36 results on '"*DELAY differential equations"'
Search Results
2. Mathematical modelling of Banana Black Sigatoka Disease with delay and Seasonality.
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Agouanet, Franklin Platini, Tankam-Chedjou, Israël, Etoua, Remy M., and Tewa, Jean Jules
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BASIC reproduction number , *PLANTAIN banana , *DELAY differential equations , *BANANAS , *MATHEMATICAL models , *PLANT diseases - Abstract
• We proposed a mathematical pathogen-host model with a time delay for the dynamics of the banana black leaf streak disease. • Model accounted for the two reproduction means of the pathogen spores, seasonality and time delay to describe the incubation. • The basic reproduction number R0 does not depend on the time delay and is related to both sexual and asexual spore production. • The stability of the system was shown to not depend on the time delay, i.e. on the duration of the incubation period. • Results proved that the control of sexual spore production is not sufficient. We provide numerical simulations. Black Sigatoka Disease, also called Black Leaf Streak Disease (BLSD), is caused by the fungus Mycosphaerella fijiensis and is arguably one of the most important pathogens affecting the banana and plantain industries. Theoretical results on its dynamics are rare, even though theoretical descriptions of epidemics of plant diseases are valuable steps toward their efficient management. In this paper, we propose a mathematical model describing the dynamics of BLSD on banana or plantain leaves within a whole field of plants. The model consists of a system of periodic non-autonomous differential equations with a time delay that accounts for the time of incubation of M. fijiensis ' spores. We compute the basic reproduction number of the disease and show that it does not depend on the time delay, meaning that the persistence of BLSD would not qualitatively change even if the incubation period of the pathogen is perturbed. We derive local and global long-term dynamics of the disease and provide numerical simulations to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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3. Existence and Stability Results of Impulsive Stochastic Partial Neutral Functional Differential Equations with Infinite Delays and Poisson Jumps.
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Anguraj, A. and Ravikumar, K.
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FUNCTIONAL differential equations , *DELAY differential equations , *STOCHASTIC difference equations , *IMPULSIVE differential equations , *STOCHASTIC differential equations , *INTEGRO-differential equations , *MATHEMATICAL models - Published
- 2020
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4. Global dynamics in sea lice model with stage structure.
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Tian, Yun, Al-Darabsah, Isam, and Yuan, Yuan
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MATHEMATICAL models , *DELAY differential equations , *BRANCHIURA (Crustacea) , *STABILITY of linear systems , *NONLINEAR analysis - Abstract
Sea lice infection is one of the major threats in the marine fishery, especially for farmed salmon. In this paper, we propose a mathematical model for the growth of sea lice with a three-stage structure: non-infectious immature, infectious immature and adults where the level of non-infectious immature development depends on the size of the adult population. We first describe the nonlinear dynamics by a system of partial differential equations, then, by mathematical techniques and an appropriate change of variables transform it into a system of delay differential equations with constant delay. We address the system threshold dynamics in the established model with respect to the adult reproduction number R s , including the global stability of the trivial steady state when R s < 1 , persistence and global attractivity of the unique positive steady state when R s > 1 . Numerical simulations are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]
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- 2018
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5. Permanence for a class of non-autonomous delay differential systems.
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Faria, Teresa
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DIFFERENTIAL algebra , *LINEAR systems , *MATHEMATICAL models , *STABILITY (Mechanics) , *DYNAMICAL systems - Abstract
We are concerned with a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of delay differential equations. Sufficient conditions for the exponential asymptotic stability of the linear system are established. By using this stability, the permanence of the perturbed nonlinear system is studied. Under more restrictive constraints on the coefficients, the system has a cooperative type behaviour, in which case explicit uniform lower and upper bounds for the solutions are obtained. As an illustration, the asymptotic behaviour of a non-autonomous Nicholson system with distributed delays is analysed. [ABSTRACT FROM AUTHOR]
- Published
- 2018
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6. Stability analysis of some equilibria in a time-delay model for cell dynamics in leukemia including the action of the immune system.
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Cândea, D., Halanay, A., and Rădulescu, I. R.
