Existence Of Hopf Bifurcation Research Articles

Analysis of stability and bifurcation and design of optimal control for phytoplankton–zooplankton–fish model with additional food and fish harvesting

ABSTRACT This paper investigates the dynamics of a four-dimensional plankton-fish system that contains three communities of plankton: non-toxic phytoplankton, toxic-producing phytoplankton, and zooplankton. The zooplankton consumes both non-toxic and toxic-producing phytoplankton. First, the positivity and boundedness of the solutions in the plankton-fish model are evaluated. Then, we derive the conditions for the existence of ecologically possible equilibria and their local and global asymptotic nature. Furthermore, we evaluate the criteria for the existence of Hopf-bifurcation and transcritical bifurcation. By applying Pontryagin’s maximum principle, we investigate the optimal control of fish harvesting, where fishing effort is taken as a control parameter. By using phase-space diagrams, time responses, bifurcation diagrams, and numerical analyses validate the theoretical results. The numerical results including maximum Lyapunov exponent verify chaotic dynamics of the system. We notice that a suitable choice of harvesting, phytoplankton growth rate, and additional food parameters can control the chaotic behavior of the presented model. The proposed novel mathematical model gives deeper insights to theoretical ecologists on how toxic-producing phytoplankton affects zooplankton and fish populations in aquatic systems. Furthermore, the proposed model is more appropriate for fish harvesting to maximize profit as well as species conversion in the presence of toxic phytoplankton.

Hopf bifurcation analysis of a two‐delayed diffusive predator–prey model with spatial memory of prey

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.

Global dynamical behavior of a delayed cytokine-enhanced viral infection model with nonlinear incidence

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.

Delay induced Hopf bifurcation and its control in a fractional p53 protein oscillator model

The p53 protein is essential for cancer prevention. Research on p53 mathematical models is popular since its oscillatory dynamics were observed. However, there is little research on fractional-order p53 oscillators, as well as on Hopf bifurcation control. We therefore introduce fractional-order into a p53 oscillator model and propose a delayed state feedback bifurcation controller. The stability of positive equilibrium and the existence of Hopf bifurcation are studied analytically regarding delay as the bifurcation parameter in both controlled and uncontrolled scenarios. Numerical simulations are consistent with the analytical results. We reveal that a smaller fractional-order facilitates the stability of fixed point and a larger feedback gain advances the Hopf bifurcation point. The work may inspire cancer therapy by artificially controlling the occurrence of p53 oscillations.

Hopf Bifurcation and Turing Instability of a Delayed Diffusive Zooplankton–Phytoplankton Model with Hunting Cooperation

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results.

Optimal control and bifurcation analysis of a delayed fractional-order epidemic model for the COVID-19 pandemic

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

Stability and Bifurcation of a Gordon–Schaefer Model with Additive Allee Effect

The rarity of species increases its market price, consequently leading to the overexploitation of the species and even the extinction of the species. We study how the harvest intensity and the additive Allee effect impact on the Gordon–Schaefer model. In addition, by Sotomayor’s theorem and Poincaré–Andronov theorem, we prove the existence of Hopf bifurcation, saddle-node bifurcation and transcritical bifurcation, respectively. Finally, we illustrate our results by numerical simulations. We find that both the cost per unit of harvest and the additive Allee effect have a significant impact on human exploitation of the population. As the additive Allee effect reduces to the weak Allee effect, the lower harvest cost encourages humans to increase the exploitation of species. This threshold is a switch that controls the strong Allee effect. If it exceeds its threshold, then the motivation of humans to exploit the species increases.

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic

The coronavirus disease 2019 (COVID-19) remains serious around the world and causes huge deaths and economic losses. Understanding the transmission dynamics of diseases and providing effective control strategies play important roles in the prevention of epidemic diseases. In this paper, to investigate the effect of delays on the transmission of COVID-19, we propose a delayed SEIR model to describe COVID-19 virus transmission, where two delays indicating the incubation and recovery periods are introduced. For this system, we prove its solutions are nonnegative and ultimately bounded with the nonnegative initial conditions. Furthermore, we calculate the disease-free and endemic equilibrium points and analyze the asymptotical stability and the existence of Hopf bifurcations at these equilibrium points. Then, by taking the weighted sum of the opposite number of recovered individuals at the terminal time, the number of exposed and infected individuals during the time horizon, and the system cost of control measures as the cost function, we present a delay optimal control problem, where two controls represent the social contact and the pharmaceutical intervention. Necessary optimality conditions of this optimal control problem are exploited to characterize the optimal control strategies. Finally, numerical simulations are performed to verify the theoretical analysis of the stability and Hopf bifurcations at the equilibrium points and to illustrate the effectiveness of the obtained optimal strategies in controlling the COVID-19 epidemic.

Bifurcation Analysis of a Class of Two-Delay Lotka–Volterra Predation Models with Coefficient-Dependent Delay

In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation.

