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Dynamic Analysis of an Algae-Bacteria Ecological Model

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Journal of Applied Mathematics and Physics 2024年第1期12卷 362-382页

作者： Gaopeng Sun Hengguo Yu School of Mathematics and Physics Wenzhou University Wenzhou China

In the paper, under the framework of exploring the interaction between algae and bacteria, an algae-bacteria ecological model was established to analyze the interaction mechanism and growth coexistence mode between algicidal bacteria and algae. Firstly, mathematical work mainly provided some threshold conditions to ensure the occurrence of transcritical bifurcation and saddle-node bifurcation, which could provide certain theoretical support for selecting key ecological environmental factors and numerical simulations. Secondly, the numerical simulation work dynamically displayed the evolution process of the bifurcation dynamic behavior of the model (2.1) and the growth coexistence mode of algae and algicidal bacteria. Finally, it was worth summarizing that intrinsic growth rate and combined capture effort of algae population had a strong influence on the dynamic behavior of the model (2.1). Furthermore, it must also be noted that transcritical bifurcation and saddle-node bifurcation were the inherent driving forces behind the formation of steady-state growth coexistence mode between algicidal bacteria and algae. In summary, it was hoped that the results of this study would contribute to accelerating the study of the interaction mechanism between algicidal bacteria and algae.

关键词： Algae Algicidal Bacteria Transcritical bifurcation saddle-node bifurcation Coexistence Mode

bifurcation control of nonlinear oscillator in primary and secondary resonance

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Journal of Central South University of Technology 2007年第6期14卷 826-831页

作者：李克安萧寒崔荣繁 Department of Mechanical and Electrical Engineering Hunan Institute of Science and Technology College of Mechanics and Aerospace Hunan University Shenyang Liming Aero-Engine (Group) Corporation

A weakly nonlinear oscillator was modeled by a sort of differential equation, a saddle-node bifurcation was found in case of primary and secondary resonance. To control the jumping phenomena and the unstable region of the nonlinear oscillator, feedback controllers were designed. bifurcation control equations were obtained by using the multiple scales method. And through the numerical analysis, good controller could be obtained by changing the feedback control gain. Then a feasible way of further research of saddle-node bifurcation was provided. Finally, an example shows that the feedback control method applied to the hanging bridge system of gas turbine is doable.

关键词： nonlinear oscillator saddle-node bifurcation feedback controller primary resonance secondary resonance gas turbine

bifurcation analysis for vibrations of a turbine blade excited by air flows

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Science China(Technological Sciences) 2016年第8期59卷 1217-1231页

作者： WANG Dan CHEN YuShu HAO ZhiFeng CAO QingJie School of Astronautics Harbin Institute of Technology Harbin 150001 China

A reduced three-degree-of-freedom model simulating the fluid-structure interactions （FSI） of the turbine blades and the on- coming air flows is proposed. The equations of motions consist of the coupling of bending and torsion of a blade as well as a van der Pol oscillation which represents the time-varying of the fluid. The 1：1 internal resonance of the system is analyzed with the multiple scale method, and the modulation equations are derived. The two-parameter bifurcation diagrams are computed. The effects of the system parameters, including the detuning parameter and the reduced frequency, on responses of the struc- ture and fluid are investigated. bifurcation curves are computed and the stability is determined by examining the eigenvalues of the Jacobian matrix. The results indicate that rich dynamic phenomena of the steady-state solutions including the sad- dle-node and Hopf bifurcations can occur under certain parameter conditions. The parameter region where the unstable solu- tions occur should be avoided to keep the safe operation of the blades. The analytical solutions are verified by the direct nu- merical simulations.

关键词： fluid-structure interaction (FSI) internal resonance two-parameter bifurcation diagram saddle-node bifurcation Hopf bifurcation direct numerical simulation

bifurcation analysis and optimal control of an epidemic model with limited number of hospital beds

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International Journal of Biomathematics 2023年第4期16卷 229-253页

作者： A.K.Misra Jyoti Maurya Department of Mathematics Institute of Science Banaras Hindu UniversityVaranasi 221005India

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are *** feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is *** show that the system exhibits transcritical *** show the existence of Bogdanov-Takens bifurcation,we have derived the normal *** have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient *** show the existence of saddle-node bifurcation,we used Sotomayor's ***,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible *** optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.

关键词： Hospital beds Hopf-bifurcation saddle-node bifurcation transcritical bifurcation Bogdanov-Takens bifurcation optimal control

bifurcation and catastrophe of seepage flow system in broken rock

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Mining Science and Technology 2009年第1期19卷 1-7页

作者： MIAO Xie-xing LI Shun-cai CHEN Zhan-qing School of Sciences China University of Mining ＆ Technology Xuzhou Jiangsu 221008 China School of Mechanical and Electrical Engineering Xuzhou Normal University Xuzhou Jiangsu 221116 China

The study of dynamical behavior of water or gas flows in broken rock is a basic research topic among a series of key projects about stability control of the surrounding rocks in mines and the prevention of some disasters such as water inrush or gas outburst and the protection of the groundwater resource. It is of great theoretical and engineering importance in respect of promo- tion of security in mine production and sustainable development of the coal industry. According to the non-Darcy property of seepage flow in broken rock dynamic equations of non-Darcy and non-steady flows in broken rock are established. By dimensionless transformation, the solution diagram of steady-states satisfying the given boundary conditions is obtained. By numerical analysis of low relaxation iteration, the dynamic responses corresponding to the different flow parameters have been obtained. The stability analysis of the steady-states indicate that a saddle-node bifurcaton exists in the seepage flow system of broken rock. Consequently, using catastrophe theory, the fold catastrophe model of seepage flow instability has been obtained. As a result, the bifurcation curves of the seepage flow systems with different control parameters are presented and the standard potential function is also given with respect to the generalized state variable for the fold catastrophe of a dynamic system of seepage flow in broken rock.

