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The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic. [ABSTRACT FROM AUTHOR]

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model. [ABSTRACT FROM AUTHOR]

It is well known that for a large class of coagulation kernels, Smoluchowski coagulation equations have particular power law solutions which yield a constant flux of mass along all scales of the system. In this paper, we prove that for some choices of the coagulation kernels there are solutions with a constant flux of mass along all scales which are not power laws. The result is proved by means of a bifurcation argument. [ABSTRACT FROM AUTHOR]

In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problems to determine the stability of the constant steady states. We then perform bifurcation analysis by taking the perceptual diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackling a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics. [ABSTRACT FROM AUTHOR]

This paper investigates the complex dynamical behavior of a discrete prey-predator system with a fear factor, a strong Allee effect, and prey refuge. The existence and stability of fixed points in the system are discussed. By applying the central manifold theorem and bifurcation theory, we have established the occurrence of various types of bifurcations, including flip bifurcation and Neimark-Sacker bifurcation. Furthermore, to address the observed chaotic behavior in the system, three controllers were designed by employing state feedback control, OGY feedback control, and hybrid control methods. These controllers serve to control chaos in the proposed system and identify specific conditions under which chaos or bifurcations can be stabilized. Finally, the theoretical analyses have been validated through numerical simulations. [ABSTRACT FROM AUTHOR]

The paper discusses the nutrient-plankton system with effect of time delay in nutrient recycling and toxin-determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient-plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system. [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]

In this paper, we investigate the (2+1)-dimensional Kadomtsev-Petviashvili-Benjamin-Bona Mahony equation using two effective methods: the unified scheme and the advanced auxiliary equation scheme, aiming to derive precise wave solutions. These solutions are expressed as combinations of trigonometric, rational, hyperbolic, and exponential functions. Visual representations, including three-dimensional (3D) and two-dimensional (2D) combined charts, are provided for some of these solutions. The influence of the nonlinear parameter p on the wave type is thoroughly examined through diverse figures, illustrating the profound impact of nonlinearity. Additionally, we briefly investigate the Hamiltonian function and the stability of the model using a planar dynamical system approach. This involves examining trajectories, isoclines, and nullclines to illustrate stable solution paths for the wave variables. Numerical results demonstrate that these methods are reliable, straightforward, and potent tools for analyzing various nonlinear evolution equations found in physics, applied mathematics, and engineering. [ABSTRACT FROM AUTHOR]

In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order α , and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, β , changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns. [ABSTRACT FROM AUTHOR]

The dynamics and vibration reduction performance of bistable nonlinear energy sink (BNES) are studied in this paper. First, the negative stiffness of BNES is realized by geometric nonlinearity, and the dynamics model of the system is established. The slow flow equation of the system under 1:1 main resonance is analyzed based on the complexification-averaging (CX-A) method, and the boundary conditions of saddle node bifurcation and Hopf bifurcation are analyzed. Second, the slow invariant manifold (SIM) of the BNES is studied based on multiscale analysis, and the excitation threshold of strongly modulated response (SMR) is analyzed; the analysis results are verified by the numerical method. The results show that the analytical solution is highly consistent with the numerical solution, and the error is less than 1%. Finally, the influence of structural parameters on the vibration reduction performance is analyzed and optimized. The vibration reduction performance of BNES and CNES is compared, and the results show that the BNES has better vibration reduction performance in the full frequency band. [ABSTRACT FROM AUTHOR]

In this paper, we carry out some research on a predator–prey system with maturation delay, a stage structure for generalist predators and a Holling type-II functional response, which has already been proposed. First, for the delayed model, we obtain the conditions for the occurrence of stability switches of the positive equilibrium and possible Hopf bifurcation values owing to the growth of the value of the delay by applying the geometric criterion. It should be pointed out that when we suppose that the characteristic equation has a pair of imaginary roots λ = ± i ω (ω > 0) , we just need to consider i ω (ω > 0) due to the symmetry, which alleviates the computation requirements. Next, we investigate the nature of Hopf bifurcation. Finally, we conduct numerical simulations to verify the correctness of our findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to a mathematical model with diffusion and cross-diffusion to describe the interaction between vegetation and soil water. First, the existence of Hopf bifurcation and cross-diffusion-driven Turing instability are discussed. Then, based on the nonlinear analysis, we obtain the exact parameters range for stationary patterns and show the dynamical behavior near Turing bifurcation point. It is found that the model has the properties of gap, strip and spot patterns. Moreover, the small water-uptake ability of vegetation roots promotes the growth of vegetation and the transitions of vegetation pattern. But with the continuous increase of the water-uptake ability of vegetation roots, the local vegetation biomass density increases and the isolation between vegetation patches also increases, which may induce the emergence of desertification. In addition, our results reveal that the water consumption rate induces the transitions of vegetation pattern and prohibits the increase of vegetation biomass density. [ABSTRACT FROM AUTHOR]

