Search

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]

Purpose: Van der Pol (VDP) oscillator is a typical self-excited system with observable limit cycle oscillations (LCOs). To suppress LCOs, a VDP oscillator coupled nonlinear energy sink (NES) is proposed in this paper. To achieve the optimal vibration reduction effect, the sensitive parameters of the VDP-NES system are investigated and its vibration reduction mechanism is revealed. Methods: First, the dynamic model of the coupled NES system is established by utilizing Newton's law. Then we reveal the vibration reduction mechanism based on stability and bifurcation theory, which may be a new angle of view to investigate the VDP oscillator with NES. Finally, the influence of different parameters on the vibration reduction effect is discussed based on numerical simulation. Results: The LCOs of VDP oscillator are caused by Hopf bifurcation, and its amplitude is closely related to the eigenvalue of the equilibrium point. When the NES is coupled, the unstable equilibrium point becomes stable, then the LCOs are completely suppressed. Although the stability of the equilibrium point of the system with NES keeps invariant, the real part of the eigenvalue decreases. As a result, the LCOs are partially suppressed, and the amplitude decreases by about 70%. In addition, the self-excited intensity and damping ratio are sensitive to the vibration reduction effect. Conclusion: Coupling NES can have a better vibration reduction effect on the LCOs, and vibration reduction is closely related to Hopf bifurcation and the equilibrium point of the system. The research shows that putting off the Hopf bifurcation to widen the parameter range of the stable equilibrium point can obtain the optimal vibration control effect. At the same time, decreasing the real part of the eigenvalue also can reduce the amplitude of LCOs. It may provide some ideas for vibration control strategies in practical engineering. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the celebrated Haken--Kelso--Bunz model, describing the dynamics of bimanual coordination, in the presence of delay. We study the linear dynamics, stability, nonlinear behavior, and bifurcations of this model by both theoretical and numerical analysis. We calculate in-phase and antiphase limit cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis and center manifold reduction. Moreover, we uncover further details on the global dynamic behavior by numerical continuation, including the occurrence of limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

This paper provides for the first time correct third-order homoclinic predictors in n-dimensional ODEs near a generic Bogdanov--Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt--Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt--Poincaré) to approximate the homoclinic solution near Bogdanov--Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont. [ABSTRACT FROM AUTHOR]

This paper investigates the complex dynamics of a ratio-dependent predator–prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation; (c) The derivation of the normal form near the Turing–Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyse a predator-prey model with disease in prey. We assume that a portion of healthy prey takes refuge to avoid predation. We find the biologically feasible equilibrium points and their stability criteria by using linearization technique. We also perform Hopf bifurcation analysis around the co-existing equilibrium point. We use substantial numerical simulation to verify our theoretical results and to investigate rich dynamics that are not possible to achieve analytically. We illustrate rich dynamics such as Hopf bifurcation. chaos, bistability, and others using one and two parameter bifurcation diagrams. We find that disease invasion in prey can produce chaos by inducing period-doubling bifurcation. but refuge can reduce chaos by causing period-halving bifurcation. We also observe that refuge can reduce the prevalence of disease in the Prey population. [ABSTRACT FROM AUTHOR]

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

In 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]

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]

Stability of a complex DC bus system (DBS) is a significant design consideration because of its constant power load (CPL) characteristics of the tightly‐regulated load converter. In this paper, the dynamic properties of a stand‐alone buck‐type DBS are investigated based upon the closed‐form large‐signal‐averaged model. The stability‐analysis‐in‐large around the operation point shows that CPLs create a destabilizing effect on the system that can lead to severe voltage oscillation caused by discontinuity‐induced Hopf bifurcations, which are qualitatively different from the bifurcations we know for smooth dynamical systems. Instead, this paper also explores the application of "amplitude death (AD) technique" to stabilize the coupled oscillatory DBSs and show that how this technique with its necessary modifications, for example, the heterogeneity‐induced AD technique characterized by two or more coupled DBSs with mismatched frequency of oscillations can provide an efficient open‐loop control solution for stabilizing the coupled DBSs. Finally, simulations and experimental results are provided to validate all these concepts. [ABSTRACT FROM AUTHOR]

