Direction Of Hopf Bifurcation Research Articles

Stability and Hopf Bifurcation Analysis of a Continuous Neural Network With Mixed Delays

A three-neuron network model with mixed delays involving multiple discrete and distributed delays is considered in this paper. Taking the discrete and distribute delays as bifurcation parameters respectively, we investigate the stability of the system structure and the conditions for the generation of Hopf bifurcation from the perspective of the distribution of the root of the characteristic equation of the linearized system at the equilibrium state of the nonlinear system. The intervals of parameters that make the system stable and unstable are also given. In addition, when the conditions of Hopf bifurcation theorem are satisfied, the calculation formulas for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution are presented by means of the central manifold theorem and normal form method. Finally, numerical experiments are carried out to support the correctness of the theoretical results in different cases. It can be concluded that the oscillation and instability caused by delays obviously affect the stability of the network.

Stationary pattern and Hopf bifurcation of a diffusive predator–prey model

This paper is concerned with a predator–prey model with prey-taxis and linear prey harvesting under the homogeneous Neumann boundary condition. The stability of the unique positive constant solution of the predator–prey model without prey-taxis is derived. Also, the emergence of Hopf bifurcation is concluded by choosing the proper Hopf bifurcation parameters. By the center manifold theorem and normal form, we compute the direction of Hopf bifurcation and the stability of the bifurcating solution. Moreover, the stationary pattern with prey-taxis is investigated. The conclusions show that prey harvesting and prey-taxis can enrich the dynamics.

Stability and bifurcations analysis in an ecoepidemic system with prey group defense and two infectious routes

In this paper, we explore an ecoepidemic model of a prey–predator type in which the prey population is infected by an SI type disease with two infectious routes (horizontal and vertical). Specifically, we consider that both susceptible and infected preys have emergent carrying capacities, which implies that both preys are involved in intraspecific and interclass competitions. In addition, we assume that the susceptible prey gathers together to defend the predator. It is found that the proposed system exhibits abundant dynamics and admits nine equilibria: three boundary equilibria, three plane equilibria, and three interior equilibria. At least one of the three populations can be eliminated from the ecosystem based on the stability of the previous two types of equilibria. According to the stability of the interior equilibria, the given ecosystem can allow the coexistence of susceptible prey–infected prey–predator. We then investigate the stability and direction of Hopf bifurcation. A sensitivity analysis is also conducted to show the effects of the six parameters which are relevant to the emergent carrying capacity and infectious routes on the studied system. Numerical simulations are conducted to verify our theoretical conclusions.

Modeling the virus-induced tumor-specific immune response with delay in tumor virotherapy

It is urgently required to design novel cancer therapies which lead to permanent cancer eradication or cancer control. Oncolytic virotherapy is a promising cancer treatment strategy using genetically engineered viruses which can selectively infect, replicate in and kill tumor cells without harming normal cells. Due to the variable interactions between tumor cells and oncolytic viruses (OVs) as well as the accordingly immune response, the impact of tumor virotherapy on tumor control is prominent. We propose a novel mathematical modeling framework based on delay differential equation to study tumor virotherapy with mediated antitumor immunity by OVs, which incorporates complex tumor–virus–immune system interactions. We initially study the existence and local stability of equilibria, and theoretically and numerically investigate the local Hopf bifurcation from the positive equilibrium by considering the time delay as a bifurcation parameter. Further, we analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution. Main results show that the time delay can induce Hopf bifurcation and result in periodic oscillations, indicating that the delayed tumor-specific CTL response induced by OVs leads to complex dynamics and may significantly influence the development process of tumor growth. Our findings also provide insight into important aspects of the virotherapy, including the dependence of the efficacy on key factors. We find that enhancing the induction of antitumor immunity by OVs can reduce the complexity of dynamics by strengthening the stability and show considerable effects on combating tumor. Further, proper choice of OVs with relative strong ability to lyse tumor cells is beneficial to ultimately control the development process of tumor growth.

Bifurcations, chaos, and multistability in a nonautonomous predator-prey model with fear.

