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Chen L, Wang J. Bifurcation analysis in an epidemic model on adaptive networks. CHAOS (WOODBURY, N.Y.) 2023; 33:033135. [PMID: 37003833 DOI: 10.1063/5.0130068] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2022] [Accepted: 03/01/2023] [Indexed: 06/19/2023]
Abstract
In this paper, we study a delayed adaptive network epidemic model in which the local spatial connections of susceptible and susceptible individuals have time-delay effects on the rate of demographic change of local spatial connections of susceptible and susceptible individuals. We prove that the Hopf bifurcation occurs at the critical value τ0 with delay τ as the bifurcation parameter. Then, by using the normal form method and the central manifold theory, the criteria for the bifurcation direction and stability are derived. Finally, numerical simulations are presented to show the feasibility of our results.
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Affiliation(s)
- Liang Chen
- Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
| | - JinRong Wang
- Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
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2
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Luo L, Liu H, Yan F. Dynamic behavior of P53-Mdm2-Wip1 gene regulatory network under the influence of time delay and noise. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:2321-2347. [PMID: 36899536 DOI: 10.3934/mbe.2023109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
The tumor suppressor protein P53 can regulate the cell cycle, thereby preventing cell abnormalities. In this paper, we study the dynamic characteristics of the P53 network under the influence of time delay and noise, including stability and bifurcation. In order to study the influence of several factors on the concentration of P53, bifurcation analysis on several important parameters is conducted; the results show that the important parameters could induce P53 oscillations within an appropriate range. Then we study the stability of the system and the existing conditions of Hopf bifurcation by using Hopf bifurcation theory with time delays as the bifurcation parameter. It is found that time delay plays a key role in inducing Hopf bifurcation and regulating the period and amplitude of system oscillation. Meanwhile, the combination of time delays can not only promote the oscillation of the system but it also provides good robustness. Changing the parameter values appropriately can change the bifurcation critical point and even the stable state of the system. In addition, due to the low copy number of the molecules and the environmental fluctuations, the influence of noise on the system is also considered. Through numerical simulation, it is found that noise not only promotes system oscillation but it also induces system state switching. The above results may help us to further understand the regulation mechanism of the P53-Mdm2-Wip1 network in the cell cycle.
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Affiliation(s)
- LanJiang Luo
- Department of Mathematics, Yunnan Normal University, Kunming 650500, China
- Key Laboratory of Complex System Modeling and Application for Universities in Yunnan, Kunming 650500, China
| | - Haihong Liu
- Department of Mathematics, Yunnan Normal University, Kunming 650500, China
- Key Laboratory of Complex System Modeling and Application for Universities in Yunnan, Kunming 650500, China
| | - Fang Yan
- Department of Mathematics, Yunnan Normal University, Kunming 650500, China
- Key Laboratory of Complex System Modeling and Application for Universities in Yunnan, Kunming 650500, China
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3
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Yang Y, Huang G, Dong Y. Stability and Hopf bifurcation of an HIV infection model with two time delays. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:1938-1959. [PMID: 36899516 DOI: 10.3934/mbe.2023089] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
This work focuses on an HIV infection model with intracellular delay and immune response delay, in which the former delay refers to the time it takes for healthy cells to become infectious after infection, and the latter delay refers to the time when immune cells are activated and induced by infected cells. By investigating the properties of the associated characteristic equation, we derive sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model. Based on normal form theory and center manifold theorem, the stability and the direction of the Hopf bifurcating periodic solutions are studied. The results reveal that the intracellular delay cannot affect the stability of the immunity-present equilibrium, but the immune response delay can destabilize the stable immunity-present equilibrium through the Hopf bifurcation. Numerical simulations are provided to support the theoretical results.