Purpose: This paper mainly studies the double Hopf bifurcation for coupled van der Pol–Rayleigh system with time delay. It aims to see how time delay and coupling factors affect the system, then avoid unstable or harmful vibrations and look for steady state in engineering. Methods: This paper mainly uses the normal form and center manifold theory and the Runge–Kutta method. First, we analyze the stability switching of the coupled van der Pol–Rayleigh system with time delay and the dynamical behavior caused by the double Hopf bifurcation. Secondly, we derive the normal form of double Hopf bifurcation by the normal form and center manifold theory. Then the dynamical phenomena of the system are investigated. Finally, we perform numerical simulations to verify the theoretical analysis by the Runge–Kutta method. Results: The stability switching region of the system with respect to the delay is obtained. And the parameter regions of amplitude death, stable periodic motion and quasi-periodic motion are obtained. And as parameter values change, the steady state of the periodic solution follows. Conclusions: In the field of engineering, it is more practical to consider systems with time delay. So we consider system with time delay, obtain the stability switching region about the time delay and discuss the dynamics of each bifurcation region. When doing some system design, it is necessary to avoid the occurrence of bifurcation behavior, because it will lead to changes in system stability. We can adjust the system parameters to avoid bifurcation according to stability region. Or some engineering problem requires the avoidance of harmful vibrations and the pursuit of steady state. With the change of time delay and coupling factors, we can find the amplitude death region and the region of stable periodic motion and quasi-periodic motion, thus avoiding harmful vibration in practical engineering by adjusting the parameters. [ABSTRACT FROM AUTHOR]

This paper exploits the concept of nonlinear sliding surfaces to be used as a basis in the development of aerial path planning projects involving aerobatic three-dimensional path curves in the presence of disturbances. This approach can be used for any kind of unmanned aerial vehicle aimed at performing aerobatic maneuvers. Each maneuver is associated with a nonlinear surface on which an aerial vehicle could be driven to slide. The surface design exploits the properties of Viviani's curve and the Hopf bifurcation. A vector form of the super twisting algorithm steers the vehicle to the prescribed surfaces. A suitable switching control law is proposed to shift between surfaces at different time instants. A practical stability analysis that involves the descriptor approach allows for determining the controller gains. Numerical simulations are developed to illustrate the accomplishment of the suggested aerobatic flight. [ABSTRACT FROM AUTHOR]

The protection of forests and the mitigation of pest damage to trees play a crucial role in mitigating the greenhouse effect. In this paper, we first establish a delayed differential equation model for Ips subelongatus Motschulsky-Larix spp., where the delay parameter represents the time required for trees to undergo curing. Second, we analyze the stability of the equilibrium of the model and derive the normal form of Hopf bifurcation using a multiple-time-scales method. Then, we analyze the stability and direction of Hopf bifurcating periodic solutions. Finally, we conduct simulations to analyze the changing trends in pest and tree populations. Additionally, we investigate the impact of altering the rate of artificial planting on the system and provide corresponding biological explanations. [ABSTRACT FROM AUTHOR]

In this paper, a generalized fractional three-species food chain model with delay is investigated. First, the existence of a positive equilibrium is discussed, and the sufficient conditions for global asymptotic stability are given. Second, through selecting the delay as the bifurcation parameter, we obtain the sufficient condition for this non-control system to generate Hopf bifurcation. Then, a nonlinear delayed feedback controller is skillfully applied to govern the system's Hopf bifurcation. The results indicate that adjusting the control intensity or the control target's age can effectively govern the bifurcation dynamics behavior of this system. Last, through application examples and numerical simulations, we confirm the validity and feasibility of the theoretical results, and find that the control strategy is also applicable to eco-epidemiological systems. [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]

