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Journal articles on the topic 'Functional equilibrium equations'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Murakami, Satoru. "Stable equilibrium point of some diffusive functional differential equations." Nonlinear Analysis: Theory, Methods & Applications 25, no. 9-10 (November 1995): 1037–43. http://dx.doi.org/10.1016/0362-546x(95)00097-f.

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2

Tian, Xiaohong, and Rui Xu. "Global dynamics of a predator-prey system with Holling type II functional response." Nonlinear Analysis: Modelling and Control 16, no. 2 (April 25, 2011): 242–53. http://dx.doi.org/10.15388/na.16.2.14109.

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In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The existence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the glob

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3

Lotfi, El Mehdi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi. "Partial Differential Equations of an Epidemic Model with Spatial Diffusion." International Journal of Partial Differential Equations 2014 (February 10, 2014): 1–6. http://dx.doi.org/10.1155/2014/186437.

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The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the ba

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4

Meng, Xin-You, and Jiao-Guo Wang. "Analysis of a delayed diffusive model with Beddington–DeAngelis functional response." International Journal of Biomathematics 12, no. 04 (May 2019): 1950047. http://dx.doi.org/10.1142/s1793524519500475.

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In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington–DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theor

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5

BENKHALTI, R., and K. EZZINBI. "A HARTMAN-GROBMAN THEOREM FOR SOME PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 10, no. 05 (May 2000): 1165–69. http://dx.doi.org/10.1142/s0218127400000839.

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We show that the flow of some partial functional differential equations has a global attractor. As a conseqsuence we prove that the flow near a hyperbolic equilibrium is equivalent to its variational equation.

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Arora, Vivek K., and George J. Boer. "Simulating Competition and Coexistence between Plant Functional Types in a Dynamic Vegetation Model." Earth Interactions 10, no. 10 (May 1, 2006): 1–30. http://dx.doi.org/10.1175/ei170.1.

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Abstract The global distribution of vegetation is broadly determined by climate, and where bioclimatic parameters are favorable for several plant functional types (PFTs), by the competition between them. Most current dynamic global vegetation models (DGVMs) do not, however, explicitly simulate inter-PFT competition and instead determine the existence and fractional coverage of PFTs based on quasi-equilibrium climate–vegetation relationships. When competition is explicitly simulated, versions of Lotka–Volterra (LV) equations developed in the context of interaction between animal species are alm

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7

Faria, Teresa, and Luis T. Magalhães. "Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 4 (1995): 759–76. http://dx.doi.org/10.1017/s030821050003033x.

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This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around

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8

Wang, Hanxiao, and Jiongmin Yong. "Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 22. http://dx.doi.org/10.1051/cocv/2021027.

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An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate

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9

MOAWAD, S. M. "Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field." Journal of Plasma Physics 79, no. 5 (June 14, 2013): 873–83. http://dx.doi.org/10.1017/s0022377813000627.

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AbstractThe equilibrium and stability properties of ideal magnetohydrodynamics (MHD) of compressible flow in a gravitational field with a translational symmetry are investigated. Variational principles for the steady-state equations are formulated. The MHD equilibrium equations are obtained as critical points of a conserved Lyapunov functional. This functional consists of the sum of the total energy, the mass, the circulation along field lines (cross helicity), the momentum, and the magnetic helicity. In the unperturbed case, the equilibrium states satisfy a nonlinear second-order partial diff

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10

Hénot, Olivier. "On polynomial forms of nonlinear functional differential equations." Journal of Computational Dynamics 8, no. 3 (2021): 307. http://dx.doi.org/10.3934/jcd.2021013.

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<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial different

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11

VLADIMIROV, V. A., and K. I. ILIN. "On the energy instability of liquid crystals." European Journal of Applied Mathematics 9, no. 1 (February 1998): 23–36. http://dx.doi.org/10.1017/s0956792597003288.

