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The paper is focused on the analysis of effect of coupling strength and time delay for a pair of connected neurons on the dynamics of the system. The FitzHugh–Nagumo model is used as a neuron model. The article contains analytical conditions for Hopf bifurcations in the system. A numerical verification of the results is given. Several examples of global bifurcation in the system were analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a multiple delayed HIV-1 infection model which describes the interactions of humoral, cytotoxic T lymphocyte (CTL) immune responses, and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission are investigated. The model incorporates three delays, including the delays of cell infection, virus production and activation of immune response. We first prove the well-posedness of the model, and calculate the biological existence of equilibria and the reproduction numbers, which contain virus infection, humoral immune response, CTL immune response, CTL immune competition, and humoral immune competition. Further, the threshold conditions for the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response, and interior are established by utilizing linearization method and the Lyapunov functionals. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Lyapunov stability framework for fractionally damped mechanical systems. In particular, we study the asymptotic stability of a linear single degree-of-freedom oscillator with viscous and fractional damping. We prove that the total mechanical energy, including the stored energy in the fractional element, is a Lyapunov functional with which one can prove stability of the equilibrium. Furthermore, we develop a strict Lyapunov functional for asymptotic stability, thereby opening the way to a nonlinear stability analysis beyond an eigenvalue analysis. A key result of the paper is a Lyapunov stability condition for systems having negative viscous damping but a sufficient amount of positive fractional damping. This result forms the stepping stone to the study of Hopf bifurcations in fractionally damped mechanical systems. The theory is demonstrated on a stick-slip oscillator with Stribeck friction law leading to an effective negative viscous damping. [ABSTRACT FROM AUTHOR]

In this paper, a delayed eco-epidemiological model including susceptible migratory birds, infected migratory birds and predator population is proposed by us. The interaction between predator and prey is represented by functional response of Leslie–Gower Holling-type II. Fear effect is considered in the model. We assume that the growth rate and activity of prey population can be reduced because of fear effect of predator, and this series of behaviors will indirectly slow down the spread of diseases. Positivity, boundedness, persistence criterion, and stability of equilibrium points of the system are analyzed. Transcritical bifurcation and Hopf-bifurcation respect to important parameters of the system have been discussed both analytically and numerically (e.g. fear of predator, disease transmission rate of prey, and delay). Numerical simulation results show that fear can not only eliminate the oscillation behavior caused by high disease transmission rate and long delay in the model system, but also eliminate the disease. [ABSTRACT FROM AUTHOR]

In this paper, a malaria disease transmission model has been developed. Here, the disease transmission rates from mosquito to human as well as human to mosquito and death rate of infected mosquito have been constituted by two variabilities: one is periodicity with respect to time and another is based on some control parameters. Also, total vector population is divided into two subpopulations such as susceptible mosquito and infected mosquito as well as the total human population is divided into three subpopulations such as susceptible human, infected human and recovered human. The biologically feasible equilibria and their stability properties have been discussed. Again, the existence condition of the disease has been illustrated theoretically and numerically. Hopf-bifurcation analysis has been done numerically for autonomous case of our proposed model with respect to some important parameters. At last, a optimal control problem is formulated and solved using Pontryagin's principle. In numerical simulations, different possible combination of controls have been illustrated including the comparisons of their effectiveness. [ABSTRACT FROM AUTHOR]

The stability and Hopf bifurcation of a delayed SIS epidemic model with double epidemic hypothesis are investigated in this paper. We first study the stability of the unique positive equilibrium of the model in four cases, and we obtain the stability conditions through analyzing the distribution of characteristic roots of the corresponding linearized system. Moreover, we choosing the delay as bifurcation parameter and the existence of Hopf bifurcation is investigated in detail. We can derive explicit formulas for determining the direction of the Hopf bifurcation and the stability of bifurcation periodic solution by center manifold theorem and normal form theory. Finally, we perform the numerical simulations for justifying the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stability and Hopf bifurcation of a biological economic system with harvesting effort on prey. The model we consider is described by differential-algebraic equations because of economic revenue. We choose economic revenue as a positive bifurcation parameter here. Different from previous researchers' models, this model with nonlinear harvesting rate is more general. Furthermore, the improved calculation process of parameterization is much simpler and it can handle more complex models which could not be dealt with by their algorithms because of enormous calculation. Finally, by MATLAB simulation, the validity and feasibility of the obtained results are illustrated. [ABSTRACT FROM AUTHOR]

