hopf bifurcation

During an epidemic, existing mass media plays a fundamental role in promoting effective, trustworthy and convenient information regarding disease symptoms and prevention measures against the infection. In this research paper, we aim to explore the impact of media awareness projected to a SEIV compartmental model incorporating newly modulated saturated incidence function and discrete-time delay during an epidemic. We considered the time lag in between the process while unaware susceptible individuals would be aware through the campaign media. Sensitivity analysis reveals the influence of the model parameters in the progression of the epidemic. Numerical simulations enable us to visualize the importance of media awareness to convey predictions regarding the mitigation and apparent eradication of the epidemic.

This paper is concerned with a cross-diffusion prey-predator system in which the prey species is equipped with the group defense ability under the Neumann boundary conditions. The tendency of the predator to pursue the prey is expressed in the cross-diffusion coefficient, which can be positive, zero, or negative. We first select the environmental protection of the prey population as a bifurcation parameter. Next, we discuss the Turing instability and the Hopf bifurcation analysis on the proposed cross-diffusion system. We show that the system without cross-diffusion is stable at the constant positive stationary solution but it becomes unstable when the cross-diffusion appears in the system. Furthermore, the stability of bifurcating periodic solutions and the direction of Hopf bifurcation are examined.

In this paper, the local stability of the endemic equilibrium and existence of a Hopf bifurcation in a Susceptible-ExposedInfected-Recovered (SEIR) delayed mathematical model for COVID-19 pandemic are investigated. By using time-delay as a bifurcationparameter, the associated characteristic equation is analyzed to reveal dynamics of the model. Finally, numerical simulations areperformed with suitable parameters choice to illustrate the theoretical results of the model.

This article investigates a fractional-order predator-prey model incorporating prey refuge and anti-predator behaviour on predator species. For our proposed model, we prove the existence, uniqueness, non-negativity and boundedness of solutions. Further, all biologically possible equilibrium points and their stability analysis for the proposed system are carried out with the linearization process. Moreover, by using an appropriate Lyapunov function, the global stability of the co-existence equilibrium point is studied. Finally, we provide numerical simulations to demonstrate how the theoretical approach  is  consistent.

This article investigates the activity regimes of a realistic neuron model (as a slow-fast system). The authors study this model using the dynam-ical systems theory, for example, qualitative theory methods of slow-fast systems. The authors obtain the stability conditions of equilibria in leech heart interneurons under defined pharmacological conditions and following Hodgkin–Huxley formalism. Although in neuronal models, the membrane is usually considered  capacitance as a fixed parameter, the membrane ca-pacitance parameter is assumed as a control parameter to guarantee the existence of Hopf bifurcation using the Routh–Hurwitz criteria. The au-thors investigate the transition mechanism between the silent phase and tonic spiking mode. Furthermore, some simulations are provided using XPPAUT software for analytical results.

In this work, we explore the consequences of fear and time delay on the intraguild predation model. Also, the predator consumes its prey in the form of a ratio-dependent type of interaction. We consider the fear in the prey population and the gestation effect on the predator population. We analyze the existence and the local stability of the proposed model without delay near all non-negative equilibrium points. Furthermore, by taking the fear parameter, the condition to satisfy the existence of Hopf-bifurcation near the coexisting equilibrium is derived. Moreover, we also examine the local stability property and Hopf-bifurcation investigation for the corresponding model in the presence of time delay. Some simulation results were also done to support the primary analytical findings.

In this paper, a general class of fractional-order complex-valued bidirectional associative memory neural network with time delay is considered. This neural network model contains an arbitrary number of neurons, i.e. one neuron in the X-layer and other neurons in the Y-layer. Hopf bifurcation analysis of this system will be discussed. Here, the number of neurons, i.e., $n$ can be chosen arbitrarily. We study Hopf bifurcation by taking the time delay as the bifurcation parameter. The critical value of the time delay for the occurrence of Hopf bifurcation is determined. Moreover, we find two kinds of appropriate Lyapunov functions that under some sufficient conditions, global stability of the system is obtained. Finally, numerical examples are also presented.

The immune system of the cancer patient’s body and the viral lytic cycle play important roles in cancer virotherapy. Most mathematical models for virotherapy do not include these two agents simultaneously. In this paper, based on clinical observations we propose a mathematical model including the time of the viral lytic cycle, the viral burst size, and the immune system response. The proposed model is a nonlinear system of delay differential equations in which the period of the viral lytic cycle is modeled as a delay parameter and is used as the bifurcation parameter. We analyze the stability of equilibrium points and the existence of Hopf bifurcation and obtain some conditions for the stability of equilibrium points in terms of the burst size and delay parameter. Finally, we confirm the results with a numerical example and describe them from a biological point of view.