Classical predator-prey models usually emphasize direct predation as the primary means of interaction between predators and prey. However, several field studies and experiments suggest that the mere presence of predators nearby can reduce prey density by forcing them to adopt costly defensive strategies. Adoption of such kind would cause a substantial change in prey demography. The present paper investigates a predator-prey model in which the predator's consumption rate (described by a functional response) is affected by both prey and predator densities. Perceived fear of predators leads to a drop in prey's birth rate. We also consider both constant and time-varying (seasonal) forms of prey's birth rate and investigate the model system's respective autonomous and nonautonomous implementations. Our analytical studies include finding conditions for the local stability of equilibrium points, the existence, direction of Hopf bifurcation, etc. Numerical illustrations include bifurcation diagrams assisted by phase portraits, construction of isospike and Lyapunov exponent diagrams in bi-parametric space that reveal the rich and complex dynamics embedded in the system. We observe different organized periodic structures within the chaotic regime, multistability between multiple pairs of coexisting attractors with intriguing basins of attractions. Our results show that even relatively slight changes in system parameters, perturbations, or environmental fluctuations may have drastic consequences on population oscillations. Our observations indicate that the fear effect alters the system dynamics significantly and drives an otherwise irregular system toward regularity.

The impact of time delay in the transmission of Japanese encephalitis without vaccination

In this manuscript, the influence of time delay in the transmission of Japanese encephalitis without vaccination model has been studied. The time delay is because of the existence of an incubation period during which the Japanese encephalitis virus reproduces enough in the mosquitoes with the goal that it tends to be transmitted by the mosquitoes to people. The motivation behind this manuscript is to assess the influence of the time delay it takes to infect susceptible human populations after interacting with infected mosquitoes. The steadystate and the threshold value R0 of the delay model were resolved. This value assists with setting up the circumstance that ensures the asymptotic stability of relating equilibrium points. Utilizing the delay as a bifurcation parameter, we built up the circumstance for the presence of a Hopf bifurcation. Moreover, we infer an express equation to decide the stability and direction of Hopf bifurcation at endemic equilibrium by using center manifold theory and normal structure strategy. It has been seen that delay plays a vital role in stability exchanging. Furthermore, the presence of Hopf bifurcation is affected by larger values of virus transmission rate from an infected mosquito to susceptible individuals and the natural mortality of humans in a model. Finally, to understand some analytical outcomes, the delay framework is simulated numerically.

Bifurcation analysis in a predator–prey model with strong Allee effect

Abstract 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.

Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

Bifurcation Analysis of a Delay-Induced Predator–Prey Model with Allee Effect and Prey Group Defense

In this paper, we establish a predator–prey model with focus on the Allee effect and prey group defense. The positivity and boundedness of the model, existence of equilibrium point, and stability change caused by Allee effect are studied. Bifurcation (transcritical bifurcation, Hopf bifurcation) analysis is discussed, and the direction of Hopf bifurcation is determined by calculating the first Lyapunov number. Then we introduce delay into the original model and consider the influence of delay on the stability of the model. By selecting delay as the bifurcation parameter, we obtain the existence conditions of Hopf bifurcation and the direction of Hopf bifurcation. Finally, we verify the theoretical analysis by numerical simulation. Considering both the Allee effect and the prey group defense, the dynamic behavior near the origin becomes more complex than only considering Allee effect or prey group defense in the model. Allee effect can bring the risk of extinction and the change of stability, and the delay effect can make the stable coexistence equilibrium unstable and lead to periodic oscillation.

Chaos control in a multiple delayed phytoplankton-zooplankton model with group defense and predator's interference.

Phytoplankton-zooplankton interaction is a topic of high interest among the interrelationships related to marine habitats. In the present manuscript, we attempt to study the dynamics of a three-dimensional system with three types of plankton: non-toxic phytoplankton, toxic producing phytoplankton, and zooplankton. We assume that both non-toxic and toxic phytoplankton are consumed by zooplankton via Beddington-DeAngelis and general Holling type-IV responses, respectively. We also incorporate gestation delay and toxic liberation delay in zooplankton's interactions with non-toxic and toxic phytoplankton correspondingly. First, we have studied the well-posedness of the system. Then, we analyze all the possible equilibrium points and their local and global asymptotic behavior. Furthermore, we assessed the conditions for the occurrence of Hopf-bifurcation and transcritical bifurcation. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also studied. Various time-series, phase portraits, and bifurcation diagrams are plotted to confirm our theoretical findings. From the numerical simulation, we observe that a limited increase in inhibitory effect of toxic phytoplankton against zooplankton can support zooplankton's growth, and rising predator's interference can also boost zooplankton expansion in contrast to the nature of Holling type IV and Beddington-DeAngelis responses. Next, we notice that on variation of toxic liberation delay, the delayed system switches its stability multiple times and becomes chaotic. Furthermore, we draw the Poincaré section and evaluate the maximum Lyapunov exponent in order to verify the delayed system's chaotic nature. Results presented in this article might be helpful to interpret biological insights into phytoplankton-zooplankton interactions.

