Search

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

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

In this paper, we formulate and study a fractional-order network model with four neurons, bidirectional ring structure and self-delay feedback. For the scenario of non-identical neurons, we develop a new algebraic technique to deal with the characteristic equation with e − 4 s τ (τ is the self-feedback delay) term and thus establish the easy-to-check criteria to determine the Hopf bifurcation point of self-feedback delay by fixing communication delay in its stable interval. For the scenario of identical neurons, we apply the crossing curves method to the fractional functional equations and thus procure the Hopf bifurcation curve. The obtained results accommodate the fact that the model cannot preserve its stability behavior when the self-feedback delay crosses the Hopf bifurcation point in the positive direction. Finally, we deliberate on the correctness of our methodology through two demonstration examples. [ABSTRACT FROM AUTHOR]

In this paper, the predator–prey model with strong Allee and fear effects is considered. The existence of the equilibria and their stability are established. Especially it is found that there is an interesting degenerate point, which is a cusp point with codimension 2 or higher codimension, or an attracting (repelling)-type saddle-node, subject to different conditions. Then the Hopf bifurcation and its direction, the saddle-node bifurcation and the Bogdanov–Tankens bifurcation are further explored. Afterwards, with the help of the energy estimates and the Leray–Schauder degree, the nonexistence and existence of the nonconstant steady states of the model are presented. From the obtained results, we find that strong Allee effect will cause the per capita growth rate of prey species from negative to positive; both the fear and Allee effects could affect the existence of equilibria and bifurcations; meanwhile, the diffusion rates will affect the existence of the nonconstant steady states. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a reaction-diffusion Nicholson's blowflies equation with advection. The stability of positive steady state and existence of Hopf bifurcation are obtained by analyzing the distribution of the eigenvalues. Moreover, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction and stability of the Hopf bifurcation is derived. Meanwhile, we find out that the bifurcation value is increasing with respect to the advection rate. Finally, numerical results demonstrate that the advection term causes the population to move from upstream to downstream, which also indicates that advection term plays a key role in the description and interpretation of some common natural phenomena. [ABSTRACT FROM AUTHOR]

This paper expounds the bifurcations of two-delayed fractional-order neural networks (FONNs) with multiple neurons. Leakage delay or communication delay is viewed as a bifurcation parameter, stability zones and bifurcation conditions with respect to them are commendably established, respectively. It declares that both leakage delay and communication delay immensely influence the stability and bifurcation of the developed FONNs. The explored FONNs illustrate superior stability performance if selecting a lesser leakage delay or communication delay, and Hopf bifurcation generates once they overstep their critical values. The verification of the feasibility of the developed analytic results is implemented via numerical experiments. [ABSTRACT FROM AUTHOR]

Since the stochastic bifurcation theory is still in its infancy, we try to analyze some stochastic bifurcation phenomenon from a simple mathematical model. Thus, this paper mainly focuses on studying the stochastic bifurcation of a stochastic logistic model with distributed delay in the strong kernel case, which is affected by noise. Therefore, we use the intrinsic growth rate as a bifurcation parameter. First, we study the stochastic D-bifurcation and stochastic P-bifurcation for stochastic logistic model. Furthermore, by deriving the corresponding Fokker–Planck equation, we obtain the expression of the joint density function of the stochastic logistic system near the positive equilibrium point. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic delayed differential algebraic predator–prey model with Michaelis–Menten-type prey harvesting is proposed. Due to the influence of gestation delay and stochastic fluctuations, the proposed model displays a complex dynamics. Criteria on the local stability of the interior equilibrium are established, and the effect of gestation delay on the model dynamics is discussed. Taking the gestation delay and economic profit as bifurcation parameters, Hopf bifurcation and singularity induced bifurcation can occur as they cross through some critical values, respectively. Moreover, the solution of the model will blow up in a limited time when delay τ > τ 0 . Then, we calculate the fluctuation intensity of the stochastic fluctuations by Fourier transform method, which is the key to illustrate the effect of stochastic fluctuations. Finally, we demonstrate our theoretical results by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a cancer virotherapy model with virus lytic cycle and diffusion term. Spatiotemporal dynamic properties of the cancer virotherapy system are studied. First, by analyzing the roots distribution of the characteristic equation and transcendental equation, the conditions for the local stability of the constant equilibria of system are given. Second, we select delay as the bifurcation parameter, the existence conditions of Hopf bifurcation are given. By using the center manifold theory and normal form method of partial functional differential equation, the detailed formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are given. Finally, some numerical simulations are given. [ABSTRACT FROM AUTHOR]