This paper focusses on developing two species, where only prey species suffers by a contagious disease. We consider the logistic growth rate of the prey population. The interaction between susceptible prey and infected prey with predator is presumed to be ruled by Holling type II and I functional response, respectively. A healthy prey is infected when it comes in direct contact with infected prey, and we also assume that predator-dependent disease spreads within the system. This research reveals that the transmission of this predator-dependent disease can have critical repercussions for the shaping of prey–predator interactions. The solution of the model is examined in relation to survival, uniqueness and boundedness. The positivity, feasibility and the stability conditions of the fixed points of the system are analysed by applying the linearization method and the Jacobian matrix method. [ABSTRACT FROM AUTHOR]

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In this paper, a three species food chain model has been developed among the interaction of prey, predator and super predator. It is assumed that the predator shows refuge behavior to the super predator. It is also assumed that a certain amount of additional food will be supplied to the super predator. It is considered that the predator population is benefiting partially from the additional food. To get optimal harvesting of super predator the Pontryagin's maximum principle has been used. It is found that super predator may be extinct if harvesting rate increase. It is observed that as the refuge rate increases, predator population gradually increases, but super predator population decreases. Also, it is found that our proposed system undergoes oscillatory or periodic behavior as the value of refuge rate (m1), harvesting rate (E), the intrinsic growth rate of prey (r), carrying capacity of prey (k) and conservation rate of prey (c1) varies for some certain range of these parameters. It is found that this study may be useful for the increase of harvesting of a super predator by supplying the additional food to our proposed system. [ABSTRACT FROM AUTHOR]

The present paper constructs a family of three-sector models of optimal endogenous growth, and conducts exact bifurcation analysis. In so doing, original six-dimensional equilibrium dynamics is decomposed into five-dimensional stationary autonomous dynamics and one-dimensional endogenously growing component. It is shown that the stationary dynamics thus decomposed undergoes supercritical Hopf bifurcation. It is inferred from the convex structure of our model that the dimension of a stable manifold of each closed orbit thus bifurcated in this five-dimensional dynamics should be two. [ABSTRACT FROM AUTHOR]

In ecology, one of the simplest representation of population dynamics is the logistic equation. This basic view can be enriched by considering two important variables: (1) the maximal population density Nature can support (carrying capacity) and (2) the critical density threshold under which the population disappears (Allee effect). The economic literature on biodiversity and renewable resources ignores both these variables. Evidence suggests also that these variables are affected by the pollution level due to economic activity. Indeed, a degraded environment is unsuitable for wildlife and reduces the carrying capacity, while the climate change entails the habitat fragmentation and, lowering the wildlife reproduction possibilities, raises the Allee effect. The present paper aims to incorporate both endogenous carrying capacity and Allee effect in a Ramsey model augmented with biodiversity as a renewable resource. Our extended framework enables us to study the effect of a Pigouvian tax on anthropogenic mass extinction. We find that, when the household overvalues biodiversity with respect to consumption, a higher green-tax rate is beneficial in three respects entailing: (1) a lower pollution and a higher biodiversity, (2) a welfare improvement and (3) a less likely mass extinction. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior of a discrete SIR epidemic model with fractional-order with non-monotonic incidence rate is discussed. The sufficient conditions of the locally asymptotic stability and bifurcation analysis of the equilibrium points are also discussed. The numerical simulations come to illustrate the dynamical behaviors of the model such as flip bifurcation, Hopf bifurcation and chaos phenomenon. The results of numerical simulation verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings. [ABSTRACT FROM AUTHOR]

