The threat of malware in wireless sensor network has stimulated some activities to model and analyze the malware prevalence.To understand the dynamics of malware propagation in wireless sensor network,we propose a nov...The threat of malware in wireless sensor network has stimulated some activities to model and analyze the malware prevalence.To understand the dynamics of malware propagation in wireless sensor network,we propose a novel epidemic model named as e-SEIR(susceptible-exposed-infectious-recovered)model,which is a set of delayed differential equations,in this paper.The model has taken into account the following two factors:1 Multi-state antivirus measures;2 Temporary immune period.Then,the stability and Hopf bifurcation at the equilibria of linearized model are carefully analyzed by considering the distribution of eigenvalues of characteristic equations.Both mathematical analysis and numerical simulations show that the dynamical features of the proposed model rely on the basic reproduction number R0 and time delayτ.This novel model can help us to better understand and predict the propagation behaviors of malware in wireless sensor networks.展开更多
An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existen...An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations,we show that the SIR(susceptible-infected-recovered)epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper.展开更多
A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attracti...A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attractivity of the model is studied using Lyapunov functions and LaSalle＇s invariance principle. By the uniform persistence theories, the permanence of the system and the existence of the positive equilibrium are obtained. Moreover, by the normal form theory and the center manifold presented by Hassard, a stability and Hopf bifurcation analysis of the system around positive equilibrium from a local perspective are performed. Numerical simulation is carried out to illustrate our results.展开更多
Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with...Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state.Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E＊ and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.展开更多
In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington-DeAngelis functional response and...In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington-DeAngelis functional response and discrete delay are studied. The Hopf bifurcations can be shown when the delay crosses the critical value. Furthermore, based on the normal form and the center manifold theorem, the type, stability and other properties of the bifurcating periodic solutions are determined. Finally, some numerical simulations are given to illustrate the results.展开更多
A differential-algebraic prey--predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey predator model int...A differential-algebraic prey--predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey predator model into its normal form and study its dynamics in terms of local analysis and Hopf bifurcation. By analyzing the associated characteristic equation, it is observed that the model undergoes a Hopf bifurcation at some critical value of time delay. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.展开更多
Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is...Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.展开更多
This paper analyzes the mechanism and principle of diffusion of technology diffusion on the basis of quantitative analysis. Then it sets up the diffusion model of innovation incorporating price, advertising and distri...This paper analyzes the mechanism and principle of diffusion of technology diffusion on the basis of quantitative analysis. Then it sets up the diffusion model of innovation incorporating price, advertising and distribution, the diffusion model of innovation including various kinds of consumers, and the substitute model between the new technology and the old one applied systems dynamics, optimization method, probabilistic method and simulation method on computer. Finally this paper concludes with some practical observations from a case study.展开更多
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 beha...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.展开更多
We find that the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of the fractional order in this study.There exists a clear difference in the bifurcation diagra...We find that the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of the fractional order in this study.There exists a clear difference in the bifurcation diagram between the fractional-order Hindmarsh-Rose model and the corresponding integer-order model even though the neuron undergoes a Hopf bifurcation to oscillation and then starts a period-doubling cascade to chaos with the decrease of the externally applied current.Interestingly,the discharge frequency of the fractional-order Hindmarsh-Rose model neuron is greater than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic or chaotic firing.Then we demonstrate that the firing behavior of the fractional-order Hindmarsh-Rose model neuron has a higher complexity than that of the integer-order counterpart.Also,the synchronization phenomenon is investigated in the network of two electrically coupled fractional-order model neurons.We show that the synchronization rate increases as the fractional order decreases.展开更多