Addressing the issue of system stability is a crucial step towards the successful integration of memristive devices in power electronics applications. This paper focuses on fluorescent lamps possessing memristive characteristics and investigates the instability phenomena and mechanisms within a full-bridge inverter with a fluorescent lamp load. Based on the memristive characteristics of fluorescent lamps, this paper establishes the averaged model of the system, whose coefficient matrix is nonlinear, periodic, and time-varying. This study identifies the occurrence of low-frequency oscillations within the system and elucidates the fundamental mechanism underlying the emergence of low-frequency oscillations. Furthermore, this paper establishes the stability boundaries of the system across different parameter planes. The research findings indicate that the low-frequency oscillations within the system are attributed to the occurrence of Hopf bifurcations in a frequency range higher than line frequency but significantly lower than switching frequency. Lastly, the PSpice circuit of the system is designed, and simulation results are provided for validation. This study can offer guidance on parameters and control strategies for ensuring the stable operation of a full-bridge inverter with fluorescent lamps. Moreover, it can facilitate the comprehension of instability mechanisms in systems incorporating memristive devices, thereby offering a foundation for the expansion of memristor applications. [ABSTRACT FROM AUTHOR]

A nonlinear phenomenon in nature is often modeled by a system of differential equations with parameters. The bifurcation occurs when a parameter varies in such systems, causing a qualitative change in its solution. In this paper, we study one of the most exciting bifurcations, which is Hopf bifurcation. We use tools from algebraic topology to analyze and reveal supercritical and subcritical Hopf bifurcations. [ABSTRACT FROM AUTHOR]

Memristor is a kind of nonlinear resistance with memory function, which is a typical nonlinear circuit component. In this paper, the problem of a simple chaotic circuit based on the memristor was studied. We constructed a new simple circuit of multiple memorizing components, summarized the mathematical model and found seven types of chaotic attractors without reference to classical circuits that substituted memristors for elements in the original circuit. Traditional methods of dynamic analysis are used to analyze the equilibrium point and stability of the system. Furthermore, we analyzed the dynamical behavior with the varying coefficient of the system specifically. In order to reduce the deviation from the actual physical circuit system as much as possible so as to facilitate the follow-up research of scholars, we solved the problem of Hopf bifurcation in this system under the condition of time delay through the use of the canonical form and Hopf bifurcation theory. Theoretical analysis and simulation results show that there existed the state transition in seven chaotic attractors of system and the bifurcation was stable when we changed the parameter τ slightly. As a small unit circuit, this paper lays a foundation for the research and control of large-scale system with memorizing components. [ABSTRACT FROM AUTHOR]

We consider a mathematical continuous-time model for biodegradation of 4-chlorophenol and sodium salicylate mixture by the microbial strain Pseudomonas putida in a chemostat. The model is described by a system of three nonlinear ordinary differential equations and is proposed for the first time in the paper [Y.-H. Lin, B.-H. Ho, Biodegradation kinetics of phenol and 4-chlorophenol in the presence of sodium salicylate in batch and chemostat systems, Processes, 10:694, 2022], where the model is only quantitatively verified. This paper provides a detailed analysis of the system dynamics. Some important basic properties of the model solutions like existence, uniqueness and uniform boundedness of positive solutions are established. Computation of equilibrium points and study of their local asymptotic stability and bifurcations in dependence of the dilution rate as a key model parameter are also presented. Thereby, particular intervals for the dilution rate are found, where one or three interior (with positive components) equilibrium points do exist and possess different types of local asymptotic stability/instability. Hopf bifurcations are detected leading to the occurrence of stable limit cycles around some interior equilibrium points. A transcritical bifurcation also exists and implies stability exchange between an interior and the boundary (washout) equilibrium. The results are illustrated by lots of numerical examples. [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]