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The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is

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12

Zakharov, Anatoly Yu. "Manifestations of Short-Range and Long-Range Parts of Interatomic Potentials In Rearrangement Processes of Multicomponent Condensed Systems." Solid State Phenomena 138 (March 2008): 347–54. http://dx.doi.org/10.4028/www.scientific.net/ssp.138.347.

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Helmholtz free energy functional for generalized lattice model is reduced to Ginzburg- Landau-like form. Connections between interatomic potentials characteristics and parameters of Ginzburg-Landau-like functional are established. Equations for equilibrium distributions of species in multicomponent systems are derived. Equations of rearrangement kinetics of multicomponent systems are obtained. Description of the rearrangement processes via non-classical partial differential equations is proposed.

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13

Seguin, Brian, Yi-chao Chen, and Eliot Fried. "Closed Unstretchable Knotless Ribbons and the Wunderlich Functional." Journal of Nonlinear Science 30, no. 6 (May 22, 2020): 2577–611. http://dx.doi.org/10.1007/s00332-020-09630-z.

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Abstract In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is

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14

Sun, Caixia, Lele Li, and Jianwen Jia. "Hopf bifurcation of an HIV-1 virus model with two delays and logistic growth." Mathematical Modelling of Natural Phenomena 15 (2020): 16. http://dx.doi.org/10.1051/mmnp/2019038.

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The paper establish and investigate an HIV-1 virus model with logistic growth, which also has intracellular delay and humoral immunity delay. The local stability of feasible equilibria are established by analyzing the characteristic equations. The globally stability of infection-free equilibrium and immunity-inactivated equilibrium are studied using the Lyapunov functional and LaSalles invariance principle. Besides, we prove that Hopf bifurcation will occur when the humoral immune delay pass through the critical value. And the stability of the positive equilibrium and Hopf bifurcations are inv

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15

LI, ZHE, and RUI XU. "STABILITY ANALYSIS OF A RATIO-DEPENDENT CHEMOSTAT MODEL WITH TIME DELAY AND VARIABLE YIELD." International Journal of Biomathematics 03, no. 02 (June 2010): 243–53. http://dx.doi.org/10.1142/s1793524510000921.

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A chemostat model with time delay, variable yield and ratio-dependent functional response is investigated. By analyzing the corresponding characteristic equations, the local stability of a boundary equilibrium and a positive equilibrium is discussed and the existence of Hopf bifurcation is established. By using the comparison arguments, sufficient conditions are obtained for the global stability of the boundary equilibrium. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium. Finally, numerical simula

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16

Yan, Caijuan, and Jianwen Jia. "Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/109372.

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We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. Ifℛ0>1, we obtain sufficient conditions under which the

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17

ZOU, WEI, JIEHUA XIE, and ZUOLIANG XIONG. "STABILITY AND HOPF BIFURCATION FOR AN ECO-EPIDEMIOLOGY MODEL WITH HOLLING-III FUNCTIONAL RESPONSE AND DELAYS." International Journal of Biomathematics 01, no. 03 (September 2008): 377–89. http://dx.doi.org/10.1142/s179352450800031x.

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In this paper, a system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. The invariance of non-negativity, nature of boundary equilibrium and global stability are analyzed. It also shows that positive equilibrium is locally asymptotically stable when time delay τ = τ1 + τ2 is suitable small, while a loss of stability by a Hopf bifurcation can occur around the positive equilibrium as the delays increase.

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18

Liu, Chao, and Qingling Zhang. "Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/184174.

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We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for

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19

Saxena, Prashant, and Basant Lal Sharma. "On equilibrium equations and their perturbations using three different variational formulations of nonlinear electroelastostatics." Mathematics and Mechanics of Solids 25, no. 8 (April 27, 2020): 1589–609. http://dx.doi.org/10.1177/1081286520911073.