This paper deals with the stability and the globally bifurcated periodic response of a delayed network of four neurons with a short-cut connection. It presents a set of sufficient conditions for the existence of periodic responses arising from a Hopf bifurcation first, and then gives the global continuation of the periodic responses according to the theorem of global Hopf bifurcation. Afterwards, the paper focuses on the validation of theoretical results through some numerical simulations and a circuit experiment. Both numerical and experimental results reach an agreement with theoretical ones. [ABSTRACT FROM AUTHOR]

This paper studies Hopf bifurcation of a kind of complex economic systems with rich elasticity. The conditions for the presence of bifurcation, the stability of periodic orbit before the emergence of Hopf bifurcation and the critical parameter value of the system are obtained. According to Taken's estimation, the evolvement situation of the complex system is also given. Numerical examples are given to verify the validity of the present theory. The obtained result is of theoretical importance and has practical applications to exploring the inherence mechanism of the complicated continuous economic systems and establishing a reasonable macro control policy. [ABSTRACT FROM AUTHOR]

In this paper, strong Allee effects on the bifurcation of the predator–prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The existence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor's theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occurrence of Bogdanov–Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexistence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation. Numerical simulations and phase portraits are also given to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a delayed predator–prey model (PPM) with nonlinear harvesting efforts by using imprecise biological parameters are studied. A method is proposed to handle these imprecise parameters by using a parametric form of interval numbers. The proposed PPM is presented with Crowley–Martin type of predation and Michaelis–Menten type prey harvesting. The existence of various equilibrium points and the stability of the system at these equilibrium points are investigated. Analytical study reveals that the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate the main analytical findings. [ABSTRACT FROM AUTHOR]

Numerous field data and experiments on the perching birds or songbirds show that the fear of predators can cause significant changes in the prey population. Fear of predatory populations increases the chances of survival of the prey population, and this can greatly reduce the reproduction of the prey population. The influence of fear has contributed a leading role in both the environmental biology and theoretical ecology. Taking into account the interaction of predator–prey with non-overlapping generations, a discrete-time model is proposed and studied. Keeping in mind the biological feasibility of species, the existence of fixed points is studied along with the local asymptotic behavior of the proposed model around these fixed points. Furthermore, taking into account the oscillatory behavior of the model, various types of bifurcations are analyzed about biologically feasible fixed points with an application of center manifold theory and bifurcation theory of normal forms. Existence of chaos is discussed, and fluctuating and chaotic behavior of the system is controlled through implementation of different chaos control procedures. The illustration of theoretical discussion is carried out via validation of observed experimental field data and appropriate numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a prey-predator model with a reserve region of predator where generalist predator cannot enter. Based on the intake capacity of food and other factors, we introduce the predator population which consumes the prey population with Holling type-II functional response; and generalist predator population consumes the predator population with Beddington-DeAngelis functional response. The density-dependent mortality rate for prey and generalist predator are considered. The equilibria of proposed system are determined. Local stability for the system are discussed. The environmental carrying capacity is considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighbourhood at an interior equilibrium point. Here the fishing effort is used as a control parameter to harvest the generalist predator population of the system. With the help of this control parameter, a dynamic framework is developed to investigate the optimal utilization of resources, sustainability properties of the stock and the resource rent. Finally, we present a numerical simulation to verify the analytical results, and the system is analyzed through graphical illustrations. The main findings with future research directions are described at last.

