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In this paper, a new delayed fractional-order model including susceptible migratory birds, infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds. Fear of predators is considered in the model, as fear can reduce the reproduction rate and disease transmission rate among prey. First, some basic mathematical results of the proposed model are discussed. Then, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such an uncontrolled system are established. A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon. It is found that the control scheme can successfully suppress the bifurcation behavior of the system, and the pulse width can be arbitrarily selected on the premise of ensuring the control effect. Compared with the traditional time-delay feedback controller, the control scheme proposed in this paper has more advantages in practical application, which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment. The proposed control scheme, in particular, remains effective even after the system has been disrupted by a constant. Numerical simulation verifies the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points. [ABSTRACT FROM AUTHOR]

This paper is concerned with the sustained periodic oscillation phenomenon in a three-species delayed predation system with Holling type II functional response and age structure in top predator. The top predator fertility function is regarded as a piecewise continuous smooth function with regard to their maturation period τ 2 . The complicated dynamic behavior is proved by integrated semigroup theory and Hopf bifurcation theorem for semilinear equations with non-dense domain. Through qualitative analysis and bifurcation study of the system, we yield that this system has a nontrivial periodic solution that bifurcates from the positive equilibrium age distribution when bifurcation parameter τ passes through some critical values. Numerical simulations are also provided to illustrate theoretically analytical results. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]

In this paper, the impact of the strong Allee effect and ratio-dependent Holling–Tanner functional response on the dynamical behaviors of a predator–prey system is investigated. First, the positivity and boundedness of solutions of the system are proved. Then, stability and bifurcation analysis on equilibria is provided, with explicit conditions obtained for Hopf bifurcation. Moreover, global dynamics of the system is discussed. In particular, the degenerate singular point at the origin is proved to be globally asymptotically stable under various conditions. Further, a detailed bifurcation analysis is presented to show that the system undergoes a codimension- 1 Hopf bifurcation and a codimension- 2 cusp Bogdanov–Takens bifurcation. Simulations are given to illustrate the theoretical predictions. The results obtained in this paper indicate that the strong Allee effect and proportional dependence coefficient have significant impact on the fundamental change of predator–prey dynamics and the species persistence. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of interacting Tilapia fish and Pelican bird population in the Salton Sea. We assume that the diseases spread in Tilapia fish follows the Holling type II response function, and the interaction between Tilapia and Pelican follows the Beddington–DeAngelis response function. The dynamics of diffusive and delayed system are discussed separately. Analytically, all the feasible equilibria and their stability are discussed. The criterion for Turing instability is derived. Based on the normal form theory and center manifold arguments, the existence of stability criterion and the direction of Hopf bifurcation are obtained. Numerical simulation shows the occurrence Hopf bifurcation, double Hopf bifurcation and transcritical bifurcation scenarios. The snap shot shows the spot, spot-strip mix patterns in the whole domain. Further, the stability switching phenomena is observed in the delayed system. Our comprehensive study highlights the effect of different parameters, multiple time delay and extinction in Pelican populations. [ABSTRACT FROM AUTHOR]

In this paper, a stage structure predator–prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed. The prey populations are divided into two parts: juvenile prey and adult prey. From extensive experimental data, it has been found that prey fear of predators can alter the physiological behavior of individual prey, and the fear effect reduces their reproductive rate and increases their mortality. In addition, we also consider the presence of constant ratio refuge in adult prey populations. Moreover, we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors. We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria. Next by choosing the intra-specific competition coefficient of adult prey as bifurcation parameter, we demonstrate that Hopf bifurcation may occur near the positive equilibrium point. Then by taking the gestation delay as bifurcation parameter, the sufficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given. And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory. What's more, numerical experiments are performed to test the theoretical results obtained in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the dynamic behaviors of fractional- order logistic model with Allee effects in Caputo–Fabrizio sense. First of all, we apply the two-step Adams–Bashforth scheme to discretize the fractional-order logistic differential equation and obtain the two-dimensional discrete system. The parametric conditions for local asymptotic stability of equilibrium points are obtained by Schur–Chon criterion. Moreover, we discuss the existence and direction for Neimark–Sacker bifurcations with the help of center manifold theorem and bifurcation theory. Numerical simulations are provided to illustrate theoretical discussion. It is observed that Allee effect plays an important role in stability analysis. Strong Allee effect in population enhances the stability of the coexisting steady state. In additional, the effect of fractional-order derivative on dynamic behavior of the system is also investigated. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission. [ABSTRACT FROM AUTHOR]