In this paper, we proposed and studied a delayed HIV pathogenesis model with saturation incidence, both virus-to-cell and cell-to-cell transmission. We address the basic reproduction number R0, the characteristic equations, and local stability of feasible equilibria are established. Where the delay incorporates both virus-to-cell and cell-to-cell transmission. Moreover, we discuss the existence of Hopf Bifurcation when a delay is used as a bifurcation parameter. Numerical simulations are performed to satisfy our theoretical results.

The contemporary theoretical inquest concerns itself with an updated mathematical model involving intraguild (IG) predation in which the IG predator acts as a generalist predator with the inclusion of harvesting in the resource population. Due attention is paid to the positivity and boundedness of the outcomes of the system under consideration. All the conceivable ecologically feasible equilibria are explored for their existence and stability under certain conditions. Special emphasis is put forward on the consequence of harvesting for the present model system. The occurrences of Hopf-bifurcation with respect to harvesting parameters involved in the harvesting effort of the model system are captured. The subsistence of the possible bionomic equilibria is, however, not ruled out from the present pursuit. The optimal harvesting policy is initiated and duly carried out with Pontryagin’s maximum principle. Numerical simulations are performed towards the end to comply with the objectives of the agreement of the numerical outcomes with their analytical counterparts and the applicability of the model is validated thereby.

‎In this paper we study the covid-19 disease with treatment and control to spread it with different measures‎. ‎The model equations are analysed from the general MC Kendrick equations for age structured populations‎. ‎The existence,positiveness,boundedness and stability of equilibria are studied as they depend on the prey's natural carrying capacity‎. ‎The main result of this paper is the three age group population‎, ‎how to control and avoid to infect the disease from predator with local,global stability and Hopf bifurcation method also utilised.Finally the result of this model prey predator where numerical examples using maple software of Rossler type.

This paper deals with an epidemiological system with stage-structured, harvesting and refuge for only prey, the disease of type (SIS) is just in the immature of the prey. The sufficient conditions guaranteeing the occurrence of local bifurcation and the Hopf bifurcation for the system are obtained. Further, the validity of our main results was demonstrated by numerical analysis.

In this paper, the local bifurcation conditions that occur near each of the equilibrium points of the eco-epidemiological system of one prey population apparition with two diseases in the same population of predator have been studied and analyzed, near E1,E2,E3,E4 and E5, a transcritical bifurcation can occur, a saddle-node bifurcation happened near E5. Pitchfork bifurcation was occurrences at E2,E3,E4 and E5. Moreover conditions for Hopf- bifurcation was studied near both of one disease stable point E3,E4 and E5 . About elucidation of the status of local bifurcation the associated of the set of hypothetical parameters with numerical results which assert our analytical results of this model.

In this study, the mathematical model of four differential equations for organisms that describe the effect of anti-predation behavior, age stage and toxicity have been analyzed. Local bifurcation and Hopf bifurcation have been studied by changing a parameter of a model to study the dynamic behavior determined by bifurcation curves and the occurrence states of bifurcation saddle node, transcritical and pitch fork bifurcation. The potential equilibrium point at which Hopf bifurcation occurs has been determined and the results of the bifurcation behavior analysis have been fully presented using numerical simulation.

‎The present study aims are to analyze a delay tumor-immune fractional-order system to describe the rivalry among the immune system and tumor cells. Given that the dynamics of this system depend on the time delay parameter, we examine the impact of time delay on this system to attain better compatibility with actuality. For this purpose, we analytically evaluated the stability of the system’s equilibrium points. It is shown that Hopf bifurcation occurs in the fractional system when the delay parameter passes a certain value. Finally, by using numerical simulations, the analytical results were compared to the numerical results to acquire several dynamical behaviors of this system.

A predator-prey model was extended to include nonlinear harvesting of the predator guided by its population, such that harvesting is only implemented if the predator population exceeds an economic threshold. Theoretical results showed that the harvesting system undergoes multiple bifurcations, including fold, supercritical Hopf, Bogdanov-Takens and cusp bifurcations. We determine stability and dynamical behaviors of the equilibrium of this system. Numerical simulation results are given to support our theoretical results.

We present and investigate the delayed model of HIV infection for drug therapy. The stability of the equilibrium states, disease free and infected equilibrium states are derived and the existence of Hopf bifurcation analysis is studied. We show that the system is asymptotically stable and the stability is lost in a range due to length of the delay, then Hopf bifurcation occurs when τ exceeds the critical value. At last numerical simulations are provided to verify the theoretical results.

The stability and Hopf bifurcation of a nonlinear mathematical model are described by the delay differential equation proposed by Wodarz for interaction between uninfected tumor cells and infected tumor cells with the virus. By choosing τ as a bifurcation parameter, we show that the Hopf bifurcation can occur for a critical value τ. Using the normal form theory and the center manifold theory, formulas are given to determine the stability and the direction of bifurcation and other properties of bifurcating periodic solutions. Then, by changing the infection rate to two nonlinear infection rates, we investigate the stability and existence of a limit cycle for the appropriate value of τ, numerically. Lastly, we present some numerical simulations to justify our theoretical results.