In addition, in both cases, the larger compromise bacterial infection steady state can either approach an equilibrium or can oscillate around it via Hopf bifurcation depending on the value of the ratio of leukocyte mortality to phagocytosis rates. Numerical results are used to provide illustrative examples of these different dynamical patterns observed in the model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new alcoholism model in which alcoholics have age structure. We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state. By linearizing the model at steady state and analyzing the associated characteristic transcendental equations, we study the local asymptotic stability of the steady state. Furthermore, by using Hopf bifurcation theorem in Liu et al. (Z. Angew. Math. Phys. 62 (2011) 191-222), we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief conclusion. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results. [ABSTRACT FROM AUTHOR]