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey's growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter (θ) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter? [ABSTRACT FROM AUTHOR]

The sterile insect technique (SIT) has been applied as an alternative method to reduce or eradicate mosquito-borne diseases. To explore the impact of the sterile mosquitoes on controlling the wild mosquito populations, in this paper, we further extend the work in [J. Li, New revised simple models for interactive wild and sterile mosquito populations and their dynamics, J. Biol. Dyn. 11(S2) (2017) 316–333] and formulate delayed models for interactive wild and sterile mosquitoes, which can depict wild mosquito population undergoing distinct stages of development during a lifetime. By performing mathematical analysis, the threshold dynamics of the proposed models are explored, respectively. In particular, Hopf bifurcation phenomena are observed as the delay τ is varying. Numerical examples illustrate our findings. [ABSTRACT FROM AUTHOR]

The security of mobile wireless sensor networks has captivated extensive attention of researchers because of their wide range of applications and vulnerability to attacks caused by malware. In this paper, we investigate a delayed malware propagation model on mobile wireless sensor network incorporating nonlinear incidence rate, logistic growth rate and recovery rate. Local asymptotic stability of the endemic equilibrium and existence of Hopf bifurcation at crucial value of the time delay are analyzed. Then, properties of Hopf bifurcation are explored. Specifically, global exponential stability is investigated via linear matrix inequality. An example is presented finally to underline the effectiveness of findings in our paper numerically and graphically. [ABSTRACT FROM AUTHOR]

In this paper, we consider the complicated dynamics of a delay-based photonic reservoir computing system. Since conventional computer architectures are approaching their limit, it is imperative to find new, efficient and fast ways of data processing. Photonic reservoir computing (RC) is a promising way which combines the computational capabilities of recurrent neural networks with high processing speed and energy efficiency of photonics. As the RC system is very promising, we analyze its dynamics so that we can make better use of it. In this paper, we mainly focus on its double Hopf bifurcation. We first analyze the existence of double Hopf bifurcation points. Then we use DDE-BIFTOOL to draw the bifurcation diagrams with respect to two bifurcation parameters, i.e. feedback strength η and delay τ , and give a clear picture of the double Hopf bifurcation points of the system. These figures show stability switches and the existence of double Hopf bifurcation points. Finally, we employ the method of multiple scales to obtain their normal forms, use the method of normal form to unfold and classify their local dynamics. The classification and unfolding of these double Hopf bifurcation points are obtained. Three types of double Hopf bifurcations are found. We verify the results by numerical simulations and find its complicated behavioral dynamics. For example, there exist stable equilibrium, stable periodic and quasi-periodic solutions in distinct regions. The discovered rich dynamical phenomena can help us to choose suitable values of parameters to achieve excellent performance of RC. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long-term sustainability of the northeast Korean pine. We propose a class of natural Korean pine population system with time delay and diffusion term. First, by analyzing the roots distribution of the characteristic equation, we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation. Second, we introduce lactation time delay into a population model with a diffusion term, based on stability theory of ordinary differential equation, norm form methods and center manifold theorem, the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given. Finally, some numerical simulations are given. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate a delayed diffusive predator–prey model affected by toxic substances. We first study the boundedness and persistence property of the model. By analyzing the associated characteristic equation, we obtain the conditions for the existence of steady state bifurcation, Hopf bifurcation and Turing bifurcation. Furthermore, we also study the Hopf bifurcation induced by the delay. Finally, our theoretical results are verified by numerical simulation. The numerical observation results are in good agreement with the theoretically predicted results. Theoretical and numerical simulations indicate that toxic substances have a great impact on the dynamics of the system. [ABSTRACT FROM AUTHOR]

The spread of infectious diseases often presents the emergent properties, which leads to more difficulties in prevention and treatment. In this paper, the SIR model with both delay and network is investigated to show the emergent properties of the infectious diseases' spread. The stability of the SIR model with a delay and two delay is analyzed to illustrate the effect of delay on the periodic outbreak of the epidemic. Then the stability conditions of Hopf bifurcation are derived by using central manifold to obtain the direction of bifurcation, which is vital for the generation of emergent behavior. Also, numerical simulation shows that the connection probability can affect the types of the spatio-temporal patterns, further induces the emergent properties. Finally, the emergent properties of COVID-19 are explained by the above results. [ABSTRACT FROM AUTHOR]