In this paper, a delayed differential model of citrus Huanglongbing infection is analyzed, in which the latencies of the citrus tree and Asian citrus psyllid are considered as two time delay factors. We compute the equilibrium points and the basic reproductive numbers with and without time delays, i.e. R 0 and R ̃ 0 , and then show that R ̃ 0 completely determines the local stability of the disease-free equilibrium. Moreover, the conditions for the existence of transcritical bifurcation are derived from Sotomayor's Theorem. The stability of the endemic equilibrium and the existence of Hopf bifurcation are investigated in four cases: (1) τ 1 = τ 2 = 0 , (2) τ 1 > 0 , τ 2 = 0 , (3) τ 1 = 0 , τ 2 > 0 and (4) τ 1 > 0 , τ 2 > 0. Optimal control theory is then applied to the model to study two time-dependent treatment efforts and minimize the infection in citrus and psyllids, while keeping the implementation cost at a minimum. Numerical simulations of the overall systems are implemented in MatLab for demonstration of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic model given by three-dimensional ordinary differential equations is studied to determine how the dynamics of tumor growth is controlled by some key parameters. By varying the competition coefficient between healthy host cells and tumor cells, a Hopf bifurcation occurs in this system, leading to the creation of a stable limit cycle. Through numerical analysis of the continuity of this limit cycle, we find that the cascade of period-doubling bifurcations leads to the generation of a chaotic attractor. Moreover, the region of attractors is shown in the parameter space. Numerical simulations, bifurcation diagrams, Lyapunov exponent graph and phase portraits permit to highlight the rich and complex phenomena presented by the model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a stochastic delayed epidemic model with double epidemic hypothesis and vaccination incorporating Lévy noise. First, we show that this model has a unique global positive solution. Furthermore, we give sufficient conditions for extinction and persistence in the mean of the two epidemics and we prove that the two diseases can coexist under some conditions. Finally, we present some examples to illustrate the analytical results by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel fractional-order proportional-derivative (PD) strategy to achieve the control of bifurcation of a fractional-order gene regulatory model with delays. The stability theory of fractional differential equations proved that with delays, some explicit conditions for the local asymptotical stability and Hopf bifurcation are given for the controlled fractional-order genetic model. It is demonstrated that the fractional-order gene regulatory model becomes controllable by adjusting the control gain parameters. In addition, the effect of fractional-order parameter on the dynamical behaviors is shown. Finally, numerical simulations are carried out to testify the validity of the main results and the availability of the fractional-order PD controller. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex. [ABSTRACT FROM AUTHOR]

Understanding the dynamical mechanism of the host-parasite interactions is one of important issues on host-parasite association. In this paper, we formulate a three-dimensional host-macroparasite system to describe the host-parasite interactions, which includes the logistic growth rate of host population, the important free-living stage and the host fecundity reduction due to parasite infection. The purpose of the paper is to investigate the asymptotical behavior of the system. By using the properties of the solution to non-autonomous linear system, the basic production number is proved to be a threshold which determines the outcome of the parasites. If , the parasite will eventually die out, and if the parasite will be uniformly persistent. Hopf bifurcation of the system is further studied, and sufficient conditions for the Hopf bifurcation are obtained. By using the singular perturbation techniques, the system is separated into two time scales with a faster time scale for the free-living infective particles and a slower time scale for the population dynamics of host and parasite, and then a complete analysis of the dynamics on the slow manifold is conducted. The theoretical results show that the level of aggregation of parasites within host may influence the persistence and stability of the system. [ABSTRACT FROM AUTHOR]

In this paper, an eco-economic model with harvesting on biological population is established, which takes the form of a differential-algebra system. The impact of the economic profit from harvesting upon the dynamics of the model is studied. By using a suitable parameterization for the differential-algebra system, we derive an equivalent parameterized system which gives the stability results for the positive equilibrium point of our model. Moreover, based on the parameterized system as well as the approaches of normal form and formal series, the conditions on the Hopf bifurcation and the stability of center are obtained. Several numerical simulations for demonstrating the theoretical results are also presented. Lastly, according to the dynamical analysis, we provide a threshold value for the economic profit, which can maintain the sustainable development of our eco-economic system. [ABSTRACT FROM AUTHOR]