The notion of competing HT ht -Laplacians with weights is considered for the first time. They are driven by a degenerated HT ht -Laplacian with weights and a competing HT ht -Laplacian with weights, respectively. Finally, the authors indicate how the result can be extended to HT ht -equations ( HT ht ). This research aims to present a linear operator HT ht by utilizing the I q i -Mittag-Leffler function and to introduce the subclass of harmonic HT ht -convex functions HT ht related to the Janowski function. [Extracted from the article]

This paper is concerned with a delayed reaction–diffusion equation on a unit disk. By means of the singularity theory and Lyapunov–Schmidt reduction, we not only derive universal conclusions about the existence of inhomogeneous steady-state solutions and the equivariant Hopf bifurcation theorems, but also obtain some more extraordinary properties of bifurcating solutions, which are produced by the radial symmetry through abstract methods based on the Lie group representation theory. Meanwhile, we illustrate our results by an application to a population model with a time delay. Furthermore, the methods established in this paper are applicable to specific delayed reaction–diffusion models with other symmetries. [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 study a predator–prey system with the simplified Holling IV functional response and antipredator behavior such that the adult prey can attack vulnerable predators. The model has been investigated by Tang and Xiao, and the existence and stability of all possible equilibria are determined. In addition, they performed a bifurcation analysis and showed that the system undergoes a Codimension 2 Bogdanov–Takens bifurcation. In this paper, for the same model, we further show that the cusp‐type Bogdanov–Takens bifurcation can be of Codimension 3, which acts as an organizing center for the whole bifurcation set. In addition, we propose the existence of Hopf bifurcation of Codimension 2 and the coexistence of stable limit cycle and unstable limit cycle. In particular, we show that the antipredator behavior has great effect on the dynamics of the model, it may cause the predator population to extinct while the prey population will increase up to the carrying capacity. Numerical simulations including bifurcation diagrams and phase portraits are performed to illustrate and confirm the theoretical results. These results may enrich the dynamics of predator–prey systems. [ABSTRACT FROM AUTHOR]

This paper reports a new 3D sub-quadratic Lorenz-like system and proves the existence of two pairs of heteroclinic orbits to two pairs of nontrivial equilibria and the origin, which are completely different from the existing ones to the unstable origin and a pair of stable nontrivial equilibria in the published literature. This motivates one to further explore it and dig out its other hidden dynamics: Hopf bifurcation, invariant algebraic surface, ultimate boundedness, singularly degenerate heteroclinic cycle and so on. Particularly, numerical simulation illustrates that the Lorenz-like chaotic attractors coexist with one saddle in the origin and two stable nontrivial equilibria, which are created through the broken infinitely many singularly degenerate heteroclinic cycles and explosions of normally hyperbolic stable foci E z. [ABSTRACT FROM AUTHOR]