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Affiliation(s)
- Yu Yang
- School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
| | - Gang Huang
- School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
| | - Yueping Dong
- School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
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4
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Nangue A, Tacteu Fokam WA. A class of diffusive delayed viral infection models with general incidence function and cellular proliferation. ARABIAN JOURNAL OF MATHEMATICS 2022; 12:173-199. [PMID: 36532511 PMCID: PMC9735094 DOI: 10.1007/s40065-022-00412-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/02/2022] [Accepted: 11/23/2022] [Indexed: 06/17/2023]
Abstract
We propose and analyze a new class of three dimensional space models that describes infectious diseases caused by viruses such as hepatitis B virus (HBV) and hepatitis C virus (HCV). This work constructs a Reaction-Diffusion-Ordinary Differential Equation model of virus dynamics, including absorption effect, cell proliferation, time delay, and a generalized incidence rate function. By constructing suitable Lyapunov functionals, we show that the model has threshold dynamics: if the basic reproduction number R 0 ( τ ) ≤ 1 , then the uninfected equilibrium is globally asymptotically stable, whereas if R 0 ( τ ) > 1 , and under certain conditions, the infected equilibrium is globally asymptotically stable. This precedes a careful study of local asymptotic stability. We pay particular attention to prove boundedness, positivity, existence and uniqueness of the solution to the obtained initial and boundary value problem. Finally, we perform some numerical simulations to illustrate the theoretical results obtained in one-dimensional space. Our results improve and generalize some known results in the framework of virus dynamics.
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Affiliation(s)
- Alexis Nangue
- Department of Mathematics, Higher Teachers’ Training College, University of Maroua, P.O.Box : 55, Maroua, Cameroon
| | - Willy Armel Tacteu Fokam
- Department of Mathematics and Computer Sciences, Faculty of Sciences, University of Maroua, P.O.Box : 814, Maroua, Cameroon
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5
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Gao X, Zhang Y. Bifurcation analysis and optimal control of a delayed single-species fishery economic model. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:8081-8106. [PMID: 35801458 DOI: 10.3934/mbe.2022378] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
In this paper, a single-species fishery economic model with two time delays is investigated. The system is shown to be locally stable around the interior equilibrium when the parameters are in a specific range, and the Hopf bifurcation is shown occur as the time delays cross the critical values. Then the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are discussed. In addition, the optimal cost strategy is obtained to maximize the net profit and minimize the waste by hoarding for speculation. We also design controls to minimize the waste by hoarding for the speculation of the system with time delays. The existence of the optimal controls and derivation from the optimality conditions are discussed. The validity of the theoretical results are shown via numerical simulation.
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Affiliation(s)
- Xin Gao
- College of Sciences, Northeastern University, Shenyang 110004, China
| | - Yue Zhang
- College of Sciences, Northeastern University, Shenyang 110004, China
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6
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Prousalis D, Wetzel L. Synchronization in the presence of time delays and inertia: Stability criteria. Phys Rev E 2022; 105:014210. [PMID: 35193231 DOI: 10.1103/physreve.105.014210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2021] [Accepted: 12/27/2021] [Indexed: 06/14/2023]
Abstract
Linear stability of synchronized states in networks of delay-coupled oscillators depends on the type of interaction, the network, and oscillator properties. For inert oscillator response, found ubiquitously from biology to engineering, states with time-dependent frequencies can arise. These generate side bands in the frequency spectrum or lead to chaotic dynamics. The time delay introduces multistability of synchronized states and an exponential term in the characteristic equation. Stability analysis using the resulting transcendental characteristic equation is a difficult task and is usually carried out numerically. We derive criteria and conditions that enable fast and robust analytical linear stability analysis based on the system parameters. These apply to arbitrary network topologies, identical oscillators, and delays.
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Affiliation(s)
| | - Lucas Wetzel
- Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
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7
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Yang L, Sun W, Turcotte M. Coexistence of Hopf-born rotation and heteroclinic cycling in a time-delayed three-gene auto-regulated and mutually-repressed core genetic regulation network. J Theor Biol 2021; 527:110813. [PMID: 34144050 DOI: 10.1016/j.jtbi.2021.110813] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2021] [Revised: 05/28/2021] [Accepted: 06/10/2021] [Indexed: 11/28/2022]
Abstract
In this work, we study the behavior of a time-delayed mutually repressive auto-activating three-gene system. Delays are introduced to account for the location difference between DNA transcription that leads to production of messenger RNA and its translation that result in protein synthesis. We study the dynamics of the system using numerical simulations, computational bifurcation analysis and mathematical analysis. We find Hopf bifurcations leading to stable and unstable rotation in the system, and we study the rotational behavior as a function of cyclic mutual repression parameter asymmetry between each gene pair in the network. We focus on how rotation co-exists with a stable heteroclinic flow linking the three saddles in the system. We find that this coexistence allows for a transition between two markedly different types of rotation leading to strikingly different phenotypes. One type of rotation belongs to Hopf-induced rotation while the other type, belongs to heteroclinic cycling between three saddle nodes in the system. We discuss the evolutionary and biological implications of our findings.