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CYTOLOGICAL research , *LEUKEMIA , *IMMUNE system , *DELAY differential equations , *MATHEMATICAL models - Abstract
In this work we analyze the stability properties of a complex, strongly nonlinear model of delay-differential equations with multiple delays, for cell evolution in leukemia. The model includes the action of the immune system. The competition on space between healthy and leukemic cell populations is taken into consideration. Three types of division that a stem-like cell can undergo, asymmetric division, self-renew and differentiation are also considered. Numerical results and simulations are discussed in relation to clinical implications of the proposed model. [ABSTRACT FROM AUTHOR]
- Published
- 2016
7. Targeting the quiescent cells in cancer chemotherapy treatment: Is it enough?
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Tridane, Abdessamad, Yafia, Radouane, and Aziz-Alaoui, M.A.
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CANCER chemotherapy , *DELAY differential equations , *BIFURCATION theory , *MATHEMATICAL models , *HOPF algebras , *HOPF bifurcations , *FUNCTIONAL differential equations - Abstract
In this work, we develop a mathematical model to study the effect of drug on the development of cancer including the quiescent compartment. The model is governed by a system of delay differential equations where the delay represents the time that the cancer cell take to proliferate. Our analytical study of the stability shows that by considering the time delay as a parameter of bifurcation, it is possible to have stability switch and oscillations through a Hopf bifurcation. Moreover, by introducing the drug intervention term, the critical delay value increases. This indicates that the system can tolerate a longer delay before oscillations start. In the end, we present some numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]
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- 2016
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8. STABILITY IN THE CLASS OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS.
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GSELMANN, ESZTER and KELEMEN, ANNA
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DELAY differential equations , *CONTINUOUS functions , *MATHEMATICAL models , *FUNCTIONAL differential equations , *TIME delay systems , *ELECTRODYNAMICS - Abstract
The main aim of this paper is the investigation of the stability problem for ordinary delay differential equations. More precisely, we would like to study the following problem. Assume that for a continuous function a given delay differential equation is fulfilled only approximately. Is it true that in this case this function is close to an exact solution of this delay differential equation?. [ABSTRACT FROM AUTHOR]
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- 2016
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9. AN ADVECTION AND AGE-STRUCTURED APPROACH TO MODELING BIRD MIGRATION AND INDIRECT TRANSMISSION OF AVIAN INFLUENZA.
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GOURLEY, STEPHEN A., JENNINGS, RACHEL, and RONGSONG LIU
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ADVECTION , *BIRD migration , *AVIAN influenza , *MIGRATORY birds , *DELAY differential equations , *MATHEMATICAL models , *INFECTIOUS disease transmission - Abstract
We model indirect transmission, via contact with viruses, of avian influenza in migratory and nonmigratory birds, taking into account age structure. Migration is modeled via a reaction-advection equation on a closed loop parametrized by arc length (the migration flyway) that starts and ends at the location where birds breed in summer. Our modeling keeps the birds together as a flock, the position of which is implicitly determined and known for all future time. Births occur when the flock passes the breeding location and are modeled using ideas of impulsive differential equations. For a migratory species the model derivation starts from age-structured reaction-advection equations with location-dependent parameters that describe local conditions. In the derivation of delay equations for the time-dependent variables representing numbers of juvenile and adult birds, these location-dependent parameters are evaluated at the flock's position, so that seasonal effects are captured indirectly but through rigorous modeling whereby we keep track of the flock's exact position and local conditions there. Sufficient conditions are obtained for the local stability of the disease-free equilibrium (for a nonmigratory species) and for the disease-free periodic solution (for a migratory species). [ABSTRACT FROM AUTHOR]
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- 2015
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10. Mathematical analysis of coral reef models.