The mammalian cochlear amplifier displays a unique nonlinear characteristic of amplifying or compressing the acoustic stimuli based on their level. Designing control algorithms for artificial hair cells (AHCs) with active sensing capabilities could mimic such dynamics. The present work focuses on developing a novel self-sensing active AHC scheme that implements the amplification/compression function of the AHCs and extends the authors’ previous designs for future cochlear implants or sensor design applications. The AHC functions near a Hopf bifurcation and varies the output piezoelectric voltage by a power-law relationship with the input. It is the first time that a self-sensing AHC is modeled as a quadmorph cantilever controlled by a phenomenological cubic damping controller. The AHC’s control signal is supplied to a pair of its piezoelectric layers and the output of the AHC is the sensed voltage of the second piezoelectric pair. This is in contrast to the previous studies where the AHC’s tip-velocity was measured with external sensors. The requirement of permanent external sensors in the system was a strict limitation that is eliminated in this work. An in-depth study of this novel AHC is conducted in this paper and AHC’s implementation is examined experimentally in Part II of the paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of ecosystem based fisheries management. The model represents the interaction between prey and predator population with Holling II functional response consisting of different carrying capacities and constant intrinsic growth rates. We have considered the continuous harvesting of predator only. It is observed that if the intrinsic growth rate of predator population crosses a certain critical value, the system enters into Hopf bifurcation. Our observations indicate that tax, the management object in fisheries system play huge impacts on this system. The optimal harvesting policy is disposed by imposing a tax per unit of predator biomass. The optimal harvest strategy is determined using Pontryagin's maximum principle, which is subject to state equations and control limitations. The implications of tax are also examined. We have derived different bifurcations and global stability of the system. Finally, numerical simulations are used to back up the analytical results. [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 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]

At present, there are few studies on the delayed kernel function of hyper-strong kernel. This paper attempts to analyze the stability and bifurcation of neural networks with distributed delayed hyper-strong kernels. Firstly, considering the average delay as a bifurcation parameter, the study discusses the characteristic equations of delayed kernels with weak kernel, strong kernel and hyper-strong kernel to provide sufficient conditions for the stability and generation of Hopf bifurcation. Secondly, it applies the normal theory and the center manifold theory to derive the formulas for determining the stability and direction of the bifurcating periodic solution. Finally, it verifies the correctness of the calculation results by numerical simulation with an example. [ABSTRACT FROM AUTHOR]

This paper studies an Internet congestion control system with two time delays, which are accessed by a single resource and considers both discrete and distributed delays of the system. By designing a new hybrid controller containing negative feedback control and time delay feedback control to control the system, and taking discrete time delay variables as bifurcation parameters, the local stability and Hopf bifurcation of the system are studied. The results show that the Hopf bifurcation can be effectively delayed or avoided by adjusting the value of the feedback control parameter β . The global asymptotically stable dynamic characteristics of the system are ideal, which has important functional significance for optimizing network congestion control. Finally, a large number of simulation examples verify the correctness of the conclusions. [ABSTRACT FROM AUTHOR]

In this essay, we introduce a bioeconomic predator–prey model which incorporates the square root functional response and nonlinear prey harvesting. Due to the introduction of nonlinear prey harvesting, the model demonstrates intricate dynamic behaviors in the predator–prey plane. Economic profit serves as a bifurcation parameter for the system. The stability and Hopf bifurcation of the model are discussed through normal forms and bifurcation theory. These results reveal richer dynamic features of the bioeconomic predator–prey model which incorporates the square root functional response and nonlinear prey harvesting, and provides guidance for realistic harvesting. A feedback controller is introduced in this paper to move the system from instability to stability. Moreover, we discuss the biological implications and interpretations of the findings. Finally, the results are validated by numerical simulations. [ABSTRACT FROM AUTHOR]

Analysing the prey-predator model is the purpose of this paper. In interactions of the Beddington–DeAngelis type, the predator consumes its prey. Researchers first examine the existence and local stability of potential unbalanced equilibrium boundaries for the model. In addition, for the suggested model incorporating the prey refuge, we investigate the Hopf bifurcation inquiry. To emphasise our key analytical conclusions, we show some numerical simulation results at the end. [ABSTRACT FROM AUTHOR]

In this paper, we develop a three-species food web model that incorporates the use of interactions between diseased predator–prey models. The logistically growing prey populations are susceptible and diseased prey. Prey populations are assumed to grow logistically in the absence of predators. We investigate the effect of fear on susceptible prey through infected prey populations. In Crowley–Martin-type interactions, it is assumed that interdependence between predators happens regardless of whether an individual predator is searching for prey or handling prey at the time. Also, the prey harvesting of susceptible and infected prey has been considered. The existence of all possible equilibrium points for biological systems has been established. The criteria for the local and global stability of equilibrium points are examined. Additionally, we look at Hopf-bifurcation analysis for the suggested model in relation to the existence of harvesting rate (h 1) . Numerical simulations are provided in order to explain the phenomenon and comprehend the complex interactions between predators and prey. [ABSTRACT FROM AUTHOR]