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Affiliation(s)
- Lei Yang
- Hangzhou Dianzi University, Hangzhou, Zhejiang, China
| | - Weigang Sun
- Hangzhou Dianzi University, Hangzhou, Zhejiang, China
| | - Marc Turcotte
- Hangzhou Dianzi University, Hangzhou, Zhejiang, China.
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8
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On Optimization Techniques for the Construction of an Exponential Estimate for Delayed Recurrent Neural Networks. Symmetry (Basel) 2020. [DOI: 10.3390/sym12101731] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
This work is devoted to the modeling and investigation of the architecture design for the delayed recurrent neural network, based on the delayed differential equations. The usage of discrete and distributed delays makes it possible to model the calculation of the next states using internal memory, which corresponds to the artificial recurrent neural network architecture used in the field of deep learning. The problem of exponential stability of the models of recurrent neural networks with multiple discrete and distributed delays is considered. For this purpose, the direct method of stability research and the gradient descent method is used. The methods are used consequentially. Firstly we use the direct method in order to construct stability conditions (resulting in an exponential estimate), which include the tuple of positive definite matrices. Then we apply the optimization technique for these stability conditions (or of exponential estimate) with the help of a generalized gradient method with respect to this tuple of matrices. The exponential estimates are constructed on the basis of the Lyapunov–Krasovskii functional. An optimization method of improving estimates is offered, which is based on the notion of the generalized gradient of the convex function of the tuple of positive definite matrices. The search for the optimal exponential estimate is reduced to finding the saddle point of the Lagrange function.
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9
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Chen M, Liu H, Yan F. Modelling and analysing biological oscillations in quorum sensing networks. IET Syst Biol 2020; 14:190-199. [PMID: 32737277 DOI: 10.1049/iet-syb.2019.0079] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022] Open
Abstract
Recent experiments have shown that the biological oscillation of quorum sensing (QS) system play a vital role not only in the process of bacterial synthesis but also in the treatment of cancer by releasing drugs. As known, these five substances TetR, CI, LacI, AiiA and AI are the core components of the QS system. However, the effects of AiiA and protein synthesis time delay on QS system are often ignored in the theoretical model, which is taken as a priority in the proposed research. Therefore, the authors developed a new mathematical model to explore the effects of AiiA and time delay on the dynamical behaviour of QS system theoretically and numerically. The results show that time delay can induce oscillation of QS system. Concretely, there exists a time delay threshold [inline-formula removed]. When time delay is less than [inline-formula removed], the system is stable. With the increasing of time delay and once it passes [inline-formula removed], oscillation behaviour occurs. Moreover, the length of time delay determines the amplitude and period of the QS oscillation. In addition, the value of [inline-formula removed] is sensitive to AiiA. These results may enhance the understanding of QS oscillations and provide new insights for bacterial release drugs to treat cancer.
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Affiliation(s)
- Menghan Chen
- Department of Mathematical, Yunnan Normal University, Kunming, People's Republic of China
| | - Haihong Liu
- Department of Mathematical, Yunnan Normal University, Kunming, People's Republic of China
| | - Fang Yan
- Department of Mathematical, Yunnan Normal University, Kunming, People's Republic of China.
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10
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Thakur NK, Ojha A. Complex dynamics of delay-induced plankton–fish interaction exhibiting defense. SN APPLIED SCIENCES 2020. [DOI: 10.1007/s42452-020-2860-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022] Open
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11
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Zhang T, Wang J, Li Y, Jiang Z, Han X. Dynamics analysis of a delayed virus model with two different transmission methods and treatments. ADVANCES IN DIFFERENCE EQUATIONS 2020; 2020:1. [PMID: 32226454 PMCID: PMC7100214 DOI: 10.1186/s13662-019-2438-0] [Citation(s) in RCA: 119] [Impact Index Per Article: 29.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2019] [Accepted: 11/29/2019] [Indexed: 05/07/2023]
Abstract
In this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.