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We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character – considering either the electric field [Formula: see text], the electric displacement [Formula: see text] or the electric polarization [Formula: see text]. The first variation of the energy functional results in the set of Euler–Lagrange partial differential equations, which are the equilibrium equations, boundary conditions and certain constitutive equations for the electroelastic syst

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20

Iqbal, Naveed, and Ranchao Wu. "Pattern formation by fractional cross-diffusion in a predator–prey model with Beddington–DeAngelis type functional response." International Journal of Modern Physics B 33, no. 25 (October 10, 2019): 1950296. http://dx.doi.org/10.1142/s0217979219502965.

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In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has b

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21

Chen, Mengye, Liang You, Jie Tang, Shasha Su, and Ruiming Zhang. "Analysis of a Viral Infection Model with Delayed Nonlytic Immune Response." Discrete Dynamics in Nature and Society 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/235420.

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We investigate the dynamical behavior of a virus infection model with delayed nonlytic immune response. By analyzing corresponding characteristic equations, the local stabilities of two boundary equilibria are established. By using suitable Lyapunov functional and LaSalle’s invariance principle, we establish the global stability of the infection-free equilibrium. We find that the infection free equilibriumE0is globally asymptotically stable whenR0⩽1, and the infected equilibrium without immunityE1is local asymptotically stable when1<R0⩽1+bβ/cd. Under the conditionR0>1+bβ/cdwe obtain the

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22

Huang, T., and S. Chucheepsakul. "Large Displacement Analysis of a Marine Riser." Journal of Energy Resources Technology 107, no. 1 (March 1, 1985): 54–59. http://dx.doi.org/10.1115/1.3231163.

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A method of static analysis for a marine riser experiencing large displacements is presented. The method is suitable for analyzing a riser having a known top tension and a possible slippage at the top slip joint. Utilizing the stationary condition of a functional coupled with an equilibrium equation, one can conveniently obtain the equilibrium configuration numerically. The configuration is expressed in terms of the rectangular coordinates. The functional representing the energy and work of the riser system is expressed in terms of the horizontal coordinate which is parameterized in terms of t

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23

Zhao, Huitao, Yiping Lin, and Yunxian Dai. "Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays." Abstract and Applied Analysis 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/321930.

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A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obt

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24

Zuo, Wenjie, and Junjie Wei. "Stability and bifurcation in a ratio-dependent Holling-III system with diffusion and delay." Nonlinear Analysis: Modelling and Control 19, no. 1 (January 20, 2014): 132–53. http://dx.doi.org/10.15388/na.2014.1.9.

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A diffusive ratio-dependent predator-prey system with Holling-III functional response and delay effects is considered. Global stability of the boundary equilibrium and the stability of the unique positive steady state and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated in detail, by the maximum principle and the characteristic equations. Ratio-dependent functional response exhibits rich spatiotemporal patterns. It is found that, the system without delay is dissipative and uniformly permanent under certain conditions, the delay can destabilize the po

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25

Mei, Hongwei, and Jiongmin Yong. "Equilibrium strategies for time-inconsistent stochastic switching systems." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 64. http://dx.doi.org/10.1051/cocv/2018051.

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An optimal control problem is considered for a stochastic differential equation containing a state-dependent regime switching, with a recursive cost functional. Due to the non-exponential discounting in the cost functional, the problem is time-inconsistent in general. Therefore, instead of finding a global optimal control (which is not possible), we look for a time-consistent (approximately) locally optimal equilibrium strategy. Such a strategy can be represented through the solution to a system of partial differential equations, called an equilibrium Hamilton–Jacob–Bellman (HJB) equation whic

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26

Kolesnichenko, Aleksandr Vladimirovich. "On the construction of a family of anomalous-diffusion Fokker–Planck−Kolmogorov’s equations based on the Sharma–Taneja–Mittal entropy functional." Mathematica Montisnigri 51 (August 2021): 74–95. http://dx.doi.org/10.20948/mathmontis-2021-51-6.