In this paper, we provide a rigorous description of the birth of canard limit cycles in slow-fast systems in R 3 through the folded saddle-node of type II and the singular Hopf bifurcation. In particular, we prove – in the analytic case only – that for all 0 < ϵ ≪ 1 there is a family of periodic orbits, born in the (singular) Hopf bifurcation and extending to O (1) cycles that follow the strong canard of the folded saddle-node. Our results can be seen as an extension of the canard explosion in R 2 , but in contrast to the planar case, the family of periodic orbits in R 3 is not explosive. For this reason, we have chosen to call the phenomena in R 3 , the "dud canard". The main difficulty of the proof lies in connecting the Hopf cycles with the canard cycles, since these are described in different scalings. As in R 2 , we use blowup to overcome this, but we also have to compensate for the lack of uniformity near the Hopf bifurcation, due to its singular nature; it is a zero-Hopf bifurcation in the limit ϵ = 0. In the present paper, we do so by imposing analyticity of the vector-field. This allows us to prove existence of an invariant slow manifold, that is not normally hyperbolic. [ABSTRACT FROM AUTHOR]

This paper studies a two-age structured SIRS epidemic model with logistic growth of susceptible population and two-time delays. We simultaneously introduce two-time delays, i.e., the immunity and incubation periods, into this dynamic system and investigate their impact on different dynamic behaviors for the model. By means of the C 0 -semigroup theory, the model is transformed into a non-densely defined abstract Cauchy problem, and the condition of the existence and uniqueness of the endemic equilibrium is obtained. Following the spectral analysis, the characteristic equation technique, and the Hopf bifurcation theorem, we show that different combinations of the two delays perform a vital role in the instability/stability as well as the Hopf bifurcation results of equilibrium solutions. We numerically provide some graphical representations to check the main theoretical results and show the rich dynamics by varying the two delay parameters. [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]

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

The study of harvesting mechanisms in predator-prey systems with an Allee effect on prey has always garnered significant attention. In this paper, the dynamics and harvesting strategies of a predator-prey system are investigated, where the prey is subject to the Allee effect. The positivity and boundedness of solutions, the existence and stability of equilibria are further studied. The existence of a Hopf bifurcation at the interior equilibrium point of the system is investigated and verified by numerical simulations. Furthermore, we investigate the maximum sustainable yield (MSY), maximum sustainable total yield (MSTY) and the optimal economic profit of the proposed system. We find that MSY can be attained through predator harvesting, while MSTY is observed when harvesting efforts are uniform across species. In these situations, the biological system maintains stability. Using the method of control parametrization, the optimal economic profit and harvesting strategy are obtained. The results show that the harvesting efforts can affect the stability of the system, resulting in several interesting biological phenomena. This research provides a theoretical basis for biological resource management. [ABSTRACT FROM AUTHOR]

In this paper, a fractional model of a special structure of bidirectional associative memory (BAM) neural networks called tri-diagonal BAM neural networks (TdBAMNNs) is considered. The Hopf bifurcation analysis is made for the proposed fractional system in the presence of leakage and communication delays. The feasibility of the obtained theoretical results is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multiple periodic solutions of the system are given. Secondly, the multiple periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric Hopf bifurcation theory. The normal form of the system on the central manifold and the bifurcation direction of the bifurcating periodic solutions are derived. Finally, numerical simulations are attached to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Crowley–Martin functional response and nonlocal competition into a reaction–diffusion immunosuppressive infection model. First, we analyze the existence and stability of the positive constant steady states of the systems with nonlocal competition and local competition, respectively. Second, we deduce the conditions for the occurrence of Turing, Hopf, and Turing–Hopf bifurcations of the system with nonlocal competition, as well as the conditions for the occurrence of Hopf bifurcations of the system with local competition. Furthermore, we employ the multiple time scales method to derive the normal forms of the Hopf bifurcations reduced on the center manifold for both systems. Finally, we conduct numerical simulations for both systems under the same parameter settings, compare the impact of nonlocal competition, and find that the nonlocal term can induce spatially inhomogeneous stable periodic solutions. We also provide corresponding biological explanations for the simulation results. [ABSTRACT FROM AUTHOR]