In this paper, we study a diffusive predator–prey model with hyperbolic mortality and prey-taxis under homogeneous Neumann boundary condition. We first analyze the influence of prey-taxis on the local stability of constant equilibria. It turns out that prey-taxis has influence on the stability of the unique positive constant equilibrium, but has no influence on the stability of the trivial equilibrium and the semi-trivial equilibrium. We then derive Hopf bifurcation and steady state bifurcation related to prey-taxis, which imply that the prey-taxis plays an important role in the dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear ratio-dependent prey–predator model with constant prey refuge in the prey population. Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population. The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail. Hopf bifurcation including its direction and stability for the model is also studied. We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system. It is also observed that for a fixed strength of fear, an increase in the Allee parameter makes the system unstable, whereas an increase in prey refuge drives the system toward stability. However, higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction. Further, we explore the variation of densities of the populations in different bi-parameter spaces, where the coexistence equilibrium point remains stable. Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software. [ABSTRACT FROM AUTHOR]

A nonlinear infectious disease model with information-influenced vaccination behavior and contact patterns is proposed in this paper, and the impact of information related to disease prevalence on increasing vaccination coverage and reducing disease incidence during the outbreak is considered. First, we perform the analysis for the existence of equilibria and the stability properties of the proposed model. In particular, the geometric approach is used to obtain the sufficient condition which guarantees the global asymptotic stability of the unique endemic equilibrium E e when the basic reproduction number R 0 > 1. Second, mathematical derivation combined with numerical simulation shows the existence of the double Hopf bifurcation around E e . Third, based on the numerical results, it is shown that the information coverage and the average information delay may lead to more complex dynamical behaviors. Finally, the optimal control problem is established with information-influenced vaccination and treatment as control variables. The corresponding optimal paths are obtained analytically by using Pontryagin's maximum principle, and the applicability and validity of virous intervention strategies for the proposed controls are presented by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a size-stage-structured cooperation model which has two distinct life stages in facultative cooperator. The primary feature of this model is to consider size structure, stage structure and obligate and facultative symbiosis at the same time in a cooperation system. We use the method of characteristic to show that this new model can be reduced to a threshold delay equations (TDEs) model, which can be further transformed into a functional differential equations (FDEs) model by a simple change of variables. Such simplification allows us to apply the classical theory of FDEs and establish a set of sufficient conditions to investigate the qualitative analysis of solutions of the FDEs model, including the global existence and uniqueness, positivity and boundedness. What's more, we use the geometric criteria to get the conclusions about stability and Hopf bifurcation of positive equilibrium because the coefficients of the characteristic equation depend on the bifurcation parameter. Finally, numerical simulations are carried out as supporting evidences of our analytical results. Our results show that the presence of size structure and stage structure plays an important role in the dynamic behavior of the model. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)- C ∞ -smooth and (non)-analytic periodic center manifolds. [ABSTRACT FROM AUTHOR]

This paper focuses on the Hopf bifurcation of a fractional tri-neuron network with both leakage delay and communication delay under different fractional orders. By applying fractional Laplace transform, the stability theorem of linear autonomous system and Hopf bifurcation theorem, we obtain a class of asymptotic stability criterion of zero solution as well as delay-induced Hopf bifurcation conditions for the considered system. Simultaneously, the stability and Hopf bifurcation for tri-neuron network with single fractional order are also discussed as a special case of our proposed neural network model. Finally, a simulation example is given to illustrate the efficiency of the presented theoretical results in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider a delayed single population model with patch structure, and investigate Hopf bifurcations when the dispersal rate or the scaling parameter of the growth rate is small. The effect of dispersal patterns is analyzed. We show that dispersal patterns affect the occurrence of Hopf bifurcations when the scaling parameter of the growth rate is small, and affect the values of Hopf bifurcations when the dispersal rate is small. [ABSTRACT FROM AUTHOR]