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Li, Xiong, Wang, Hao, Zhang, Zheng, and Hastings, Alan
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CORAL reefs & islands , *MATHEMATICAL analysis , *MATHEMATICAL models , *STABILITY theory , *DELAY differential equations - Abstract
Abstract: It is acknowledged that coral reefs are globally threatened. P.J. Mumby et al. [10] constructed a mathematical model with ordinary differential equations to investigate the dynamics of coral reefs. In this paper, we first provide a detailed global analysis of the coral reef ODE model in [10]. Next we incorporate the inherent time delay to obtain a mathematical model with delay differential equations. We consider the grazing intensity and the time delay as focused parameters and perform local stability analysis for the coral reef DDE model. If the time delay is sufficiently small, the stability results remain the same. However, if the time delay is large enough, macroalgae only state and coral only state are both unstable, while they are both stable in the ODE model. Meanwhile, if the grazing intensity and the time delay are endowed some suitable values, the DDE model possesses a nontrivial periodic solution, whereas the ODE model has no nontrivial periodic solutions for any grazing rate. We study the existence and property of the Hopf bifurcation points and the corresponding stability switching directions. [Copyright &y& Elsevier]
- Published
- 2014
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11. A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis.
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Rihan, F.A., Abdel Rahman, D.H., Lakshmanan, S., and Alkhajeh, A.S.
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TIME delay systems , *IMMUNE system , *PARAMETER estimation , *SENSITIVITY analysis , *NUMBER theory , *DELAY differential equations , *TUMORS , *MATHEMATICAL models - Abstract
Abstract: Recently, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. In this paper, we present a delay differential model to describe the interactions between the effector and tumour cells. The existence of the possible steady states and their local stability and change of stability via Hopf bifurcation are theoretically and numerically investigated. Parameter estimation problem for given real observations, using least squares approach, is studied. The global stability and sensitivity analysis are also numerically proved for the model. The stability and periodicity of the solutions may depend on the time-lag parameter. The model is qualitatively consistent with the experimental observations of immune-induced tumour dormancy. The model also predicts dormancy as a transient period of growth which necessarily results in either tumour elimination or tumour escape. [Copyright &y& Elsevier]
- Published
- 2014
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12. Stability analysis of equilibria in a delay differentialequations model of CML including asymmetric division and treatment.
- Author
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Halanay, A., Cândea, D., and Rădulescu, I.R.
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DELAY differential equations , *STABILITY theory , *MATHEMATICAL models , *STEM cells , *LEUCOCYTES , *CELL cycle - Abstract
A two dimensional two-delays differential system modeling the dynamics of stem-like cells and white-blood cells in Chronic Myelogenous Leukemia under treatment is considered. Stability of equilibria is investigated and emergence of periodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventually shown. All three types of stem cell division (asymmetric division, symmetric renewal and symmetric differentiation) are present in the model. The effect of drug resistance is considered through the Goldie–Coldman law. [ABSTRACT FROM AUTHOR]
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- 2015
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13. Modeling the bursting effect in neuron systems.
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Glyzin, S., Kolesov, A., and Rozov, N.
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DELAY differential equations , *NEURONS , *NONLINEAR difference equations , *PERTURBATION theory , *MATHEMATICAL models - Abstract
We propose a new method for modeling the well-known phenomenon of 'bursting behavior' in neuron systems by invoking delay equations. Namely, we consider a singularly perturbed nonlinear difference-differential equation with two delays describing the functioning of an isolated neuron. Under a suitable choice of parameters, we establish the existence of a stable periodic motion with any prescribed number of spikes on a closed time interval equal to the period length. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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14. Bautin bifurcation in a delay differential equation modeling leukemia
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Ion, Anca Veronica and Georgescu, Raluca Mihaela