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A logical scheme for constructing thermodynamics of anomalous stochastic systems based on the nonextensive two-parameter (κ, ς) -entropy of Sharma–Taneja–Mittal (SHTM) is considered. Thermodynamics within the framework (2 - q) -statistics of Tsallis was constructed, which belongs to the STM family of statistics. The approach of linear nonequilibrium thermodynamics to the construction of a family of nonlinear equations of Fokker−Planck−Kolmogorov (FPK), is used, correlated with the entropy of the STM, in which the stationary solution of the diffusion equation coincides with the corresponding ge

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27

An, Qiguang, and Qingfeng Zhu. "Partially Observed Nonzero-Sum Differential Game of BSDEs with Delay and Applications." Mathematical Problems in Engineering 2020 (June 19, 2020): 1–10. http://dx.doi.org/10.1155/2020/3518961.

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A class of partially observed nonzero-sum differential games for backward stochastic differential equations with time delays is studied, in which both game system and cost functional involve the time delays of state variables and control variables under each participant with different observation equations. A necessary condition (maximum principle) for the Nash equilibrium point to this kind of partially observed game is established, and a sufficient condition (verification theorem) for the Nash equilibrium point is given. A partially observed linear quadratic game is taken as an example to il

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28

Wu, Jianhong, and H. I. Freedman. "Monotone Semiflows Generated by Neutral Functional Differential Equations With Application to Compartmental Systems." Canadian Journal of Mathematics 43, no. 5 (October 1, 1991): 1098–120. http://dx.doi.org/10.4153/cjm-1991-064-1.

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AbstractThis paper is devoted to the machinery necessary to apply the general theory of monotone dynamical systems to neutral functional differential equations. We introduce an ordering structure for the phase space, investigate its compatibility with the usual uniform convergence topology, and develop several sufficient conditions of strong monotonicity of the solution semiflows to neutral equations. By applying some general results due to Hirsch and Matano for monotone dynamical systems to neutral equations, we establish several (generic) convergence results and an equivalence theorem of the

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29

Montanino, Andrea, Gianluca Alaimo, and Ettore Lanzarone. "A gradient-based optimization method with functional principal component analysis for efficient structural topology optimization." Structural and Multidisciplinary Optimization 64, no. 1 (March 25, 2021): 177–88. http://dx.doi.org/10.1007/s00158-021-02872-9.

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AbstractStructural topology optimization (STO) is usually treated as a constrained minimization problem, which is iteratively addressed by solving the equilibrium equations for the problem under consideration. To reduce the computational effort, several reduced basis approaches that solve the equilibrium equations in a reduced space have been proposed. In this work, we apply functional principal component analysis (FPCA) to generate the reduced basis, and we couple FPCA with a gradient-based optimization method for the first time in the literature. The proposed algorithm has been tested on a l

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30

Moon, Jun, and Wonhee Kim. "Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models." Mathematics 8, no. 10 (September 28, 2020): 1669. http://dx.doi.org/10.3390/math8101669.

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We consider the indefinite, linear-quadratic, mean-field-type stochastic zero-sum differential game for jump-diffusion models (I-LQ-MF-SZSDG-JD). Specifically, there are two players in the I-LQ-MF-SZSDG-JD, where Player 1 minimizes the objective functional, while Player 2 maximizes the same objective functional. In the I-LQ-MF-SZSDG-JD, the jump-diffusion-type state dynamics controlled by the two players and the objective functional include the mean-field variables, i.e., the expected values of state and control variables, and the parameters of the objective functional do not need to be (posit

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31

Wang, Lingshu, and Guanghui Feng. "Global Stability and Hopf Bifurcation of a Predator-Prey Model with Time Delay and Stage Structure." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/431671.