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

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

This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response p (x) = m x 2 a x 2 + b x + 1 where b > - 2 a . We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp singularity and of a nilpotent saddle of order 3 with a fixed invariant line are developed. The dependence of codimension of degenerate Hopf bifurcation on b is thoroughly investigated. It is proven that there exist a homoclinic loop of order 2 and a heteroclinic loop of order 2 for - 2 a < b < 0 , and three limit cycles for b > 0 . Together with existing work for Holling type I, II and IV functional responses, our results complement the analysis of the classical predator–prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of nilpotent saddle, through which the existence and order of the heteroclinic loop can be easily obtained for a general class of predator–prey systems with any smooth functional response. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, multiplicity, stability, and Hopf bifurcation of steady-state solutions of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and nonlinear boundary condition. It is found that there is no Hopf bifurcation when the interior reaction term is weaker than the boundary reaction term, and that the interior reaction delay determines the existence of Hopf bifurcation only when the interior reaction term is stronger than the boundary reaction term. This observation helps us to understand the nonlinear balance between the interior reaction and boundary flux in nonlinear boundary problems. Moreover, the general results are illustrated by applications to a model with homogeneous kernels. [ABSTRACT FROM AUTHOR]

The tea green leafhopper Empoasca onukii is one kind of insect pest threatening the tea production, and the mite Anystis baccarum has been used as an agent for pest control. In this paper, we introduce a generalist predator-prey model to study the dynamics for informing biological control. There have been some bifurcation studies of the generalist predator-prey model in the last few years. Except for the bifurcations include saddle-node bifurcation of codimension 1 and 2, Hopf bifurcations, and Bogdanov-Takens bifurcation of codimension 2 and 3, we also present the bifurcations of nilpotent singularities of elliptic and focus type of codimension 3. We find that the nilpotent singularities are associated with a cubic Liénard system, and the nilpotent bifurcations are three-parameter bifurcations of a codimension 4 nilpotent focus. Furthermore, we show that the nilpotent focus serves as an organizing center to connect all the codimension 3 bifurcations in the system. We also present the bifurcation diagrams to unfold the nilpotent singularities of codimension 3. One interesting observation is that we show numerically the existence of three limit cycles in the system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single species population model over patches with delay and nonlocal interactions, for which no symmetry for the dispersion (connection) matrix is assumed. We show that there exists a positive equilibrium when the dispersal rate is large. We also discuss the stability/instability of this positive equilibrium, establish the threshold dynamics and explore the associated Hopf bifurcation. Moreover, we demonstrate our theoretical results by a nonlocal logistic population model and by the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

Using a semidiscretization method, we derive in this paper a discrete slow-fast predator-prey system with ratio-dependent functional response. First of all, a detailed study for the local stability of fixed points of the system is obtained by invoking an important lemma. In addition, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the transcritical bifurcation and Neimark-Sacker bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the corresponding theoretical results and reveal some new dynamics of the system. Our results clearly demonstrate that the system is very sensitive to its fast time scale parameter variable. [ABSTRACT FROM AUTHOR]

This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R 0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness. [ABSTRACT FROM AUTHOR]

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion. [ABSTRACT FROM AUTHOR]