Canards are a type of transient dynamics that occur in singularly perturbed systems, and they are specific types of solutions with varied dynamic behaviours at the boundary region. This paper introduces the emergence and development of canard phenomena in a neuron model. The singular perturbation system of a general neuron model is investigated, and the link between the transient transition from a neuron model to a canard is summarised. First, the relationship between the folded saddle-type canard and the parabolic burster, as well as the firing-threshold manifold, is established. Moreover, the association between the mixed-mode oscillation and the folded node type is unique. Furthermore, the connection between the mixed-mode oscillation and the limit-cycle canard (singular Hopf bifurcation) is stated. In addition, the link between the torus canard and the transition from tonic spiking to bursting is illustrated. Finally, the specific manifestations of these canard phenomena in the neuron model are demonstrated, such as the singular Hopf bifurcation, the folded-node canard, the torus canard, and the "blue sky catastrophe". The summary and outlook of this paper point to the realistic possibility of canards, which have not yet been discovered in the neuron model. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of bifurcation analysis and bifurcation control of a fractional-order gene regulatory network with sRNA. Firstly, the process of stability change of system equilibrium under the influence of the sum of time delay is discussed, the critical condition of Hopf bifurcation is explored, and the effect of fractional order on the system stability domain. Secondly, aiming at the system's instability caused by a large time delay, we design a controller to improve the system's stability and derive the parameter conditions that satisfy the system's stability. It is found that changing the parameter values of the controller within a certain range can control the system's nonlinear behaviours and effectively expand the stability range. Then, a numerical example is given to illustrate the results of this paper. [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]

Forest pests and diseases can diminish forest biodiversity, damage forest ecosystem functions, and have an impact on water conservation. Therefore, it is necessary to analyze the interaction mechanism between plants and pests. In this paper, the prevention and control of a specific pest—namely the larva of Paranthrene tabaniformis (Rott.) (hereinafter referred to as larva)—are studied. Based on the invasion mechanism of the larva in poplar, we establish a delayed differential equation and analyze the existence and stability of equilibria. Next, we assess the existence of a Hopf bifurcation to determine the range of parameters that ensures that the equilibria are stable. Then, we select a set of parameters to verify the results of the stability analysis. Finally, we provide biological explanations and effective theoretical control methods for poplar pests and diseases. [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]

A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics of a diffusive nutrientphytoplankton model with delayed nutrient recycling. We first study the stability of positive equilibrium and Turing instability induced by diffusion. We then investigate the effect of delay, and it turns out that the value of the rate of recycling k plays an important role in the Hopf bifurcation induced by delay. The delay will and will not induce Hopf bifurcation with low and high level of k, respectively. To reveal the spatiotemporal dynamics, Turing–Hopf bifurcation is carried out, and normal form is derived. Many spatiotemporal dynamics are found, including the coexistence of two stable spatially inhomogeneous periodic solutions or two stable spatially inhomogeneous steadystate solutions. [ABSTRACT FROM AUTHOR]

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R 0 h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R ˜ 0 h , is rigorously investigated. More specifically, if R ˜ 0 h < 1 , the disease-free equilibrium is globally asymptotically stable; if R ˜ 0 h > 1 , the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R 0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1. [ABSTRACT FROM AUTHOR]