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Affiliation(s)
- Tongqian Zhang
- College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, P.R. China
| | - Junling Wang
- College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, P.R. China
| | - Yuqing Li
- College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, P.R. China
| | - Zhichao Jiang
- Fundamental Science Department, North China Institute of Aerospace Engineering, Langfang, P.R. China
| | - Xiaofeng Han
- College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, P.R. China
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12
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Chen M, Liu H, Yan F. Oscillatory dynamics mechanism induced by protein synthesis time delay in quorum-sensing system. Phys Rev E 2019; 99:062405. [PMID: 31330665 DOI: 10.1103/physreve.99.062405] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2018] [Indexed: 01/22/2023]
Abstract
Recent experimental evidence reports that the oscillatory behavior of quorum sensing plays an extremely important role in the process of bacterial synthesis and release drug to fight cancer. As we know, the six substances AiiA, LuxI, internal AHL, external AHL, AHL substrate, and H_{2}O_{2} are the core parts of the quorum-sensing system. Here, the effects of several important factors, including time delay, variable H_{2}O_{2}, AHL synthesis rate induced by LuxI, and AHL degradation rate induced by AiiA on the oscillatory behavior of the quorum-sensing system are studied theoretically based on a part of mathematical model describing the interaction of the above six substances proposed by Prindle et al. [Nature 508, 387 (2014)10.1038/nature13238]. The results show that the time delay is a prerequisite for inducing oscillation of the quorum-sensing system. Furthermore, the length of time delay can determine the amplitude and period of oscillation. As a further matter, the change of H_{2}O_{2} concentration can induce the oscillatory behavior of the quorum-sensing system. In addition, under the regulation of H_{2}O_{2}, the period robustness of the quorum-sensing system is increased. Similarly, the quorum-sensing system exhibits periodic oscillation when AHL synthesis rate induced by LuxI less than a certain critical value, unless it displays a steady state. Additionally, a too-high or too-low level of AHL degradation rate induced by AiiA will fail to generate oscillation of the quorum-sensing system, only the intermediate level will cause oscillation. Finally, the two and three parameter regions in which the quorum-sensing system exhibits oscillation behavior are generated.
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Affiliation(s)
- Menghan Chen
- Department of Mathematics, Yunnan Normal University, Kunming 650500, China
| | - Haihong Liu
- Department of Mathematics, Yunnan Normal University, Kunming 650500, China
| | - Fang Yan
- Department of Mathematics, Yunnan Normal University, Kunming 650500, China
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13
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Abstract
In this paper, a model with time delay describing biodegradation of Microcystins (MCs) is investigated. Firstly, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Furthermore, an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the applications of the results.
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Affiliation(s)
- Keying Song
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China
| | - Wanbiao Ma
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China
- Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, University of Science and Technology Beijing, Beijing 100083, P. R. China
| | - Zhichao Jiang
- Fundamental Science Department, North China Institute of Aerospace Engineering, Langfang 065000, P. R. China
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14
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Li Z, Dai B. Global dynamics of delayed intraguild predation model with intraspecific competition. INT J BIOMATH 2019. [DOI: 10.1142/s1793524518501164] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A delayed intraguild predation (IGP) model with intraspecific competition is considered. It is shown that the delay has a destabilizing effect and induces oscillations. The global existence results of periodic solutions bifurcating from the positive equilibrium are established. It is shown that there exists at least one nontrival periodic solution when the delay passes through a certain critical value. Numerical simulations are performed to illustrate our theoretical results and show that intraspecific competition can also affect the stability of the positive equilibrium of the system.
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Affiliation(s)
- Zhenzhen Li
- School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P. R. China
| | - Binxiang Dai
- School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P. R. China
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15
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Du Y, Niu B, Wei J. Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system. CHAOS (WOODBURY, N.Y.) 2019; 29:013101. [PMID: 30709153 DOI: 10.1063/1.5078814] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Accepted: 12/09/2018] [Indexed: 06/09/2023]
Abstract
In this paper, the dynamics of a modified Leslie-Gower predator-prey system with two delays and diffusion is considered. By calculating stability switching curves, the stability of positive equilibrium and the existence of Hopf bifurcation and double Hopf bifurcation are investigated on the parametric plane of two delays. Taking two time delays as bifurcation parameters, the normal form on the center manifold near the double Hopf bifurcation point is derived, and the unfoldings near the critical points are given. Finally, we obtain the complex dynamics near the double Hopf bifurcation point, including the existence of quasi-periodic solutions on a 2-torus, quasi-periodic solutions on a 3-torus, and strange attractors.