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A delayed predator-prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of persistence theory on infinite dimensional systems, it is proved that the system is permanent. By using Lyapunov functions and the LaSalle invariant principle, the global stability of each of the feasible equilibria of

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32

Fruman, Mark D., та Theodore G. Shepherd. "Symmetric Stability of Compressible Zonal Flows on a Generalized Equatorial β Plane". Journal of the Atmospheric Sciences 65, № 6 (1 червня 2008): 1927–40. http://dx.doi.org/10.1175/2007jas2582.1.

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Abstract Sufficient conditions are derived for the linear stability with respect to zonally symmetric perturbations of a steady zonal solution to the nonhydrostatic compressible Euler equations on an equatorial β plane, including a leading order representation of the Coriolis force terms due to the poleward component of the planetary rotation vector. A version of the energy–Casimir method of stability proof is applied: an invariant functional of the Euler equations linearized about the equilibrium zonal flow is found, and positive definiteness of the functional is shown to imply linear stabili

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Peng, Miao, and Zhengdi Zhang. "Bifurcation analysis and control of a delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response." Journal of Vibration and Control 26, no. 13-14 (December 30, 2019): 1232–45. http://dx.doi.org/10.1177/1077546319892144.

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A delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response is proposed and explored in this study. We discuss the positivity and the existence of equilibrium points. By choosing time delay as the bifurcation parameter and analyzing the relevant characteristic equations, the local stability of the trivial equilibrium, the predator-extinction equilibrium, and the coexistence equilibrium of the system is investigated. In accordance with the normal form method and center manifold theorem, the property analysis of Hopf bifurcation of the system is obtai

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34

HU, GUANG-PING, WAN-TONG LI, and XIANG-PING YAN. "HOPF BIFURCATION AND STABILITY OF PERIODIC SOLUTIONS IN THE DELAYED LIÉNARD EQUATION." International Journal of Bifurcation and Chaos 18, no. 10 (October 2008): 3147–57. http://dx.doi.org/10.1142/s0218127408022317.

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In this paper, the classical Liénard equation with a discrete delay is considered. Under the assumption that the classical Liénard equation without delay has a unique stable trivial equilibrium, we consider the effect of the delay on the stability of zero equilibrium. It is found that the increase of delay not only can change the stability of zero equilibrium but can also lead to the occurrence of periodic solutions near the zero equilibrium. Furthermore, the stability of bifurcated periodic solutions is investigated by applying the normal form theory and center manifold reduction for function

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35

Jiang, Zhichao, Wenzhi Zhang, Jing Zhang, and Tongqian Zhang. "Dynamical Analysis of a Phytoplankton–Zooplankton System with Harvesting Term and Holling III Functional Response." International Journal of Bifurcation and Chaos 28, no. 13 (December 12, 2018): 1850162. http://dx.doi.org/10.1142/s0218127418501626.

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A toxin-producing phytoplankton and zooplankton system is investigated. Considering that zooplankton can be harvested for food in some bodies of water, the harvesting term is introduced to zooplankton population. Firstly, from the ordinary differential equation (ODE) system, we obtain the global asymptotic stability of equilibrium and optimal capture problem. Secondly, based on the ODE system, the diffusion term is introduced and the global asymptotic stability of the steady state solution is obtained. As a result, the diffusion cannot affect the global asymptotic stability of equilibrium, and

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36

Vorotnikov, V. I. "On Problem of Partial Stability for Functional Differential Systems with Holdover." Mekhatronika, Avtomatizatsiya, Upravlenie 20, no. 7 (July 4, 2019): 398–404. http://dx.doi.org/10.17587/mau.20.398-404.

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The theory of systems of functional differential equations is a significant and rapidly developing sphere of modern mathematics which finds extensive application in complex systems of automatic control and also in economic, modern technical, ecological, and biological models. Naturally, the problems arises of stability and partial stability of the processes described by the class of the equation. The article studies the problem of partial stability which arise in applications either from the requirement of proper performance of a system or in assessing system capability. Also very effective is

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37

Liu, Junli, and Tailei Zhang. "Stability and Hopf Bifurcation Analysis of a Plant Virus Propagation Model with Two Delays." Discrete Dynamics in Nature and Society 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/7126135.