In this paper, we consider a diffusive density-dependent predator-prey model with Crowley-Martin functional responses subject to Neumann boundary condition. We ana- lyze the stability of the positive equilibrium and t...In this paper, we consider a diffusive density-dependent predator-prey model with Crowley-Martin functional responses subject to Neumann boundary condition. We ana- lyze the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions through the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the center manifold theory. Finally, numerical simulations are given to verify our theoretical analysis.展开更多
A mathematical model describing the dynamics of toxin producing phytoplankton- zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multip...A mathematical model describing the dynamics of toxin producing phytoplankton- zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings.展开更多
We develop and analyze a mathematical model for the transmission dynamics of HIV that accounts for behavioral change. The contact rate is modeled by a decreasing function （response function） of HIV prevalence to ref...We develop and analyze a mathematical model for the transmission dynamics of HIV that accounts for behavioral change. The contact rate is modeled by a decreasing function （response function） of HIV prevalence to reflect a reduction in risky behavior that results from the awareness of individuals to a higher HIV prevalence. The model also includes a distributed delay representing the time needed for individuals to reduce their risky behavior. We study mathematically and numerically the impact of the response function and the distributed delay on the model＇s dynamics. Threshold values for the delay at which the system destabilizes and periodic solutions can arise through Hopf bifurcation are determined.展开更多
In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit a...In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.展开更多
基金National Natural Science Foundation of China(No.61379125)
文摘The threat of malware in wireless sensor network has stimulated some activities to model and analyze the malware prevalence.To understand the dynamics of malware propagation in wireless sensor network,we propose a novel epidemic model named as e-SEIR(susceptible-exposed-infectious-recovered)model,which is a set of delayed differential equations,in this paper.The model has taken into account the following two factors:1 Multi-state antivirus measures;2 Temporary immune period.Then,the stability and Hopf bifurcation at the equilibria of linearized model are carefully analyzed by considering the distribution of eigenvalues of characteristic equations.Both mathematical analysis and numerical simulations show that the dynamical features of the proposed model rely on the basic reproduction number R0 and time delayτ.This novel model can help us to better understand and predict the propagation behaviors of malware in wireless sensor networks.
基金supported by National Natural Science Foundation of China (Grant Nos. 11471044 and 11371058)the Fundamental Research Funds for the Central Universities
文摘An infection-age structured epidemic model with a nonlinear incidence rate is investigated.We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness for positive age-dependent equilibrium of the model.By analyzing the associated characteristic transcendental equation and applying the normal form theory presented recently for non-densely defined semilinear equations,we show that the SIR(susceptible-infected-recovered)epidemic model undergoes Zero-Hopf bifurcation at the positive equilibrium which is the main result of this paper.
文摘A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attractivity of the model is studied using Lyapunov functions and LaSalle＇s invariance principle. By the uniform persistence theories, the permanence of the system and the existence of the positive equilibrium are obtained. Moreover, by the normal form theory and the center manifold presented by Hassard, a stability and Hopf bifurcation analysis of the system around positive equilibrium from a local perspective are performed. Numerical simulation is carried out to illustrate our results.
文摘Ever since HIV was first diagnosed in human, a great number of scientific works have been undertaken to explore the biological mechanisms involved in the infection and progression of the disease. This paper deals with stability and bifurcation analyses of mathematical model that represents the dynamics of HIV infection of thymus. The existence and stability of the equilibria are investigated. The model is described by a system of delay differential equations with logistic growth term, cure rate and discrete type of time delay. Choosing the time delay as a bifurcation parameter, the analysis is mainly focused on the Hopf bifurcation problem to predict the existence of a limit cycle bifurcating from the infected steady state.Further, using center manifold theory and normal form method we derive explicit formulae to determine the stability and direction of the limit cycles. Moreover the mitosis rate r also plays a vital role in the model, so we fix it as second bifurcation parameter in the incidence of viral infection. Our analysis shows that, while both the bifurcation parameters can destabilize the equilibrium E＊ and cause limit cycles. Numerical simulations are performed to investigate the qualitative behaviors of the inherent model.