In this paper, we consider the complex dynamics of a fiscal dynamical model, which was improved from Wolfstetter classical growth cycle model by Sportelli et al. The main work of the present paper is to study the impact of fiscal policy delays on the national income adjustment processes using a dynamical method, such as double Hopf bifurcation analysis. We first use DDE-BIFTOOL to find the double Hopf bifurcation points of the system, and draw the bifurcation diagrams with two bifurcation parameters, i.e. the tax collection delay τ 1 and the public expenditure decision-making delay τ 2 . Then we employ the method of multiple scales to obtain two amplitude equations. By analyzing these amplitude equations, we derive the classification and unfolding of these double Hopf bifurcation points. And three types of double Hopf bifurcations are found. Finally, we verify the results by numerical simulations. We find complex dynamic behaviors of the system via the analytical method, such as stable equilibrium, stable periodic, quasi-periodic and phase-locked solutions in respective regions. The dynamical phenomena can help policy makers to choose a proper range of the delays so that they could effectively formulate fiscal policies to stabilize the economy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics of the Sel'kov–Schnakenberg system. The stability of the positive constant steady state is studied by the linear stability theory. Hopf bifurcation and Turing–Hopf bifurcation are generated by varying two parameters in the model. The normal form near the Turing–Hopf singularity is calculated to explore the complex dynamics of the system. Finally, numerical simulations are carried out to verify the theoretical results. Our results show that the Sel'kov–Schnakenberg system exhibits complex dynamics near the Turing–Hopf singularity, including the existence of inhomogeneous steady states, homogeneous periodic solutions and inhomogeneous periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, a Leslie–Gower model with weak Allee effect on prey and fear effect on predator is proposed. Compared with the case without fear effect on predator, the new model undergoes richer dynamic behaviors such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Also, different from the strong Allee effect on prey, the system with weak Allee effect has bistable attractors which are a largely stable limit cycle and a stable positive equilibrium, two stable equilibria, or a stable limit cycle and a stable trivial equilibrium. When the Allee effect coefficient is intermediate, fear effect on the predator can protect the prey and the predator from being extinguished. The results in this paper can be seen as a complement to those in the literatures about the Leslie–Gower model with Allee effect and fear effect. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of a predator–prey model with cooperation and Allee effect considering both deterministic and stochastic approaches. The main aim of this paper is to investigate the effect of environmental fluctuation in a deterministic predator–prey model. During the analysis of the deterministic model, it is shown that the system has saddle-node point of co-dimension 1, Hopf bifurcation and Bogdanov–Takens bifurcation of co-dimension 2. To study the effect of environmental fluctuation, we use perturbation to the birth rate of prey and death rate of predator density by Gaussian white noise. The persistence of the model and the stationary distribution is shown by forming a suitable Lyapunov function. Finally, numerical simulations are performed to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper proposes a diffusive predator–prey model with Allee effect, time delay and anti-predator behavior. First, the existence and stability of all equilibria are analyzed and the conditions for the appearance of the Hopf bifurcation are studied. Using the normal form and center manifold theory, the formulas which can determine the direction, period and stability of Hopf bifurcation are obtained. Numerical simulations show that the Allee effect can determine the survival abundance of the prey and predator populations, and anti-predator behavior can greatly improve the stability of the coexisting equilibrium. [ABSTRACT FROM AUTHOR]

From the point of view of nonlinear dynamical systems, some neurological disorders can be indicated by bifurcations because bifurcations change the firing patterns of neurons; therefore, it is essential to control the bifurcations. We can avoid undesirable dynamical behaviors such as the behaviors of the Rulkov map model by controlling bifurcation which, then, can assist in modeling neuronal diseases. In this paper, we investigate the existence of Hopf bifurcation and analytically identify the type of bifurcation for the Rulkov map model; then, we apply a dynamic feedback controller using a washout filter to control the onset of Hopf bifurcation. Also, we can control the behaviors of the neurons, such as spiking or spiking-bursting behavior of neurons, and create the Hopf bifurcation for some parameters. The results analytically obtained in this paper can be applied to control some epileptic seizures. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of tumor–immune systems perturbed by the environmental noise and focus on the longtime behaviors. The existence and uniqueness of the globally positive solution to the tumor–immune system are proved using stochastic Lyapunov analysis and Itô's formula. We study the boundedness of moments for tumor cells and effector cells. By considering the dynamics on the boundary, applying the comparison theorem and the strong ergodic theorem, we obtain a threshold λ which is used to characterize the stochastic permanence in the sense that there is a unique invariant measure and extinction of the stochastic tumor–immune system. We also give biological interpretations about our analytical results of stochastic system. In addition, we present numerical examples and discussions to illustrate our analysis results. We find that the small noises preserve Hopf bifurcation of the deterministic system in stochastic setting and study numerically how the stochastic Hopf bifurcation with parameters occurs. [ABSTRACT FROM AUTHOR]