The Kolmogorov model is a class of significant ecological models and is initially introduced to describe the interaction between two species occupying the same ecological habitat. Limit cycle bifurcation problem is close to Hilbertis 16th problem. In this paper, we focus on investigating bifurcation of limit cycle for a class of quartic Kolmogorov model with two positive equilibrium points. Using the singular values method, we obtain the Lyapunov constants for each positive equilibrium point and investigate their limit cycle bifurcations behavior. Furthermore, based on the analysis of their Lyapunov constants' structure and Hopf bifurcation, we give the condition that each one positive equilibrium point of studied model can bifurcate 5 limit cycles, which include 3 stable limit cycles. [ABSTRACT FROM AUTHOR]

This paper focuses on a chemotaxis model with no-flux boundary conditions. We first discuss the stability of the unique positive equilibrium by treating the chemotaxis coefficient \xi as the Hopf bifurcation and the steady state bifurcation parameter. Hereafter, we perform the existence and stability of the bifurcating solution, which bifurcated from the steady state bifurcation, by using the Crandall-Rabinowitz local bifurcation theory. It is noticed that few existing literatures give a discriminant to determine the stability of the bifurcating solution for the chemotaxis models. To this end, we will fill this gap, and an explicit formula will be presented. This technique can also be applied in other ecological models with chemotaxis. [ABSTRACT FROM AUTHOR]

The malware attacks targeting the industrial control network are gradually increasing, and the nonlinear phenomenon makes it difficult to predict the propagation behavior of malware. Once the dynamic system becomes unstable, the propagation of malware will be out of control, which will seriously threaten the security of the industrial control network. So, it is necessary to model and study the propagation of malware in the industrial control network. In this paper, a SIDQR model with dual delay is proposed by fully considering the characteristics of the industrial control network. By analyzing the nonlinear dynamics of the model, the Hopf bifurcation is discussed in detail when the value of dual delay is greater than zero, and the expression for the threshold is also provided. The results of the experiments indicate that the system may have multiple bifurcation points. By comparing different immune and quarantine rates, it is found that the immune rate can be appropriately increased and the isolation rate can be appropriately reduced in the industrial control network, which can suppress the spread of malware without interrupting the industrial production. [ABSTRACT FROM AUTHOR]

This paper proposes a prey–predator model affected by fear effects and toxic substances. We used the Lipschitz condition to prove the uniqueness of the model solution and Laplace transform to prove the boundedness of the model solution. We used the fractional-order stability theorem to provide sufficient conditions for the local stability of equilibrium points, and selected fractional-order derivatives as parameters to perform Hopf bifurcation analysis on the system. Finally, the theoretical results are verified via numerical simulation. The results show that a value of α will affect the stability of the system and that the population size and the effect of toxic substances have a huge impact on the stability of the system. [ABSTRACT FROM AUTHOR]

In this paper we consider a jerk system x ˙ = y , y ˙ = z , z ˙ = j (x , y , z , α) , where j is an arbitrary smooth function and α is a real parameter. Using the derivatives of j at an equilibrium point, we discuss the stability of that point, and we point out some local codim-1 bifurcations. Moreover, we deduce jerk approximate normal forms for the most common fold bifurcations. [ABSTRACT FROM AUTHOR]

Under the background of double carbon, it is important to study forest pests and diseases to improve forest carbon sink. In this paper, we establish a delayed model associated with the larvae and adults of Saperda populnea, susceptible poplars, and infected poplars. First, we analyze the existence and stability of the equilibrium of the model. Second, we study the existence of Hopf bifurcation near the equilibrium and obtain the normal form of Hopf bifurcation by the multiple time scales method. Then, we analyze the direction and stability of Hopf bifurcating periodic solutions. Third, we analyze and conjecture some parameter values based on official data, and carry out numerical simulations to verify our results. Finally, we give some suggestions on the prevention and control of Saperda populnea. [ABSTRACT FROM AUTHOR]

This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations. [ABSTRACT FROM AUTHOR]