In this paper, a study is carried out on the spatiotemporal dynamics of a Lengyel--Epstein model describing the chlorite-iodine-malonic-acid (CIMA) reaction with time delay and the Neumann boundary condition in a two-dimensional region. The existences for Turing, Hopf, Turing--Turing, Turing--Hopf, and Bogdanov--Takens bifurcations are derived by analyzing the dispersion relation between eigenvalues and wave numbers. In particular, to study the dynamics around a Turing--Hopf bifurcation singularity, the amplitude equations near a codimension-two bifurcation point are derived by employing the weakly nonlinear analysis method. Different spatiotemporal patterns for the system in parameter space are classified and various patterns identified, including spatially homogeneous periodic solutions, mixed mode, coexistence mode, bistable phenomenon, square, hexagon, black eye, two-phase oscillating staggered hexagon lattice, and other complex spatiotemporal patterns. The theoretical predictions are verified by numerical simulations showing an excellent agreement with many reported experiment results not only in chemistry but also in physics and biology. Results presented in this article reveal the mechanism of generating the spatiotemporal patterns of the CIMA reaction. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the spatiotemporal pattern formations and bifurcations for a nondegenerate reaction-diffusion rabies SIR model which was used to explain the epidemiological patterns observed in Europe. First, by using the iteration methods, we are able to show the global existence and boundedness of in-time solutions of the parabolic system. Second, for the ODEs, we analytically prove the phenomena observed by Anderson et al. [Nature, 289 (1981), pp. 765-771]: if the carrying capacity k is smaller than some positive k*, then rabies eventually dies out; if k is larger than k*, then the rabies prevails. Moreover, if k* < k < k for some positive ... > k*, then the endemic equilibrium solution is (locally asymptotically) stable, while it is unstable if k > ... In particular, at k = ..., the loss of the stability of the endemic equilibrium leads to a Hopf bifurcation. Finally, for the PDEs, we derive sufficient conditions on the diffusion rates so that under these conditions, Turing instability of both the endemic equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions can occur. Once Turing instability of the solution (equilibrium or periodic solution) occurs, it is observed numerically that the system might have new spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the periodic pattern formations with spatial multi-peaks in a classic diffusive Holling-Tanner predator-prey model with nonlocal intraspecific prey competition. The main innovation is that a spatial dependently kernel is considered in the nonlocal effect, which mathematically complicates the linear stability analysis. We first generate the existences of Hopf, Turing, Turing-Hopf and double-Hopf bifurcations, and determine the stability of the positive equilibrium. It turns out that the stable parameter region for the positive equilibrium decreases with α increasing, which implies that the parameter region of pattern formation for such kernel is smaller than the spatial average case. For double-Hopf bifurcation, we calculate the normal form up to the third-order term restricted on the center manifold, which is expressed by the original parameters of the system. Via analyzing the equivalent amplitude equations, the system exhibits stable spatially nonhomogeneous periodic patterns, the bistability of such periodic solutions, as well as unstable spatially nonhomogeneous quasi-periodic solutions, all of them possess multiple spatial peaks. Interestingly, some possible strange attractors are found numerically near the double-Hopf singularity. Biologically, the emerging spatio-temporal patterns imply that such nonlocal intraspecific competition can promote the coexistence of the prey and predator species in the form of more complex periodic states. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations. [ABSTRACT FROM AUTHOR]

Vaccination is widely acknowledged as an affordable and cost-effective approach to guard against infectious diseases. It is important to take vaccination rate, vaccine effectiveness, and vaccine-induced immune decline into account in epidemic dynamical modeling. In this paper, an epidemic dynamical model of vaccination is developed. This model provides a framework of the infectious disease transmission dynamics model through qualitative and quantitative analysis. The result shows that the system may have multiple equilibria. We used the next-generation operator approach to calculate the maximum spectral radius, that is, basic reproduction number Rvac. Next, by dividing the model into infected and uninfected subjects, we can prove that the disease-free equilibrium is globally asymptotically stable when Rvac < 1, provided certain assumptions are satisfied. When Rvac > 1, there exists a unique endemic equilibrium. Using geometric methods, we calculate the second compound matrix and demonstrate the Lozinskii measure ... ≤ 0, which is equivalent to the unique endemic equilibrium, which is globally asymptotically stable. Then, using center manifold theory, we justify the existence of forward bifurcation. As the vaccination rate decreases, the likelihood of forward bifurcation increases. We also theoretically show the presence of Hopf bifurcation. Then, we performed sensitivity analysis and found that increasing the vaccine effectiveness rate can curb the propagation of disease effectively. To examine the influence of vaccination on disease control, we chose the vaccination rate as the optimal vaccination control parameter, using the Pontryagin maximum principle, and we found that increasing vaccination rates reduces the number of infected individuals. Finally, we ran a numerical simulation to finalize the theoretical results. [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]