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BIFURCATION theory , *TIME delay systems , *DIFFERENTIAL equations , *LEUKEMIA , *MATHEMATICAL models , *DYNAMICAL systems , *QUALITATIVE research - Abstract
Abstract: This paper continues the work contained in two previous papers of the authors, devoted to the qualitative study of the dynamical system generated by a delay differential equation that models leukemia. The problem depends on five parameters and has two equilibria. As already known, at the non-zero equilibrium solution, for certain values of the parameters, Hopf bifurcation occurs. Our aim here is to investigate the Bautin bifurcation for the considered model. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We explore the space of parameters in a zone of biological interest and find, by direct computation, points where the first Lyapunov coefficient equals zero. These are candidates for Bautin type bifurcation. For these points we compute the second Lyapunov coefficient, that determines the type of Bautin bifurcation. The computation of the second Lyapunov coefficient requires a fourth order approximation of the center manifold, that we determine. The points with null first Lyapunov coefficient obtained are given in tables and are also plotted on surfaces of Hopf bifurcation obtained by fixing two parameters. Next we vary two parameters around a point in the space of parameters with and numerically explore the behavior of the solution. The results confirm the Bautin type-bifurcation. [Copyright &y& Elsevier]
- Published
- 2013
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15. Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect
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Chen, Shanshan and Shi, Junping
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STABILITY theory , *HOPF algebras , *BIFURCATION theory , *MATHEMATICAL models , *DELAY differential equations , *KERNEL functions - Abstract
Abstract: A reaction–diffusion model with logistic type growth, nonlocal delay effect and Dirichlet boundary condition is considered, and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior. The stability of the positive spatially nonhomogeneous positive equilibrium and associated Hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel. [Copyright &y& Elsevier]
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- 2012
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16. Global existence of bifurcated periodic solutions in a commensalism model with delays
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Zhang, Jia-Fang
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GLOBAL analysis (Mathematics) , *EXISTENCE theorems , *BIFURCATION theory , *PERIODIC functions , *MATHEMATICAL models , *DELAY differential equations , *HOPF algebras - Abstract
Abstract: This paper is concerned with a commensalism model with a discrete delay and a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the interior equilibrium is investigated. It is found that the interior equilibrium is asymptotically stable when the discrete delay is less than a certain critical value and unstable when the discrete delay is greater than this critical value. By regarding the delay as the bifurcation parameter, the existence of Hopf bifurcation is also considered. Furthermore, the properties of Hopf bifurcation are determined. In particular, the global existence of bifurcated periodic solutions is established by applying the topological global Hopf bifurcation theorem, which shows that the local Hopf bifurcations imply the global ones after the second critical value of parameter. Finally, numerical simulations supporting the theoretical analysis are also included. [Copyright &y& Elsevier]
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- 2012
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17. Hopf bifurcation analysis of delayed model of thymic infection with HIV-1
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Balasubramaniam, P., Prakash, M., and Park, Ju H.
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HOPF algebras , *BIFURCATION theory , *MATHEMATICAL models , *HIV infections , *DELAY differential equations , *EXISTENCE theorems , *ASYMPTOTIC expansions , *NUMERICAL analysis - Abstract
Abstract: In this paper, a delayed differential equation model that describes infection of thymus with HIV-1 is considered. We first investigate the existence and stability of the equilibria and then we study the effect of the time delay on the stability of the infected equilibrium. Criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. Finally, by using a delay as a bifurcation parameter, the existence of Hopf bifurcation is investigated. Numerical simulations are presented to illustrate the analytical results. [Copyright &y& Elsevier]
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- 2012
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18. Mathematical Modelling of Cancer Stem Cells Population Behavior.
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Beretta, E., Capasso, V., and Morozova, N.