In this article, a mathematical model has been derived for studying the dynamics of malaria disease and the influence of awareness-based interventions, for control of the same, that depend on 'level of awareness'. We have assumed the disease transmission rates from vector to human and from human to vector, as decreasing functions of 'level of awareness'. The effect of insecticides for controlling the mosquito population is influenced by the level of awareness, modelled using a saturated term. Organizing any awareness campaign takes time. Therefore a time delay has been incorporated in the model. Some basic mathematical properties such as nonnegativity and boundedness of solutions, feasibility and stability of equilibria have been analysed. The basic reproduction number is derived which depends on media coverage. We found two equilibria of the model namely the disease-free and endemic equilibrium. Disease-free equilibrium is stable if basic reproduction number (ℛ0) is less than unity (ℛ0 < 1). Stability switches occur through Hopf bifurcation when time delay crosses a critical value. Numerical simulations confirm the main results. It has been established that awareness campaign in the form of using different control measures can lead to eradication of malaria. [ABSTRACT FROM AUTHOR]

The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of delay coupled Lotka-Volterra ring systems. Based on the symmetric bifurcation theory of delay differential equations and representation theory of standard dihedral groups, properties of phase locked periodic solutions are given. Moreover, the direction and the stability of the Hopf bifurcation periodic orbits are obtained by using normal form and center manifold theory. Finally, the research results are verified by numerical simulation.

In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Basically enzymes are biological catalysts that increase the speed of a chemical reaction without undergoing any permanent chemical change. With the application of Euler's forward scheme, a discrete-time enzyme model is presented. Further investigation related to its qualitative behaviour revealed that discrete-time model shows rich dynamics as compared to its continuous counterpart. It is investigated that discrete-time model has a unique trivial equilibrium point. The local asymptotic behaviour of equilibrium is discussed for discrete-time enzyme model. Furthermore, with the help of the bifurcation theory and centre manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. Moreover, two existing chaos control methods, that is, Ott, Grebogi and Yorke feedback control and hybrid control strategy, are implemented. In particular, a novel chaos control technique, based on state feedback control is introduced for controlling chaos under the influence of flip and Hopf bifurcations in discrete-time enzyme model. Numerical simulations are provided to illustrate theoretical discussion and effectiveness of newly introduced chaos control method. [ABSTRACT FROM AUTHOR]

In this work we study a fractional-order susceptible-infective-recovered-susceptible (SIRS) epidemic model with a nonlinear incidence rate. The incidence is assumed to be a convex function with respect to the infective class of a host population. Local and uniform stability analysis of the disease-free equilibrium is investigated. The conditions for the existence of endemic equilibria (EE) are given. Local stability of the EE is discussed. Conditions for the existence of Hopf bifurcation at the EE are given. Most importantly, conditions ensuring that the system exhibits backward bifurcation are provided. Numerical simulations are performed to verify the correctness of results obtained analytically. [ABSTRACT FROM AUTHOR]

As scarce environmental resources necessarily put a constraint on population growth, we use more realistic population growth dynamics which contemplates a feedback mechanism between population growth rate and resource availability. We examine the local stability properties in overlapping generations resource economies which takes this feedback mechanism into account. The results indicate that Hopf bifurcation may arise without requiring logistic regeneration or unconventional constraints on parameter values. In particular, Hopf bifurcation is encountered under convex-concave dependence of carrying capacity on the resource availability. [ABSTRACT FROM AUTHOR]

A nonlinear model for a balanced flexible rotor-bearing system is developed to analyze the dynamics of nonlinear rotor-bearing system. A continuation algorithm in conjunction with the Newton iteration and the shooting method respectively are applied to calculate the branches of the equilibrium solutions and periodic solutions of the system. The Hopf bifurcation and the post Hopf bifurcation behaviors of a specific balanced flexible rotor-bearing system are investigated. The results show that the supercritical Hopf bifurcation occurs. Analysis and comparison between the frequencies of periodic solutions and the eigen-frequencies of the system reveal that the model enables the mechanism identification of the self-excitation of oil film force. Moreover, by the aid of the bifurcation diagrams, Poincare maps and Lyapunov exponents, the bifurcations of periodic solutions, chaos and the route to and out of chaos in a balanced rotor-bearing system are revealed. [ABSTRACT FROM AUTHOR]