In this paper, a size-structured population model is studied. Choosing the coefficient of birth function as the varying parameter, we prove the existence of Hopf bifurcation and discuss the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The methods used in this paper include Hopf bifurcation theorem, center manifold theorem and normal form theory for the abstract Cauchy problems in a nondense domain. Numerical simulations are finally carried out to show the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a model called the k-nearest neighbor queue with the possibility of having delayed queue length feedback. We prove fluid limits for the stochastic queueing model and show that the fluid limit converges to a system of delay differential equations. Using the properties of circulant matrices, we derive a closed form expression for the value of the critical delay, which governs whether the delayed information will induce oscillations or a Hopf bifurcation in our queueing system. [ABSTRACT FROM AUTHOR]

In this paper, Myc/E2F/miR-17-92 network under Gaussian white noise is studied. Taking the time delay as the parameter, the Hopf bifurcation of the system is obtained, which causes the protein concentration to oscillate periodically. Under the influence of time delay and noise, the stochastic D-bifurcation of the system is obtained. It is worth noting that the occurrence of stochastic P-bifurcation is successfully captured. Thus a pattern of coexistence of high and low protein concentrations is founded in the network. The specific research methods of this paper are as follows: firstly, the system is reduced to a finite dimensional system by using stochastic center manifold and normal form theory. Then, using the stochastic averaging method, the Fokker–Planck–Kolmogorov equation of the system is constructed in which the statistical response in the stationary state is the probability density. Finally, the stochastic bifurcation analysis and numerical simulation are carried out. The agreements between the analytical method and those obtained numerically validate the effectiveness of the analytical investigations. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the classical SIR epidemic model by replacing the simple bilinear transmission rate by a nonlinear one. Our results show that in the presence of environmental fluctuations represented by Brownian motion and that mainly act on the transmission rate, the generalized non-concave force of infection adopted here, greatly affects the long-time behavior of the epidemic. Employing the Markov semigroup theory, we prove that the model solutions do not admit a unique stationary distribution but converge to 0 in p th moment for any p > 0. Furthermore, we prove that the disease extinguishes asymptotically exponentially with probability 1 without any restriction on the model parameters and we also determine the rate of convergence. This is an unexpected qualitative behavior in comparison with the existing literature where the studied epidemic models have a threshold dynamics behavior. It is also a very surprising behavior regarding the deterministic counterpart that can exhibit a rich qualitative dynamical behaviors such as backward bifurcation and Hopf bifurcation. On the other hand, we show by several numerical simulations that as the intensity of environmental noises becomes sufficiently small, the epidemic tends to persist for a very long time before dying out from the host population. To solve this problem and to be able to manage the pre-extinction period, we construct a new process in terms of the number of infected and recovered individuals which admits a unique invariant stationary distribution. Finally, we discuss the obtained analytical results through a series of numerical simulations. [ABSTRACT FROM AUTHOR]

In comparison with numerous available results with regard to bifurcations on account of fractional-order neural networks (FONNs) with the retarded delays, this paper is concerned with the stability and bifurcation of FONN with neutral delays. Delay-induced bifurcation results are firstly, procured and the bifurcation diagrams with respect to different delays are nicely plotted. Then it reveals that both time delay and fractional order have important effects in affecting the stability and bifurcation of the developed FONN. It further detects that FONN involving neutral delays can postpone the onset of bifurcations of the developed FONN compared with FONN with the retarded delays. It hints that neutral delays have a vital impact on the dynamics of FONNs in comparison with FONN with the retarded delays, which has outstanding performance in maintaining stability of FONNs. Eventually, numerical examples are addressed to check the availability of the developed results. The obtained criterion can be evaluated as an important contribution to the stability issue of the considered neural networks (NNs). [ABSTRACT FROM AUTHOR]