关键词： broken rock non-Darcy flow stability saddle-node bifurcation fold catastrophe

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

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Applied Mathematics and Mechanics(English Edition) 2023年第1期44卷 141-158页

作者： A.MOSLEMI M.R.HOMAEINEZHAD Faculty of Mechanical Engineering K.N.Toosi University of TechnologyTehran ***Iran

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally,but the lack of studies of real environmental conditions on these absorbers is *** present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES).In this paper,the Burgers model is assumed for the viscoelasticity in an NES,and a linear oscillator system is considered for investigating the instabilities and *** equations of motion of the coupled system are solved by using the harmonic balance and pseudo-arc-length continuation *** results show that the viscoelasticity affects the frequency intervals of the Hopf and saddle-node branches,and by increasing the stiffness parameters of the viscoelasticity,the conditions of these branches occur in larger ranges of the external force amplitudes,and also reduce the frequency range of the *** addition,increasing the viscoelastic damping parameter has the potential to completely eliminate the instability of the system and gradually reduce the amplitude of the jump phenomenon.

关键词： viscoelasticity Burgers model nonlinear energy sink(NES) saddle-node bifurcation Hopf bifurcation

Dynamical analysis of tumor-immune-help T cells system

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International Journal of Biomathematics 2019年第7期12卷 51-70页

作者： Huixia Li Shaoli Wang Fei Xu School of Mathematics and Statistics Henan University Kaifeng 475001HenanP.R.China School of Mathematics and Statistics Bioinformatics Center Henan UniversityKaifeng 475001HenanP.R.China Department of Mathematics Wilfrid Laurier University WaterlooOntarioN2L 3C5Canada

In this paper,we construct a mathematical model to investigate the interaction between the tumor cells,the immune cells and the helper T cells(HTCs).We perform math-ematical analysis to reveal the stability of the equilibria of the *** our model,the HTCs are stimulated by the identification of the presence of tumor *** investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells,which leads to the elimination of the bistable behavior of the tumor-immune system,*** equilibrium corresponding to the high steady state of tumor cells is *** immune intensity c as bifurcation parameter,there exists saddle-node ***,there exists a critical value C*,at which a Hopf bifurcation *** stability and direction of Hopf bifurcation are discussed.

关键词： Tumor-immune-helper T cells system inhibit high steady state saddle-node bifurcation Hopf bifurcation

Periodical bifurcation Analysis of a Type of Hematopoietic Stem Cell Model with Feedback Control

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International Journal of Modern Nonlinear Theory and Application 2023年第1期12卷 18-29页

作者： Suqi Ma Department of Mathematics China Agricultural University Beijing China

The delay feedback control brings forth interesting periodical oscillation bifurcation phenomena which reflect in Mackey-Glass white blood cell model. Hopf bifurcation is analyzed and the transversal condition of Hopf bifurcation is given. Both the period-doubling bifurcation and saddle-node bifurcation of periodical solutions are computed since the observed floquet multiplier overpass the unit circle by DDE-Biftool software in Matlab. The continuation of saddle-node bifurcation line or period-doubling curve is carried out as varying free parameters and time delays. Two different transition modes of saddle-node bifurcation are discovered which is verified by numerical simulation work with aids of DDE-Biftool.

关键词： bifurcation saddle-node bifurcation Period-Doubling bifurcation Hopf bifurcation Time Delay

On a reaction-diffusion model for sterile insect release method on a bounded domain

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International Journal of Biomathematics 2014年第3期7卷 119-135页

作者： Weihua Jiang Xin Li XingfuZou Department of Mathematics Harbin Institute of TechnologyHarbin~ Heilongjiang 150001~ P. R. China Department of Applied MathemaLies University of Western OntarioLondon Ontario N6A 5B7 Canada

We consider a system of partial differential equations that describes the interaction of the sterile and fertile species undergoing the sterile insect release method （SIRM）. Unlike in the previous work [M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci. 5 （1992） 221 247] where the habitat is assumed to be the one-dimensional whole space ~, we consider this system in a bounded one- dimensional domain （interval）. Our goal is to derive sufficient conditions for success of the SIRM. We show the existence of the fertile-free steady state and prove its stability. Using the releasing rate as the parameter, and by a saddle-node bifurcation analysis, we obtain conditions for existence of two co-persistence steady states, one stable and the other unstable. Biological implications of our mathematical results are that：（i） when the fertile population is at low level, the SIRM, even with small releasing rate, can successfully eradicate the fertile insects; （ii） when the fertile population is at a higher level, the SIRM can succeed as long as the strength of the sterile releasing is large enough, while the method may also fail if the releasing is not sufficient.

关键词： Sterile insect release method diffusion saddle-node bifurcation upper lowersolution method.