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CANCER stem cells , *CANCER , *CELL culture , *CANCER cells , *CELL communication , *DELAY differential equations , *MATHEMATICAL models - Abstract
Recent discovery of cancer stem cells in tumorigenic tissues has raised many questions about their nature, origin, function and their behavior in cell culture. Most of current experiments reporting a dynamics of cancer stem cell populations in culture show the eventual stability of the percentages of these cell populations in the whole population of cancer cells, independently of the starting conditions. In this paper we propose a mathematical model of cancer stem cell population behavior, based on specific features of cancer stem cell divisions and including, as a mathematical formalization of cell-cell communications, an underlying field concept. We compare the qualitative behavior of mathematical models of stem cells evolution, without and with an underlying signal. In absence of an underlying field, we propose a mathematical model described by a system of ordinary differential equations, while in presence of an underlying field it is described by a system of delay differential equations, by admitting a delayed signal originated by existing cells. Under realistic assumptions on the parameters, in both cases (ODE without underlying field, and DDE with underlying field) we show in particular the stability of percentages, provided that the delay is sufficiently small. Further, for the DDE case (in presence of an underlying field) we show the possible existence of, either damped or standing, oscillations in the cell populations, in agreement with some existing mathematical literature. The outcomes of the analysis may offer to experimentalists a tool for addressing the issue regarding the possible non-stem to stem cells transition, by determining conditions under which the stability of cancer stem cells population can be obtained only in the case in which such transition can occur. Further, the provided description of the variable corresponding to an underlying field may stimulate further experiments for elucidating the nature of “instructive" signals for cell divisions, underlying a proper pattern of the biological system. [ABSTRACT FROM PUBLISHER]
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- 2012
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19. The nature of Hopf bifurcation for the Gompertz model with delays
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Piotrowska, Monika J. and Foryś, Urszula
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BIFURCATION theory , *DELAY differential equations , *TIME delay systems , *STABILITY (Mechanics) , *MATHEMATICAL models , *DYNAMICS - Abstract
Abstract: In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of . The function describing stability of HB is periodic with respect to . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data. [Copyright &y& Elsevier]
- Published
- 2011
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20. Stability of delay integro-differential equations using a spectral element method
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Khasawneh, Firas A. and Mann, Brian P.
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INTEGRO-differential equations , *DELAY differential equations , *STABILITY (Mechanics) , *KERNEL functions , *MATHEMATICAL models , *INTEGRALS , *SPECTRAL theory - Abstract
Abstract: This paper describes a spectral element approach for studying the stability of delay integro-differential equations (DIDEs). In contrast to delay differential equations (DDEs) with discrete delays that act point-wise, the delays in DIDEs are distributed over a period of time through an integral term. Although both types of delays lead to an infinite dimensional state-space, the analysis of DDEs with distributed delays is far more involved. Nevertheless, the approach that we describe here is applicable to both autonomous and non-autonomous DIDEs with smooth bounded kernel functions. We also describe the stability analysis of DIDEs with special kernels (gamma-type kernel functions) via converting the DIDE into a higher order DDE with only discrete delays. This case of DIDEs is of practical importance, e.g., in modeling wheel shimmy phenomenon. A set of case studies are then provided to show the effectiveness of the proposed approach. [Copyright &y& Elsevier]
- Published
- 2011
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21. Global existence of periodic solutions in a six-neuron BAM neural network model with discrete delays
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Xu, Changjin, He, Xiaofei, and Li, Peiluan
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ARTIFICIAL neural networks , *ASSOCIATIVE storage , *EXISTENCE theorems , *MATHEMATICAL models , *BIFURCATION theory , *NUMERICAL solutions to delay differential equations , *NUMERICAL solutions to functional differential equations , *NUMERICAL analysis - Abstract
Abstract: In this paper, a six-neuron BAM neural network model with discrete delays is considered. Using the global Hopf bifurcation theorem for FDE due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc. 350 (1998) 4799–4838] and the Bendixson''s criterion for high-dimensional ODE due to Li and Muldowney [On Bendixson'' criterion, J. Differential Equations 106 (1994) 27–39], a set of sufficient conditions for the system to have multiple periodic solutions are derived when the sum of delays is sufficiently large. [Copyright &y& Elsevier]
- Published
- 2011
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22. Hopf bifurcation in a delayed differential–algebraic biological economic system
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Zhang, Guodong, Zhu, Lulu, and Chen, Boshan
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BIFURCATION theory , *DELAY differential equations , *DIFFERENTIAL algebra , *BIOLOGICAL systems , *ECONOMIC systems , *MATHEMATICAL models , *COMPUTER simulation , *VOLTERRA equations - Abstract
Abstract: In this paper, we consider a differential–algebraic biological economic system with time delay where the model with Holling type II functional response incorporates a constant prey refuge and prey harvesting. By considering time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the differential–algebraic biological economic system based on the new normal form approach of the differential–algebraic system and the normal form approach and the center manifold theory. Finally, numerical simulations illustrate the effectiveness of our results. [Copyright &y& Elsevier]
- Published
- 2011
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23. Mathematical model of marine protected areas.