This paper presents a brucellosis disease model with reaction–diffusion and time delay. The model takes into account both the direct and indirect transmission of infected animals and pathogens in the environment. By analyzing the associated characteristic equation, the local stability of the unique positive equilibrium point is established. The existence of Hopf bifurcations at the positive equilibrium point is also examined by considering the discrete time delay as a bifurcation parameter. Additionally, an optimal control analysis is conducted to minimize disease outbreaks and control costs. This includes reducing the exposure of susceptible animals to infected animals, removing infected animals from herds, and reducing emissions of brucella into the environment. By constructing Hamiltonian function and applying Pontryagin’s maximum principle, the necessary conditions for the existence of optimal control are given. Finally, the existence of bifurcation periodic solutions and the effectiveness of control strategies are illustrated through numerical simulations. [ABSTRACT FROM AUTHOR]

This paper proposes a novel nonuniform continuous traffic flow model, which takes into account the differences in drivers' expected headway and further improves the model by incorporating the influences of relative distance and relative velocity of vehicles through linear weighting. Additionally, the model also considers the dynamic response time of drivers for further refinement. With this model, bifurcation theory can also be applied in the stability analysis of traffic systems to study the stability mutation behavior of traffic systems at bifurcation points. Through linear and nonlinear analysis methods, the stability conditions of the model and the KdV–Burgers equation can be derived, the type of equilibrium point of the model can be judged, the conditions for the existence of Hopf bifurcations and the type of bifurcations can be proved, and the traffic flow problem can be transformed into the stability analysis problem of the system. Through numerical simulations in both high-density and low-density scenarios, it is shown that the model can well describe the actual traffic phenomena and can describe the stability abrupt behavior of the traffic system at Hopf bifurcation points and saddle-node bifurcation points through density-time and phase plane diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Leslie–Gower type reaction–diffusion predator–prey system with an increasing functional response. We mainly study the effect of three different types of diffusion on the stability of this system. The main results are as follows: (1) in the absence of prey diffusion, diffusion-driven instability can occur; (2) in the absence of predator diffusion, diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable; (3) in the presence of both prey diffusion and predator diffusion, the system can occur diffusion-driven instability and Turing patterns. At the same time, we also get the existence conditions of the Hopf bifurcation and the Turing–Hopf bifurcation, along with the normal form for the Turing–Hopf bifurcation. In addition, we conduct numerical simulations for all three cases to support the results of our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a delayed cytokine-enhanced viral infection model incorporating nonlinear incidence and immune response is proposed and studied. The global stabilities of the equilibria of the model are characterized by constructing suitable Lyapunov functionals. The existence and properties of local Hopf bifurcation are discussed by regarding the immunity delay as the bifurcation parameter. Moreover, the global existence of Hopf bifurcation has been proved by regarding the immune delay as the bifurcation parameter. Numerical simulations are carried out to validate the obtained results. The results show that ignoring the cytokine-enhanced effect makes the infection risk underevaluated. The simulations show that increasing the immune delay destabilizes the model and generates a Hopf bifurcation and stability switches occurs. Moreover, it shows that immune delay may dominate the intracellular delays in such a viral infection model which means that the immune system of the host itself is complicated during virus infection. [ABSTRACT FROM AUTHOR]