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Affiliation(s)
- Yanfei Du
- College of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021, China
| | - Ben Niu
- Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
| | - Junjie Wei
- Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
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16
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Modeling Inhibitory Effect on the Growth of Uninfected T Cells Caused by Infected T Cells: Stability and Hopf Bifurcation. COMPUTATIONAL AND MATHEMATICAL METHODS IN MEDICINE 2018; 2018:3176893. [PMID: 31191708 PMCID: PMC6525856 DOI: 10.1155/2018/3176893] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/01/2018] [Accepted: 04/30/2018] [Indexed: 01/02/2023]
Abstract
We consider a class of viral infection dynamic models with inhibitory effect on the growth of uninfected T cells caused by infected T cells and logistic target cell growth. The basic reproduction number R0 is derived. It is shown that the uninfected equilibrium is globally asymptotically stable if R0 < 1. Sufficient conditions for the existence of Hopf bifurcation at the infected equilibrium are investigated by analyzing the distribution of eigenvalues. Furthermore, the properties of Hopf bifurcation are determined by the normal form theory and the center manifold. Numerical simulations are carried out to support the theoretical analysis.
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18
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Pradeep BGSA, Ma W, Wang W. Stability and Hopf bifurcation analysis of an SEIR model with nonlinear incidence rate and relapse. JOURNAL OF STATISTICS & MANAGEMENT SYSTEMS 2017. [DOI: 10.1080/09720510.2016.1228321] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
| | - Wanbiao Ma
- Department of Applied Mathematics and Physics, University of Science and Technology Beijing, Beijing-100083, China
| | - Wei Wang
- Department of Applied Mathematics and Physics, University of Science and Technology Beijing, Beijing-100083, China
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19
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Huang J, Liu Z, Ruan S. Bifurcation and temporal periodic patterns in a plant-pollinator model with diffusion and time delay effects. JOURNAL OF BIOLOGICAL DYNAMICS 2017; 11:138-159. [PMID: 27184331 DOI: 10.1080/17513758.2016.1181802] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
This paper deals with a plant-pollinator model with diffusion and time delay effects. By considering the distribution of eigenvalues of the corresponding linearized equation, we first study stability of the positive constant steady-state and existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. We then derive an explicit formula for determining the direction and stability of the Hopf bifurcation by applying the normal form theory and the centre manifold reduction for partial functional differential equations. Finally, we present an example and numerical simulations to illustrate the obtained theoretical results.
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Affiliation(s)
- Jirong Huang
- a School of Mathematical Sciences , Beijing Normal University , Beijing , People's Republic of China
| | - Zhihua Liu
- a School of Mathematical Sciences , Beijing Normal University , Beijing , People's Republic of China
| | - Shigui Ruan
- b Department of Mathematics , University of Miami , Coral Gables , FL , USA
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20
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Abstract
In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte (CTL) cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.
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Affiliation(s)
- Jianhua Pang
- School of Science, Guangxi University of Science and Technology, Liuzhou 545006, P. R. China
| | - Jing-An Cui
- School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, P. R. China
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21
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Rehim M, Zhang Z, Muhammadhaji A. Mathematical analysis of a nutrient-plankton system with delay. SPRINGERPLUS 2016; 5:1055. [PMID: 27462503 PMCID: PMC4940349 DOI: 10.1186/s40064-016-2435-7] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/27/2015] [Accepted: 05/26/2016] [Indexed: 11/13/2022]
Abstract
A mathematical model describing the interaction of nutrient–plankton is investigated in this paper. In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system. Moreover, it is also taken into account discrete time delays which indicates the partially recycled nutrient decomposed by bacteria after the death of biomass. In the first part of our analysis the sufficient conditions ensuring local and global asymptotic stability of the model are obtained. Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions. Numerical simulations are presented to illustrate the analytical results.
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Affiliation(s)
- Mehbuba Rehim
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 Xinjiang China
| | - Zhenzhen Zhang
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 Xinjiang China
| | - Ahmadjan Muhammadhaji
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 Xinjiang China
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22
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23
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Time-delayed model of immune response in plants. J Theor Biol 2015; 389:28-39. [PMID: 26551159 DOI: 10.1016/j.jtbi.2015.10.020] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2015] [Revised: 10/01/2015] [Accepted: 10/28/2015] [Indexed: 11/21/2022]
Abstract
In the studies of plant infections, the plant immune response is known to play an essential role. In this paper we derive and analyse a new mathematical model of plant immune response with particular account for post-transcriptional gene silencing (PTGS). Besides biologically accurate representation of the PTGS dynamics, the model explicitly includes two time delays to represent the maturation time of the growing plant tissue and the non-instantaneous nature of the PTGS. Through analytical and numerical analysis of stability of the steady states of the model we identify parameter regions associated with recovery and resistant phenotypes, as well as possible chronic infections. Dynamics of the system in these regimes is illustrated by numerical simulations of the model.