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BEREZANSKY, L., IDELS, L., and KIPNIS, M.
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MATHEMATICAL models , *MARINE parks & reserves , *FISHERIES , *DIFFERENTIAL equations , *POPULATION dynamics - Abstract
We consider two regions with a fish population that is dispersing between the two areas, and fishing takes place only in region 2, with region 1 established as no-fishing zone. Marine protected areas (MPAs) have been promoted as conservation and fishery management tools, and at present, there are over 1300 MPAs in the world. A new mathematical model of an MPA that reflects the complexity of the natural setting is presented. The resulting model of an age-structured fish population belongs to a class of non-linear systems of differential equations with delay. New easily verifiable sufficient conditions for the existence, boundedness, permanence and stability of the positive internal steady-state solutions are obtained. From the point of view of fishery managers, the existence of stable solutions is necessary for planning harvesting strategies and sustaining the fishing grounds. Numerical simulations illustrate qualitative behaviour of the model, including stability switches. [ABSTRACT FROM PUBLISHER]
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- 2011
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24. On a thermomechanical milling model
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Chełminski, Krzysztof, Hömberg, Dietmar, and Rott, Oliver
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THERMOELASTICITY , *NUMERICAL solutions to delay differential equations , *STABILITY (Mechanics) , *MILLING cutters , *MATHEMATICAL models , *COMPUTER simulation , *LINEAR systems - Abstract
Abstract: This paper deals with a new mathematical model to characterize the interaction between machine and workpiece in a milling process. The model consists of a harmonic oscillator equation for the dynamics of the cutter and a linear thermoelastic workpiece model. The coupling through the cutting force adds delay terms and further nonlinear effects. After a short derivation of the governing equations it is shown that the complete system admits a unique weak solution. A numerical solution strategy is outlined and complemented by numerical simulations of stable and unstable cutting conditions. [Copyright &y& Elsevier]
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- 2011
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25. Convergence of two implicit numerical schemes for diffusion mathematical models with delay
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García, P., Castro, M.A., Martín, J.A., and Sirvent, A.
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STOCHASTIC convergence , *STABILITY (Mechanics) , *PARTIAL differential equations , *DELAY differential equations , *MATHEMATICAL models , *DIFFUSION processes , *IMPLICIT functions - Abstract
Abstract: In this paper two implicit numerical difference schemes for mixed problems for a delay diffusion model are proposed. Consistence, convergence and some properties of stability for these schemes are studied. Illustrative examples of numerical results are also included. [ABSTRACT FROM AUTHOR]
- Published
- 2010
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26. Stability of Delay Equations Written as State Space Models.
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MANN, B. P. and PATEL, B. R.
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TIME delay systems , *ENGINEERING , *STABILITY (Mechanics) , *FUNCTIONAL differential equations , *DELAY differential equations , *FINITE element method , *MATHEMATICAL models - Abstract
In this paper we describe a new approach to examine the stability of delay differential equations that builds upon prior work using temporal finite element analysis. In contrast to previous analyses, which could only be applied to second-order delay differential equations, the present manuscript develops an approach which can be applied to a broader class of systems: systems that may be written in the form of a state space model. A primary outcome from this work is a generalized framework to investigate the asymptotic stability of autonomous delay differential equations with a single time delay. Furthermore, this approach is shown to be applicable to time-periodic delay differential equations and equations that are piecewise continuous. [ABSTRACT FROM AUTHOR]
- Published
- 2010
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27. On the Use of Delay Equations in Engineering Applications.
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KYRYCHKO, Y. N. and HOGAN, S. J.