基金Acknowledgments The authors would like to thank the editors and the anonymous referees for their helpful suggestions and comments which led to the improvement of our original manuscript. This work is supported by the National Natural Science Foundation of China （Grant Nos. 11561022, 11261017）, the China Postdoctoral Science Foundation （Grant No. 2014M562008）.
文摘In this paper, regarding the time delay as a bifurcation parameter, the stability and Hopf bifurcation of the model of competition between two species in a turbidostat with Beddington-DeAngelis functional response and discrete delay are studied. The Hopf bifurcations can be shown when the delay crosses the critical value. Furthermore, based on the normal form and the center manifold theorem, the type, stability and other properties of the bifurcating periodic solutions are determined. Finally, some numerical simulations are given to illustrate the results.
基金This work was supported by National Science Foundation of China 61273008 and 61203001, Doctor Startup Fund of Liaoning Province （20131026）, Fundamental Research Funds for the Central University （N140504005） and China Scholarship Council. The authors gratefully thank referees for their valuable suggestions.
文摘A differential-algebraic prey--predator model with time delay and Allee effect on the growth of the prey population is investigated. Using differential-algebraic system theory, we transform the prey predator model into its normal form and study its dynamics in terms of local analysis and Hopf bifurcation. By analyzing the associated characteristic equation, it is observed that the model undergoes a Hopf bifurcation at some critical value of time delay. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.
基金supported by National Natural Science Foundation of China (Grant No.11171296)the Zhejiang Provincial Natural Science Foundation of China (Grant No. D7080064)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110101110010)
文摘Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.
基金This research is supported by the National Science Foundation, P. R. C., under project NSF 70171021. Supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C. The project is supp
文摘This paper analyzes the mechanism and principle of diffusion of technology diffusion on the basis of quantitative analysis. Then it sets up the diffusion model of innovation incorporating price, advertising and distribution, the diffusion model of innovation including various kinds of consumers, and the substitute model between the new technology and the old one applied systems dynamics, optimization method, probabilistic method and simulation method on computer. Finally this paper concludes with some practical observations from a case study.
文摘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.
基金supported by the National Natural Science Foundation of China(Grant Nos.11272241and 10972170)
文摘We find that the fractional-order Hindmarsh-Rose model neuron demonstrates various types of firing behavior as a function of the fractional order in this study.There exists a clear difference in the bifurcation diagram between the fractional-order Hindmarsh-Rose model and the corresponding integer-order model even though the neuron undergoes a Hopf bifurcation to oscillation and then starts a period-doubling cascade to chaos with the decrease of the externally applied current.Interestingly,the discharge frequency of the fractional-order Hindmarsh-Rose model neuron is greater than that of the integer-order counterpart irrespective of whether the neuron exhibits periodic or chaotic firing.Then we demonstrate that the firing behavior of the fractional-order Hindmarsh-Rose model neuron has a higher complexity than that of the integer-order counterpart.Also,the synchronization phenomenon is investigated in the network of two electrically coupled fractional-order model neurons.We show that the synchronization rate increases as the fractional order decreases.
文摘In this paper, we consider a diffusive density-dependent predator-prey model with Crowley-Martin functional responses subject to Neumann boundary condition. We ana- lyze the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions through the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the center manifold theory. Finally, numerical simulations are given to verify our theoretical analysis.
文摘A mathematical model describing the dynamics of toxin producing phytoplankton- zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings.
文摘We develop and analyze a mathematical model for the transmission dynamics of HIV that accounts for behavioral change. The contact rate is modeled by a decreasing function （response function） of HIV prevalence to reflect a reduction in risky behavior that results from the awareness of individuals to a higher HIV prevalence. The model also includes a distributed delay representing the time needed for individuals to reduce their risky behavior. We study mathematically and numerically the impact of the response function and the distributed delay on the model＇s dynamics. Threshold values for the delay at which the system destabilizes and periodic solutions can arise through Hopf bifurcation are determined.
文摘In this paper, a time-delayed predator-prey system is considered. The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.