In this paper, we use a semidiscretization method to derive a discrete predator–prey model with Holling type II, whose continuous version is stated in [F. Wu and Y. J. Jiao, Stability and Hopf bifurcation of a predator-prey model, Bound. Value Probl.129 (2019) 1–11]. First, the existence and local stability of fixed points of the system are investigated by employing a key lemma. Then we obtain the sufficient conditions for the occurrence of the transcritical bifurcation and Neimark–Sacker bifurcation and the stability of the closed orbits bifurcated by using the Center Manifold theorem and bifurcation theory. Finally, we present numerical simulations to verify corresponding theoretical results and reveal some new dynamics. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model to study the impact of Twitter in controlling infectious disease is proposed. The model includes the dynamics of “tweets” which may enhance awareness of the disease and cause behavioral changes among the public, thus reducing the transmission of the disease. Furthermore, the model is improved by introducing a time delay between the outbreak of disease and the release of Twitter messages. The basic reproduction number and the conditions for the stability of the equilibria are derived. It is shown that the system undergoes Hopf bifurcation when time delay is increased. Finally, numerical simulations are given to verify the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting. First, we discuss the existence and stability of all possible equilibria. Next, we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter. Furthermore, we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem. In addition, we give the optimal harvesting strategy of the delayed system based on Pontryagin's maximum principle with delay. Finally, some numerical simulations are carried out to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hopf bifurcation of predator–prey system with two delays and disease transmission. Furthermore, the global existence of bifurcated periodic solution was studied, the influence of disease transmission is given. At last, some simulations are given to support our result. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order delayed SIQR model with nonlinear incidence rate is investigated. Two time delays are incorporated in the model to describe the incubation period and the time caused by the healing cycle. By analyzing the associated characteristic equations, the stability of the endemic equilibrium and the existence of Hopf bifurcation are obtained in three different cases. Besides, the critical values of time delays at which a Hopf bifurcation occurs are obtained, and the influence of the fractional order on the dynamics behavior of the system is also investigated. Numerically, it has been shown that when the endemic equilibrium is locally stable, the convergence rate of the system becomes slower with the increase of the fractional order. Besides, our studies also imply that the decline of the fractional order may convert a oscillatory system into a stable one. Furthermore, we find in all these three cases, the bifurcation values are very sensitive to the change of the fractional order, and they decrease with the increase of the order, which means the Hopf bifurcation gradually occurs in advance. [ABSTRACT FROM AUTHOR]

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Testing–culling is one of the important prevention and control measures considered in the study of animal infectious diseases. However, the process of finding infected animals (animal testing) is still not well studied through the kinetic model. In this paper, based on the characteristics of animal testing, a time-delayed model on brucellosis transmission is established. Under the general hypothesis of biological significance, the existence and stability of equilibria are first investigated. The results find that the global stability of equilibria depends on the basic reproduction number R 0 without the information delay: if R 0 ≤ 1 , the disease dies out; if R 0 > 1 , the endemic equilibrium exists and the disease persists. Next, the impact of information delay on the dynamics of the model is analyzed and Hopf bifurcation is found in the established model when the information delay is greater than a critical value. Finally, the theoretical results are then further explained through numerical analysis and the significance of these results for the development of risk management measures is elaborated. [ABSTRACT FROM AUTHOR]

Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate. In this paper, we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system. First, the stability of the unique positive equilibrium point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory. Second, the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. In addition, the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations. The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed viral infection model with cellular infection and full logistic proliferations for both healthy and infected cells. The global asymptotic stabilities of the equilibria are studied by constructing Lyapunov functionals. Moreover, we investigated the existence of Hopf bifurcation at the infected equilibrium by regarding the possible combination of the two delays as bifurcation parameters. The results show that time delays may destabilize the infected equilibrium and lead to Hopf bifurcation. Finally, numerical simulations are carried out to illustrate the main results and explore the dynamics including Hopf bifurcation and stability switches. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a predator–prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions. First, the temporal behavior of the model has been investigated, where Hopf bifurcation has been obtained. Then, by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation, Turing instability, even Turing–Hopf bifurcation. Further, the impact of the prey herd shape on the spatiotemporal patterns has been discussed. Furthermore, by computing and analyzing the normal form associated with the Turing–Hopf bifurcation point, the spatiotemporal dynamics near the Turing–Hopf bifurcation point has been discussed and also justified by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of singular stochastic bio-economic models described by differential-algebraic equations due to the influence of economic factors. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. Stochastic stability and Hopf bifurcations can be analytically determined based on the singular boundary theory of diffusion process, the Maximal Lyapunov exponent and the invariant measure theory. The critical value of the stochastic Hopf bifurcation parameter is obtained and the position of Hopf bifurcation drifting with the parameter increase is presented as a result. Practical example is presented to verify the effectiveness of the results. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

In this paper, a human immunodeficiency virus (HIV) infection model with both the types of immune responses, the antibody and the killer cell immune responses has been introduced. The model has been made more logical by including two delays in the activation of both the immune responses, along with the combination drug therapy. The inclusion of both the delayed immune responses provides a greater understanding of long-term dynamics of the disease. The dependence of the stability of the steady states of the model on the reproduction number R 0 has been explored through stability theory. Moreover, the global stability analysis of the infection-free steady state and the infected steady state has been proved with respect to R 0 . The bifurcation analysis of the infected steady state with respect to both delays has been performed. Numerical simulations have been carried out to justify the results proved. This model is capable of explaining the long-term dynamics of HIV infection to a greater extent than that of the existing model as it captures some basic parameters involved in the system such as immunological delay and immune response. Similarly, the model also explains the basic understanding of the disease dynamics as a result of activation of the immune response toward the virus. [ABSTRACT FROM AUTHOR]