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STABILITY (Mechanics) , *ENGINEERING , *DELAY differential equations , *MATHEMATICAL models , *HYBRID systems , *SPATIAL systems - Abstract
This paper is a review of applications of delay differential equations to different areas of engineering science. Starting with a general overview of delay models, we present some recent results on the use of retarded, advanced and neutral delay differential equations. An emerging area for modeling with the help of delay equations is real-time dynamic substructuring, or hybrid testing. We introduce the main idea of this technique together with the latest advances. Special emphasis is given to the development of the theory and applications of partial delay differential equations. The review concludes with a summary of some open problems and questions concerning the analysis of spatially extended delayed systems. [ABSTRACT FROM AUTHOR]
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- 2010
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28. Oscillatory behavior in microbial continuous culture with discrete time delay
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Lian, Hansheng, Feng, Enmin, Li, Xiaofang, Ye, Jianxiong, and Xiu, Zhilong
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TIME delay systems , *MATHEMATICAL models , *ORGANIC compounds , *FERMENTATION , *DELAY differential equations , *NUMERICAL analysis , *SIMULATION methods & models , *PHASE diagrams - Abstract
Abstract: By introducing discrete time delay into the model for producing 1,3-propanediol by microbial continuous fermentation, we consider the stability and Hopf bifurcation of the delay differential system. Through numerical simulations, we get the rule of branch value changing with parameter and draw the pictures of periodic solutions and phase diagrams with specified parameters. The effect of time delay suggests that the system can qualitatively describe oscillatory phenomena occurring in the experiment. [Copyright &y& Elsevier]
- Published
- 2009
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29. Ill-posed problems in thermomechanics
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Dreher, Michael, Quintanilla, Ramón, and Racke, Reinhard
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MATHEMATICAL physics , *HEAT conduction , *MATHEMATICAL models , *MATHEMATICAL analysis , *VISCOELASTICITY , *DELAY differential equations - Abstract
Abstract: Several thermomechanical models have been proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models, among those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat conduction models can be interpreted as formal expansions of delay equations. The delay equations are shown to be ill-posed, as are the formal expansions of higher order — in contrast to lower-order expansions leading to Fourier’s or Cattaneo’s law. The ill-posedness is proved, showing the lack of continuous dependence on the data, and thus showing that these models (delay or higher-order expansion ones) are highly explosive. In this note we shall present conditions for when this happens. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
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30. Mathematical analysis of a sex-structured HIV/AIDS model with a discrete time delay
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Mukandavire, Zindoga, Chiyaka, Christinah, Garira, Winston, and Musuka, Godfrey
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MATHEMATICAL analysis , *LYAPUNOV functions , *MATHEMATICAL models , *HIV infection transmission , *TIME delay systems , *DELAY differential equations , *EPIDEMIOLOGY , *QUALITATIVE research - Abstract
Abstract: A sex-structured mathematical model for heterosexual transmission of HIV/AIDS with explicit incubation period is presented as a system of discrete delay differential equations. The epidemic threshold and equilibria for the model are determined and stabilities are examined. The disease-free equilibrium is shown to be locally and globally stable when the basic reproductive number is less than unity. We use the Lyapunov functional approach to show that the endemic equilibrium is locally asymptotically stable. Further comprehensive qualitative analysis of the model including persistence and permanence are investigated. [Copyright &y& Elsevier]
- Published
- 2009
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31. Stability analysis of delay differential equation models of HIV-1 therapy for fighting a virus with another virus
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Lv, Cuifang and Yuan, Zhaohui
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DELAY differential equations , *LYAPUNOV functions , *STABILITY (Mechanics) , *SYMMETRY (Physics) , *HIV infections , *THERAPEUTICS , *MATHEMATICAL models , *MATHEMATICAL analysis - Abstract
Abstract: Considering two kinds of delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells, we develop and analyze a mathematical model for HIV-1 therapy by fighting a virus with another virus. For the different values of the basic reproduction number and the second basic reproduction number, we investigate the stability of the infection-free equilibrium, the single-infection equilibrium and the double-infection equilibrium. We conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number. Our results have potential applications in HIV-1 therapy. The approach we use here is a combination of analysis of characteristic equations, Fluctuation Lemma and Lyapunov function. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
32. Oscillation and global asymptotic stability of a neuronic equation with two delays.
- Author
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El-Morshedy, Hassan A. and El-Matary, B. M.