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Prakash M, Balasubramaniam P. Stability and multi-parametric Hopf bifurcation analyses of viral infection model with time delay. INT J BIOMATH 2015. [DOI: 10.1142/s179352451550059x] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
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.
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Affiliation(s)
- M. Prakash
- Department of Mathematics, Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Tamil Nadu, India
| | - P. Balasubramaniam
- Department of Mathematics, Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Tamil Nadu, India
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25
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Misra OP, Sinha P, Singh C. Dynamics of one-prey two-predator system with square root functional response and time lag. INT J BIOMATH 2015. [DOI: 10.1142/s1793524515500291] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Animals grouping together is one of the most interesting phenomena in population dynamics and different functional responses as a result of prey–predator forming groups have been considered by many authors in their models. In the present paper we have considered a model for one prey and two competing predator populations with time lag and square root functional response on account of herd formation by prey. It is shown that due to the inclusion of another competing predator, the underlying system without delay becomes more stable and limit cycles do not occur naturally. However, after considering the effect of time lag in the basic system, limit cycles appear in the case of all equilibrium points when delay time crosses some critical value. From the numerical simulation, it is observed that the length of delay is minimum when only prey population survives and it is maximum when all the populations coexist.
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Affiliation(s)
- O. P. Misra
- School of Mathematics and Allied Sciences, Jiwaji University, Gwalior (M.P.)-474011, India
| | - Poonam Sinha
- Department of Mathematics, S. M. S. Govt. Model Science College, Gwalior (M.P.)-474010, India
| | - Chhatrapal Singh
- School of Mathematics and Allied Sciences, Jiwaji University, Gwalior (M.P.)-474011, India
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26
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Shu H, Hu X, Wang L, Watmough J. Delay induced stability switch, multitype bistability and chaos in an intraguild predation model. J Math Biol 2015; 71:1269-98. [PMID: 25656348 DOI: 10.1007/s00285-015-0857-4] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2014] [Revised: 01/03/2015] [Indexed: 11/25/2022]
Abstract
In many predator-prey models, delay has a destabilizing effect and induces oscillations; while in many competition models, delay does not induce oscillations. By analyzing a rather simple delayed intraguild predation model, which combines both the predator-prey relation and competition, we show that delay in intraguild predation models promotes very complex dynamics. The delay can induce stability switches exhibiting a destabilizing role as well as a stabilizing role. It is shown that three types of bistability are possible: one stable equilibrium coexists with another stable equilibrium (node-node bistability); one stable equilibrium coexists with a stable periodic solution (node-cycle bistability); one stable periodic solution coexists with another stable periodic solution (cycle-cycle bistability). Numerical simulations suggest that delay can also induce chaos in intraguild predation models.
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Affiliation(s)
- Hongying Shu
- Department of Mathematics, Tongji University, Shanghai, 200092, P. R. China
| | - Xi Hu
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, E3B 5A3, Canada
| | - Lin Wang
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, E3B 5A3, Canada.
| | - James Watmough
- Department of Mathematics and Statistics, University of New Brunswick, Fredericton, E3B 5A3, Canada.
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27
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Mackey MC, Santillán M, Tyran-Kamińska M, Zeron ES. The utility of simple mathematical models in understanding gene regulatory dynamics. In Silico Biol 2015; 12:23-53. [PMID: 25402755 PMCID: PMC4923710 DOI: 10.3233/isb-140463] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/23/2014] [Revised: 10/22/2014] [Accepted: 10/23/2014] [Indexed: 11/17/2022]
Abstract
In this review, we survey work that has been carried out in the attempts of biomathematicians to understand the dynamic behaviour of simple bacterial operons starting with the initial work of the 1960's. We concentrate on the simplest of situations, discussing both repressible and inducible systems and then turning to concrete examples related to the biology of the lactose and tryptophan operons. We conclude with a brief discussion of the role of both extrinsic noise and so-called intrinsic noise in the form of translational and/or transcriptional bursting.