Purpose: The bifurcating solution generated by Neimark-Sacker bifurcation of map exhibits a typical quasi-periodic oscillation behavior, which is beneficial for the active use of bifurcation in bifurcation control field. In this paper, an impulsive state feedback control method is proposed to create the quasi-periodic oscillation with certain desired characteristic in the vibro-impact system. Method: The linear control gains determined by using the explicit criteria of Neimark-Sacker bifurcation without directly using eigenvalues are designed to guarantee the existence of quasi-periodic oscillation. On the other hand, the non-linear control gains, which are determined by using center manifold reduction theory and normal form approach, ensure that the created quasi-periodic oscillation is stable. The amplitude of the trajectory section on the torus of quasi-periodic oscillation is analytically derived to achieve the control of the amplitude of invariant torus section by selecting appropriate non-linear control gains. Results and conclusions: Numerical experiments are provided to show the effectiveness of the proposed control method in a two-degree-of-freedom vibro-impact system. The results of quantitative analysis show that a stable quasi-periodic oscillation in the vibro-impact system can be created at a preassigned bifurcation point by desiging linear and nonlinear control gains, and the effect of the nonlinear control gains on the amplitude is either a positive relationship or a negative correlation to the amplitude. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker's type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter. [ABSTRACT FROM AUTHOR]

Given that the Crimean and Congo hemorrhagic fever is one of the deadly viral diseases that occur seasonally due to the activity of the carrier "tick," studying and developing a mathematical model simulating this illness are crucial. Due to the delay in the disease's incubation time in the sick individual, the paper involved the development of a mathematical model modeling the transmission of the disease from the carrier to humans and its spread among them. The major objective is to comprehend the dynamics of illness transmission so that it may be controlled, as well as how time delay affects this. The discussion of every one of the solution's qualitative attributes is included. According to the established basic reproduction number, the stability analysis of the endemic equilibrium point and the disease-free equilibrium point is examined for the presence or absence of delay. Hopf bifurcation's triggering circumstance is identified. Using the center manifold theorem and the normal form, the direction and stability of the bifurcating Hopf bifurcation are explored. The next step is sensitivity analysis, which explains the set of control settings that have an impact on how the system behaves. Finally, to further comprehend the model's dynamical behavior and validate the discovered analytical conclusions, numerical simulation has been used. [ABSTRACT FROM AUTHOR]

A high spike-timing precision characterized by a small variation in interspike intervals of neurons is important for information processing in various brain functions. An experimental study on fast-spiking interneurons has shown that inhibitory autapses functioning as negative self-feedback can enhance spike-timing precision. In the present paper, bifurcation and negative self-feedback mechanisms for the enhanced spike-timing precision to stochastic modulations are obtained in two theoretical models, presenting theoretical explanations to the experimental finding. For stochastic spikes near both the saddle-node bifurcation on an invariant cycle (SNIC) and the subcritical Hopf (SubH) bifurcation with classes 1 and 2 excitabilities, respectively, enhanced spike-timing precision appears in large ranges of the conductance and the decaying rate of inhibitory autapses, closely matching the experimental observation. The inhibitory autaptic current reduces the membrane potential after a spike to a level lower than that in the absence of inhibitory autapses and the threshold to evoke the next spike, making it more difficult for stochastic modulations to affect spike timings, and thereby enhancing spike-timing precision. In addition, firing frequency near the SubH bifurcation is more robust than that near the SNIC bifurcation, resulting in a higher spike-timing precision for the SubH bifurcation. The bifurcation and negative self-feedback mechanisms for the enhanced spike-timing precision present potential measures to modulate the neuronal dynamics or the autaptic parameters to adjust the spike-timing precision. [ABSTRACT FROM AUTHOR]

In this paper, the stability and bifurcation of a two–dimensional p53 gene regulatory network without and with time delay are taken into account by rigorous theoretical analyses and numerical simulations. In the absence of time delay, the existence and local stability of the positive equilibrium are considered through the Descartes' rule of signs, the determinant and trace of the Jacobian matrix, respectively. Then, the conditions for the occurrence of codimension–1 saddle–node and Hopf bifurcation are obtained with the help of Sotomayor's theorem and the Hopf bifurcation theorem, respectively, and the stability of the limit cycle induced by hopf bifurcation is analyzed through the calculation of the first Lyapunov number. Furthermore, codimension-2 Bogdanov–Takens bifurcation is investigated by calculating a universal unfolding near the cusp. In the presence of time delay, we prove that time delay can destabilize a stable equilibrium. All theoretical analyses are supported by numerical simulations. These results will expand our understanding of the complex dynamics of p53 and provide several potential biological applications. [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]