Dynamic analysis of a predator-prey system with nonlinear prey harvesting and square root functional response

Abstract In this work, the dynamics of a predator-prey system considering square root type functional response for prey herd behaviour and nonlinear prey harvesting has been analyzed. The conditions under which all equilibria exist as well as the stability of every equilibrium point of the system have been investigated. The proposed model conditionally posses two types of bifurcations, Hopf bifurcation, and saddle-node bifurcation. The saddle-node bifurcation has been analyzed, where the bifurcation parameter is harvesting rate. The existence of a maximum sustainable yield to ensure both populations coexist has been discussed. The results give a clear idea that, if the harvesting rate is chosen at a proper value lesser than the maximum sustainable yield then both populations will coexist and the ecological balance will be maintained. The calculation of the first Lyapunov number provides the Hopf bifurcation direction. To verify our analytical results, several numerical simulations have been carried out.

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay.

Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

Stability and Hopf Bifurcation of a Reaction–Diffusion System with Weak Allee Effect

A delayed three-component reaction–diffusion system with weak Allee effect and Dirichlet boundary condition is considered. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained via the implicit function theorem. Moreover, taking delay as the bifurcation parameter, the Hopf bifurcation near the spatially nonhomogeneous steady-state solution is proved to occur at a critical value. Especially, the direction of Hopf bifurcation is forward and the bifurcated periodic solutions are unstable. Finally, the general results are applied to four types of three-species population models with weak Allee effect in growth.

Stability and Hopf bifurcation analysis in a Lotka–Volterra competition–diffusion–advection model with time delay effect * *This research is supported by the National Natural Science Foundation of China (No. 11871475) and the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts040).

In this paper, we consider a classical two-species Lotka–Volterra competition–diffusion–advection model with time delay effect. By utilizing the implicit function theorem, we obtain the existence of at least one spatially nonhomogeneous positive steady state under some conditions on parameters. By analyzing the corresponding characteristic equation, we show the local stability of this spatially nonhomogeneous positive steady state and the occurrence of Hopf bifurcation from it. When there is no time delay, we also study the global stability of the positive steady state. Based on the idea of Chen et al (2018 J. Differ. Equ. 264 5333–5359), the stability and direction of Hopf bifurcation are derived by introducing a weighted inner product associated with the advection rate. Finally, numerical simulations are carried out to verify the theoretical analysis results.

Bifurcation analysis in a diffusive phytoplankton\u2013zooplankton model with harvesting

A diffusive phytoplankton–zooplankton model with nonlinear harvesting is considered in this paper. Firstly, using the harvesting as the parameter, we get the existence and stability of the positive steady state, and also investigate the existence of spatially homogeneous and inhomogeneous periodic solutions. Then, by applying the normal form theory and center manifold theorem, we give the stability and direction of Hopf bifurcation from the positive steady state. In addition, we also prove the existence of the Bogdanov–Takens bifurcation. These results reveal that the harvesting and diffusion really affect the spatiotemporal complexity of the system. Finally, numerical simulations are also given to support our theoretical analysis.

Hopf bifurcation of a diffusive SIS epidemic system with delay in heterogeneous environment

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction.

Hopf Bifurcation in a Delayed Single Species Network System

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

Dynamical behavior of a delayed prey-predator-scavenger system with fear effect and linear harvesting

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.

Modeling and numerical simulations for a prey–predator model with interference among predators

In this paper, we modeled a prey–predator system with interference among predators using the Crowley–Martin functional response. The local stability and existence of Hopf bifurcation at the coexistence equilibrium of the system in the absence of diffusion are analyzed. Further, the stability of bifurcating periodic solutions is investigated. We derived the conditions for which nontrivial equilibrium is globally asymptotically stable. In addition, we study the diffusion driven instability, Hopf bifurcation of the corresponding diffusion system with zero flux boundary conditions and the Turing instability region regarding parameters are established. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are performed to illustrate the theoretical results.

Dynamics of a delayed predator-prey system with fear effect, herd behavior and disease in the susceptible prey

In this article, a delayed predator-prey system with fear effect, disease and herd behavior in prey incorporating refuge is established. Firstly, the positiveness and boundedness of the solutions is proved, and the basic reproduction number $ R_0 $ is calculated. Secondly, by analyzing the characteristic equations of the system, the local asymptotic stability of the equilibria is discussed. Then taking time delay as the bifurcation parameters, the existence of Hopf bifurcation of the system at the positive equilibrium is given. Thirdly, the global asymptotic stability of the equilibria is discussed by constructing a suitable Lyapunov function. Next, the direction of Hopf bifurcation and the stability of the periodic solution are analyzed based on the center manifold theorem and normal form theory. What's more, the impact of the prey refuge, fear effect and capture rate on system is given. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.