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Affiliation(s)
- Michael C. Mackey
- Departments of Physiology, Physics & Mathematics, McGill University, Montreal, Quebec, Canada
| | - Moisés Santillán
- Centro de Investigación y de Estudios Avanzados del IPN, Unidad Monterrey, Parque de Investigación e Innovación Tecnológica, Apodaca NL, México
| | | | - Eduardo S. Zeron
- Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal, México DF, México
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28
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Prakash M, Balasubramaniam P. Mathematical analysis of a model for thymus infection with discrete and distributed delays. INT J BIOMATH 2014. [DOI: 10.1142/s1793524514500703] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.
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Affiliation(s)
- M. Prakash
- Department of Mathematics, Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Tamil Nadu, India
| | - P. Balasubramaniam
- Department of Mathematics, Gandhigram Rural Institute, Deemed University, Gandhigram 624 302, Tamil Nadu, India
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29
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Abstract
The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by the threshold value R0. If R0 < 1, then the infection-free equilibrium is global asymptotically stable, that is, the cholera dies out; If R0 > 1, then the unique endemic equilibrium is global asymptotically stable, which means that the infection persists. The results obtained show that the delay does not lead to periodic oscillations. Finally, some numerical simulations support our theoretical results.
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Affiliation(s)
- YUJIE WANG
- Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, P. R. China
- Department of Mathematics, Jilin Normal University, Siping 136000, Jilin, P. R. China
| | - JUNJIE WEI
- Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, P. R. China
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30
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Symmetry Breaking in a Model of Antigenic Variation with Immune Delay. Bull Math Biol 2012; 74:2488-509. [DOI: 10.1007/s11538-012-9763-8] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2011] [Accepted: 07/27/2012] [Indexed: 11/25/2022]
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31
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ZHA LIJUAN, CUI JINGAN, ZHOU XUEYONG. RATIO-DEPENDENT PREDATOR–PREY MODEL WITH STAGE STRUCTURE AND TIME DELAY. INT J BIOMATH 2012. [DOI: 10.1142/s1793524511001556] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Ratio-dependent predator–prey models are favored by many animal ecologists recently as more suitable ones for predator–prey interactions where predation involves searching process. In this paper, a ratio-dependent predator–prey model with stage structure and time delay for prey is proposed and analyzed. In this model, we only consider the stage structure of immature and mature prey species and not consider the stage structure of predator species. We assume that the predator only feed on the mature prey and the time for prey from birth to maturity represented by a constant time delay. At first, we investigate the permanence and existence of the proposed model and sufficient conditions are derived. Then the global stability of the nonnegative equilibria are derived. We also get the sufficient criteria for stability switch of the positive equilibrium. Finally, some numerical simulations are carried out for supporting the analytic results.
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Affiliation(s)
- LIJUAN ZHA
- School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, P. R. China
| | - JING-AN CUI
- School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, P. R. China
| | - XUEYONG ZHOU
- School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, Jiangsu, P. R. China
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32
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Li YL, Xu DL, Fu YM, Zhou JX. Stability and chaotification of vibration isolation floating raft systems with time-delayed feedback control. CHAOS (WOODBURY, N.Y.) 2011; 21:033115. [PMID: 21974650 DOI: 10.1063/1.3615710] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
This paper presents a systematic study on the stability of a two-dimensional vibration isolation floating raft system with a time-delayed feedback control. Based on the generalized Sturm criterion, the critical control gain for the delay-independent stability region and critical time delays for the stability switches are derived. The critical conditions can provide a theoretical guidance of chaotification design for line spectra reduction. Numerical simulations verify the correctness of the approach. Bifurcation analyses reveal that chaotification is more likely to occur in unstable region defined by these critical conditions, and the stiffness of the floating raft and mass ratio are the sensitive parameters to reduce critical control gain.