Stability and codimension 2 bifurcations in an [formula omitted] model with incubation delay

The aim of this paper is to investigate the complex nonlinear dynamics, such as the existence and effect of various types of bifurcations, in an infectious disease model system. An SIR delay model that accounts for saturated incidence, saturated treatment, and self-protection in the susceptible population is considered. The model system has a disease free equilibrium, and its local stability is determined by the basic reproduction number for any delay value. The existence of multiple endemic equilibria is obtained, and these equilibria may switch stability due to variation of bifurcation parameters. When incubation delay is zero, Bogdanov–Takens bifurcation occurs, which implies the existence of Hopf bifurcation, saddle–node, and homoclinic bifurcations. Thus, the disease will persist within or on the homoclinic loop and otherwise dies out. We also find the existence of backward bifurcation and forward (transcritical) bifurcation. The delay-induced stability switches are observed due to Hopf bifurcations, leading to endemic bubbles. Incubation delay affects the stability of equilibria when the model system has multiple endemic equilibria, so multi-stability is observed. Due to incubation delay, the two frequency Hopf–Hopf bifurcation and Bogdanov–Takens bifurcation are found. Thus, both codimension 1 and codimension 2 bifurcations in the effect of incubation delay are exhibited in this study. Numerically, we notice that with increasing incubation delay, the endemic equilibrium either changes stability (from stable to unstable or unstable to stable) or remains unstable with different frequencies of oscillations.

Stability analysis of a pine wood nematode prevention and control model with delay

Pine wilt disease is a destructive forest disease with strong infectivity, a wide spread range and high difficulty in prevention and control. Since controlling Monochamus alternatus, the vector of pine wood nematode (Bursaphelenchus xylophilus) can reduce the occurrence of pine wilt disease efficiently, the parasitic natural enemy of M. alternatus, Dastarcus helophoroides, is introduced in this paper. Considering the influence of parasitic time of D. helophoroides on the control effect, based on the mutualistic symbiosis and parasitic relationship among pine wood nematode, M. alternatus and D. heloporoides, this paper establishes a pine wood nematode prevention and control model with delay. Then, the stability of positive equilibrium and the existence of Hopf bifurcation are discussed. Besides, we obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Finally, numerical simulations with two sets of meaningful parameters selected by means of statistical analysis are carried out to support the theoretical findings. Through the comparative analysis of numerical simulations, the factors affecting the control effect of pine wilt disease are obtained, and some suggestions are put forward for practical control in the forest.

Hopf bifurcation in an age-structured predator–prey system with Beddington–DeAngelis functional response and constant harvesting

In this paper, an age-structured predator–prey system with Beddington–DeAngelis (B–D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β(a)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (a)$$\\end{document} are assumed to be piecewise functions related to their maturation period τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented.

Rich dynamics of a delayed Filippov avian-only influenza model with two-thresholds policy

A nonsmooth Filippov avian-only influenza model with threshold strategies of culling susceptible and/or infected birds under different conditions is proposed to control the spread of avian influenza. The time delay representing the incubation period of avian influenza is incorporated into our model to make it more realistic, which is different from the traditional Filippov model. The stability of various types of equilibria and the existence of Hopf bifurcation are researched. Moreover, the existence of the sliding mode and its dynamics are investigated by Filippov convexity method. The theoretical analysis and numerical simulations indicate that, according to the value of thresholds and time delay, all solutions eventually converge to the regular equilibrium, pseudoequilibrium or stable periodic solution. We also indicate that time delay has a great influence on the sliding mode. Due to the many factors involved, and our main purpose is to study the impact of time delay on system stability, we have only conducted a simple analysis of the impact of time delay on the sliding mode. Furthermore, the boundary bifurcation switching stable regular equilibrium or stable limit cycle to a stable pseudoequilibrium can be exhibited by numerical simulations. Finally, with the increase of time delay, the global bifurcations from grazing bifurcation to buckling bifurcation and then to cross bifurcation are obtained. Our results indicate that although Filippov control strategies can effectively control the number of infected birds in many cases, the existence of time delay may challenge influenza control by the emergence of buckling bifurcation and cross bifurcation.

Dynamic Analysis of a Ratio-Dependent Food Chain Model with Prey-Taxis

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results.