- Subjects
- *
OSCILLATION theory of difference equations , *DIFFERENTIAL equations , *ASYMPTOTIC theory of algebraic ideals , *NUMERICAL solutions to delay differential equations , *MATHEMATICAL models , *NEURONS - Abstract
In this paper we study the oscillatory and global asymptotic stability of a single neuron model with two delays and a general activation function. New sufficient conditions for the oscillation and nonoscillation of the model are given. We obtain both delay-dependent and delay-independent global asymptotic stability criteria. Some of our results are new even for models with one delay. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
33. HOPF BIFURCATION ANALYSIS FOR A MATHEMATICAL MODEL OF P53-MDM2 INTERACTION.
- Author
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NEAMŢU, MIHAELA, HORHAT, RAUL FLORIN, and OPRIŞ, DUMITRU
- Subjects
- *
PROTEIN-protein interactions , *MATHEMATICAL models , *BIFURCATION theory , *DELAY differential equations , *STABILITY (Mechanics) - Abstract
In this paper we analyze a simple mathematical model which describes the interaction between proteins P53 and Mdm2. For the stationary state we discuss the local stability and the existence of a Hopf bifurcation. We study the direction and stability of the bifurcating periodic solutions by choosing the delay as a bifurcation parameter. Finally, we will offer some numerical simulations and present our conclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
34. SPATIAL SPREAD OF RABIES REVISITED: INFLUENCE OF AGE-DEPENDENT DIFFUSION ON NONLINEAR DYNAMICS.
- Author
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Chunhua Ou and Jianhong Wu
- Subjects
- *
RABIES , *INFECTIOUS disease transmission , *DELAY differential equations , *MATHEMATICAL models , *POPULATION biology , *SPATIAL ecology - Abstract
We consider the spatio-temporal patterns of disease spread involving structured populations. We start with a general model framework in population biology and spatial ecology where the individual's spatial movement behaviors depend on its maturation status, and we show how delayed reaction diffusion equations with nonlocal interactions arise naturally. We then consider the impact of this delayed nonlocal interaction on the disease spread by revisiting the spatial spread of rabies in continental Europe during the period between 1945 and 1985. We show how the distinction of territorial patterns between juvenile and adult foxes, the main carriers of the rabies under consideration, yields a class of partial differential equations involving delayed and nonlocal terms that are implicitly defined by a hyperbolic-parabolic equation, and we show how incorporating this distinction into the model leads to a formula describing the relation of the minimal wave speed and the maturation time of foxes. We show how the homotopy argument developed by Chow, Lin, and Mallet-Paret can be applied to obtain the existence of a heteroclinic orbit between a disease-free equilibrium and an endemic state for the spatially averaged system of delay differential equations, and we illustrate how the technique developed by Faria, Huang, and Wu can be used to establish the existence of a family of traveling wavefronts in the neighborhood of the heteroclinic orbit for the corresponding spatial model. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
35. Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics
- Author
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Adimy, Mostafa, Crauste, Fabien, and Ruan, Shigui
- Subjects
- *
STABILITY (Mechanics) , *MATHEMATICAL models , *MATHEMATICS , *MATHEMATICAL statistics , *BONE marrow - Abstract
Abstract: We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay and show that the distributed delay can destabilize the entire system. In particular, it is shown that a Hopf bifurcation can occur. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
36. Numerical treatment of nonlinear mixed delay differential equations
- Author
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Ismail, Gamal A.F.
- Subjects
- *
MATHEMATICAL models , *MYELINATED nerve fibers , *DIFFERENTIAL equations , *BESSEL functions - Abstract
Abstract: The mathematical modeling of many problems in biology (the model for myelinated axons) has been formulated as nonlinear delay differential equations. Some basic properties of the myelinated axons are discussed. The numerical scheme for solving this model is described. The convergence and stability of numerical treatments that are discussed analytically are tested. The dependence of both the solution and the internodal delay τ on the various physical parameters are investigated and its numerical results are presented and interpreted. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
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