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Affiliation(s)
- Y L Li
- State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, People's Republic of China
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33
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Crauste F, Hbid ML, Kacha A. A delay reaction-diffusion model of the dynamics of botulinum in fish. Math Biosci 2008; 216:17-29. [DOI: 10.1016/j.mbs.2008.07.012] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2008] [Revised: 07/18/2008] [Accepted: 07/23/2008] [Indexed: 11/25/2022]
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34
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Yu W, Cao J, Chen G. Stability and Hopf Bifurcation of a General Delayed Recurrent Neural Network. ACTA ACUST UNITED AC 2008; 19:845-54. [DOI: 10.1109/tnn.2007.912589] [Citation(s) in RCA: 66] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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35
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Churilov A, Medvedev A, Shepeljavyi A. Bifurcations in a mathematical model of non-basal testosterone production. ACTA ACUST UNITED AC 2008. [DOI: 10.3182/20080706-5-kr-1001.01746] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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36
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Cao J, Xiao M. Stability and Hopf Bifurcation in a Simplified BAM Neural Network With Two Time Delays. ACTA ACUST UNITED AC 2007; 18:416-30. [PMID: 17385629 DOI: 10.1109/tnn.2006.886358] [Citation(s) in RCA: 158] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
Various local periodic solutions may represent different classes of storage patterns or memory patterns, and arise from the different equilibrium points of neural networks (NNs) by applying Hopf bifurcation technique. In this paper, a bidirectional associative memory NN with four neurons and multiple delays is considered. By applying the normal form theory and the center manifold theorem, analysis of its linear stability and Hopf bifurcation is performed. An algorithm is worked out for determining the direction and stability of the bifurcated periodic solutions. Numerical simulation results supporting the theoretical analysis are also given.
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Affiliation(s)
- Jinde Cao
- Department of Mathematics, Southeast University, Nanjing 210096, China.
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37
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Adimy M, Crauste F, Ruan S. Modelling Hematopoiesis Mediated by Growth Factors With Applications to Periodic Hematological Diseases. Bull Math Biol 2006; 68:2321-51. [PMID: 17086497 DOI: 10.1007/s11538-006-9121-9] [Citation(s) in RCA: 47] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2006] [Accepted: 03/22/2006] [Indexed: 10/24/2022]
Abstract
Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to understand some blood diseases characterized by very long period oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is proposed and analyzed. The existence of a Hopf bifurcation at a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long-period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors, for reasonable cell cycle durations. These oscillations can be related to observations on some periodic hematological diseases (such as chronic myelogenous leukemia, for example).
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Affiliation(s)
- Mostafa Adimy
- Laboratoire de Mathématiques Appliquées UMR 5142, Université de Pau et des Pays de l'Adour, Avenue de l'université, 64000, Pau, France.
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38
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Wang L, Li MY. Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. Math Biosci 2006; 200:44-57. [PMID: 16466751 DOI: 10.1016/j.mbs.2005.12.026] [Citation(s) in RCA: 206] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2004] [Revised: 12/13/2005] [Accepted: 12/22/2005] [Indexed: 11/25/2022]
Abstract
A mathematical model that describes HIV infection of CD4(+) T cells is analyzed. Global dynamics of the model is rigorously established. We prove that, if the basic reproduction number R(0) < or = 1, the HIV infection is cleared from the T-cell population; if R(0) > 1, the HIV infection persists. For an open set of parameter values, the chronic-infection equilibrium P* can be unstable and periodic solutions may exist. We establish parameter regions for which P* is globally stable.
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Affiliation(s)
- Liancheng Wang
- Department of Mathematics, Kennesaw State University, Kennesaw, GA 30144, USA
| | - Michael Y Li
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1.
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39
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Abstract
We prove the global asymptotic stability of a well-known delayed negative-feedback model of testosterone dynamics, which has been proposed as a model of oscillatory behavior. We establish stability (and hence the impossibility of oscillations) even in the presence of delays of arbitrary length.
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Affiliation(s)
- German Enciso
- Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA.
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40
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Wei JJ, Velarde MG. Bifurcation analysis and existence of periodic solutions in a simple neural network with delays. CHAOS (WOODBURY, N.Y.) 2004; 14:940-953. [PMID: 15447004 DOI: 10.1063/1.1768111] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Results are provided here about the stability and bifurcation of periodic solutions for a (neural) network with n elements where delays between adjacent units and external inputs are included. The particular cases n = 2 and n = 3 are discussed in details, to explicitly illustrate the role of the delays in the corresponding bifurcation sets and the stability properties, like a Hopf bifurcation, a pitchfork bifurcation, and a Bogdanov-Takens bifurcation.
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Affiliation(s)
- J J Wei
- Instituto Pluridisciplinar, Universidad Complutense de Madrid, Paseo Juan XXIII, n. 1, 28040 Madrid, Spain
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