Stability, period and chaos of the evolutionary game strategy induced by time-delay and mutation feedback

In the classical evolutionary game theory, mutation is usually considered as a constant, however strategy mutation is affected by strategies in the real game process. Therefore, the main purpose of this paper is to study the effects of mutation feedback and time delays on strategy dynamics, where mutation is a linear feedback related to strategy. First, we construct a co-evolutionary game model with two time delays induced by mutation feedback and analyze the existence and stability of the equilibria of the non-time delay system. The conditions for the coexistence of the two strategies and the mutation rate are obtained. Second, Sotomayor’s theorem is used to explore the transcritical bifurcation of the system. Then, the existence of Hopf bifurcation is investigated by using feedback delay and payoff delay as bifurcation parameters in the time-delay system. Furthermore, we discuss the direction of the Hopf bifurcation, stability and periodic change of the periodic solution in detail. Finally, a series of numerical simulations are used to describe the theoretical analysis. The main results are as follows: (i) Mutation causes negative feedback to cooperative strategy. (ii) As the time delay increases, the stable equilibrium point becomes unstable, and which branches a stable limit cycle. When the time delay continues to increase sufficiently, the stable limit cycle becomes unstable and produces irregular oscillation and chaos. (iii) When the two time delays are large enough, the coexistence of the two strategies becomes that defective strategy is dominant, and the mutation rate also reaches the maximum.

The role of zooplankton in the growth of algal bloom: a mathematical study

Here a nonlinear mathematical model for an aquatic ecosystem containing both non-toxic and toxin-producing algal blooms is formulated and analyzed. The presence of harmful algae impacts the aquatic population, subsequently affecting the human population that relies on zooplankton (fish, crabs, etc.). It also deteriorates the water quality, and it is advisable not to use water from lakes containing harmful algae. Here, our focus is to understand the dynamics of aquatic ecosystems when we have both types of algae, one that is non-toxic and acts as food to zooplankton, and another that is toxin-producing algae that causes death to zooplankton. The predation of zooplankton on algae influences the growth of the algae population. First, we consider a deterministic model, and later, this model is extended to stochastic model to check whether it has any impact on the dynamics of the populations under consideration. Various equilibria of the proposed model are derived, and the local stability of these equilibria is thoroughly examined. The existence of Hopf-bifurcation is also demonstrated for a suitable set of parameters. The results of deterministic and stochastic models are compared using numerical simulation. Upon examining our results and simulation data, it is evident that modifying specific key parameters, such as the growth rate of non-toxic algae due to nutrient intake and the parameter associated with the predation of non-toxic algae by zooplankton, has a substantial impact on algal bloom dynamics. Minor adjustments to these parameters can result in an augmented presence of non-toxic algae and a simultaneous decrease in toxin-producing algae, aligning with the primary objectives set by policymakers.

Stability and Hopf Bifurcation Analysis of A Fractional-Order BAM Neural Network with Two Delays Under Hybrid Control

In this paper, considering that fractional-order calculus can more accurately describe memory and genetic properties, we introduce fractional integral operators into neural networks and discuss the stability and Hopf bifurcation of a fractional-order bidirectional associate memory (BAM) neural network with two delays. In addition, the hybrid controller is proposed to achieve Hopf bifurcation control of the system. By taking two time delays as the bifurcation parameters and analyzing of the corresponding characteristic equation, stability switching curves of the controllable system for two delays are obtained. The direction of the characteristic root crossing the imaginary axis in stability switching curves is determined. Sufficient criteria are sequentially given to judge the local stability and the existence of Hopf bifurcation of a fractional-order BAM neural network system. The numerical simulation results show that the hybrid controller can effectively control Hopf bifurcation of a fractional-order BAM neural network system with two delays.

Bifurcation analysis on a river population model with varying boundary conditions

A delayed reaction-diffusion-advection equation subject to constant-flux and free-flow boundary conditions is considered, which models single population dynamics in a river. At first, we show the existence of a nonconstant steady state induced by the change of constant flux value. Then by analyzing the distribution of eigenvalues, the stability of the constant and nonconstant steady states and the existence of Hopf bifurcations are obtained. And an algorithm for determining the direction and stability of Hopf bifurcations is derived by applying the center manifold theory and normal form method for PFDEs. Finally, the effects of advection and downstream boundary condition on periodic oscillations are discussed theoretically and numerically.

Stability, Bifurcation and Dynamics in a Network with Delays

In real-world networks, due to complex topological structures and uncertainties such as time delays, uncontrolled systems may generate instability and complexity, thereby degrading network performance. This paper provides a detailed analysis of the stability, Hopf bifurcation, and complex dynamics of a networked system under delayed feedback control. Based on the linear stability method and Hopf bifurcation theorem, the stability of the equilibrium of the error system and the existence of Hopf bifurcation are studied. The stability of periodic solutions bifurcating from the trivial equilibrium is analyzed using normal form theory and central manifold theorem. Special focus is on the effects of the network topology and time delays on the stability and Hopf bifurcation. The theoretical results are also extended to the complex networks with asymmetric adjacent matrices. In addition, the controlled model exhibits complicated dynamical behavior via three types of codimension two bifurcations and period-doubling bifurcations that eventually lead to chaos. Numerical experiments have validated the theoretical results and indicated that delayed feedback control can effectively generate or annihilate the complicated behavior of complex networks.