Motivated by the study of multiple time scale system dynamics, we analyze the Murali–Lakshmanan–Chua (MLC) dissipative circuit exhibiting periodic bursting with different types of oscillations in this paper. Such patterns can be analyzed by treating the slow variable as a parameter of the nonsmooth fast subsystem. The discontinuous bifurcation structure of the subsystem can be described by the generalized differential of Clarke and the Jacobian matrix of the system. Bursting patterns as well as the oscillation mechanisms can be clarified by the discontinuous bifurcations, time series and phase portraits. For the chosen circuit parameter values, this designed circuit admits both MLC type bursting attractors and Duffing–van der Pol circuit type bursting attractors. It is found that not only the features combined with the circuit parameters of the system, but also the two switching boundaries may have an important impact on the origin of the bursting behaviors. In particular, the explicit analytic solutions of the proposed circuit are investigated in detail. We also show that the bursting can arise from either discontinuous saddle-node bifurcation or slow passage through a discontinuous Hopf bifurcation. Finally, the validity of the theoretical analysis has been well verified by the numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a spatial S − I epidemic model with nonlinear incidence rate. First, the existence of the equilibrium is discussed in different conditions. Then the main criteria for the stability and instability of the constant steady-state solutions are presented. In addition, the effect of diffusion coefficients on Turing instability is described. Next, by applying the normal form theory and the center manifold theorem, the existence and direction of Hopf bifurcation for the ordinary differential equations system and the partial differential equations system are given, respectively. The bifurcation diagrams of Hopf and Turing bifurcations are shown. Moreover, a priori estimates and local steady-state bifurcation are investigated. Furthermore, our analysis focuses on providing specific conditions that can determine the local bifurcation direction and extend the local bifurcation to the global one. Finally, the numerical results demonstrate that the intrinsic growth rate, denoted as r , has significant influence on the spatial pattern. Specifically, different patterns appear, with the increase of r. The obtained results greatly expand on the discovery of pattern formation in the epidemic model. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear stochastic Duffing oscillators with multidelay feedback. We propose an effective reduction approach with the help of center manifold theory and stochastic averaging method. Taking the initial time-delay τ as the parameter, we reduce the original system to a one-dimensional averaged Itô equation. Our analysis reveals that the original system exhibits stochastic bifurcations, including stochastic D and P bifurcations. Once we have a clear understanding of the bifurcation structure, we can use this knowledge to choose appropriate system parameters and place the system in the desired state. For instance, by adjusting the initial time-delay τ of the control system, we can stabilize the system and achieve the desired outcome. Numerical simulations also verify the theoretical results. With appropriate parameter choices, multiple time delays can destabilize the equilibrium and promote chaotic behaviors, and can also lead to more stable dynamical behavior. Remarkably, we discovered that increasing the interval of time delays and feedback numbers can enhance system stability. It may potentially serve as a novel mechanism for stabilizing stochastic systems. The study provides a solid theoretical foundation for exploring stochastic systems subject to complex time-delay feedback control, and offers a valuable framework for related fields. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a quadratic chaotic system modeling self-excited and hidden attractors which is described by a system of three nonlinear ordinary differential equations with three real parameters. The primary goal is to establish the existence of two limit cycles that bifurcate based on the system's nature as an electronic circuits model, specifically via Hopf bifurcation. Notably, the application of the first and second Lyapunov coefficients is utilized to demonstrate the bifurcation of two limit cycles from an equilibrium point near a Hopf critical point. Furthermore, employing the first-order averaging theory enables us to confirm the presence of unstable periodic orbits originating from the zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]

This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium. [ABSTRACT FROM AUTHOR]

A two-degree-of-freedom nonlinear high-speed railway wheelset model with two time delays in the lateral and yaw dampers is studied. The aim is to investigate the effect of time delays on stability and Hopf bifurcation characteristics of the wheelset model. The local stability of the trivial equilibrium under different time delay conditions is qualitatively analyzed. Analytical studies reveal that the wheelset model undergoes stability switches with the variation of the time delays. The stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Furthermore, properties of Hopf bifurcation including direction and stability of bifurcating limit cycles are studied by using the normal form theory and the center manifold theorem. Our findings indicate that time delays in both lateral dampers and yaw dampers influence the stability and direction of Hopf bifurcation. Additionally, the numerical results show that time delays in the lateral and yaw dampers not only affect the amplitude of the hunting motion of the wheelset but also the periodic and chaotic motions. If the time delays gradually increase, the wheelset will vibrate irregularly with large lateral displacements. The analytical results presented in this paper offer a theoretical reference for the stability design of wheelsets. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

In this paper, a predator–prey system with multiple anti-predator behaviors is developed and studied, where not only the prey may spread between patches but also the fear effect and counter-attack behavior of the prey are taken into account. First, the stability and existence of coexistence equilibria are presented. The unique positive equilibrium may be a saddle-node or a cusp of codimension 2. Then, various transversality conditions of bifurcations such as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation are obtained. Moreover, compared with a single strategy, the multiple anti-predator strategies are more beneficial to the persistence and the population density of prey. [ABSTRACT FROM AUTHOR]

In this paper, we designed a population model that shows how a prey species defends itself against a generalist predator by exhibiting group defence. A non-monotonic functional response is used to represent the group defence functionality. We have demonstrated the model's local stability in the vicinity of the coexisting equilibrium solution employing a local Lyapunov function. Condition for existence of Hopf bifurcation is obtained along with its normal form. The suggested model has been validated by numerical simulations, which have also been used to verify the acquired analytical results. The parameters are subjected to sensitivity analysis by utilizing partial rank correlation coefficient (PRCC) and Latin hypercube sampling (LHS). The Z-type dynamic method is used to prevent population blow-up. [ABSTRACT FROM AUTHOR]