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To understand the interaction between the insects and the plants, a system of delay differential equations is proposed and studied. We prove that if R0≤1, the disease-free equilibrium is globally asymptotically stable for any length of time delays by constructing a Lyapunov functional, and the system admits a unique endemic equilibrium if R0>1. We establish the sufficient conditions for the stability of the endemic equilibrium and existence of Hopf bifurcation. Using the normal form theory and center manifold theorem, the explicit formulae which determine the stability, direction, and other

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38

Zheng, ZuoHuan, and XiLiang Li. "Necessary and sufficient conditions for the existence of equilibrium in abstract non-autonomous functional differential equations." Science China Mathematics 53, no. 8 (June 16, 2010): 2045–59. http://dx.doi.org/10.1007/s11425-010-3012-0.

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39

Xie, Xiaoliang, and Wen Zhang. "Hopf bifurcations in a three-species food chain system with multiple delays." Open Mathematics 15, no. 1 (April 26, 2017): 508–19. http://dx.doi.org/10.1515/math-2017-0039.

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Abstract This paper is concerned with a three-species Lotka-Volterra food chain system with multiple delays. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the stability of the positive equilibrium and existence of Hopf bifurcations are investigated. Furthermore, the direction of bifurcations and the stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoreti

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Kanatnikov, A. N., and A. P. Krishchenko. "Functional Method of Localization and LaSalle Invariance Principle." Mathematics and Mathematical Modeling, no. 1 (May 4, 2021): 1–12. http://dx.doi.org/10.24108/mathm.0121.0000256.

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A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other s

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Yang, Peng, and Yuanshi Wang. "Periodic Solutions of a Delayed Eco-Epidemiological Model with Infection-Age Structure and Holling Type II Functional Response." International Journal of Bifurcation and Chaos 30, no. 01 (January 2020): 2050011. http://dx.doi.org/10.1142/s021812742050011x.

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This paper is devoted to the study of a new delayed eco-epidemiological model with infection-age structure and Holling type II functional response. Firstly, the disease transmission rate function among the predator population is treated as the piecewise function concerning the incubation period [Formula: see text] of the epidemic disease and the model is rewritten as an abstract nondensely defined Cauchy problem. Besides, the prerequisite which guarantees the presence of the coexistence equilibrium is achieved. Secondly, via utilizing the theory of integrated semigroup and the Hopf bifurcation

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Bai, Yuzhen, and Xiaopeng Zhang. "Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay." Abstract and Applied Analysis 2011 (2011): 1–22. http://dx.doi.org/10.1155/2011/463721.

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This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form theory and center manifold reduction for partial functional differential equations.

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Masoumi, S., M. Akhlaghi, and M. Salehi. "Multi-scale analysis of viscoelastic–viscoplastic laminated composite plates using generalized differential quadrature method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 227, no. 7 (October 25, 2012): 1406–16. http://dx.doi.org/10.1177/0954406212464929.

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Multi-scale analysis of laminated composite plates with viscoelastic–viscoplastic behavior of matrix is studied. Simplified unit cell method is developed to derive a new formulation for analysis of composite materials, including viscoelastic–viscoplastic matrix. The viscoelastic behavior of the matrix is modeled using Boltzmann superposition principle and the creep compliance is modeled using Prony series. Zapas–Crissman functional model is applied to obtain viscoplastic behavior of the matrix. In structural level, equations of equilibrium of laminated composite plate in terms of displacements

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Zhang, Xuebing, and Hongyong Zhao. "Harvest control for a delayed stage-structured diffusive predator–prey model." International Journal of Biomathematics 10, no. 01 (November 15, 2016): 1750004. http://dx.doi.org/10.1142/s1793524517500048.

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In this paper, we have considered a delayed stage-structured diffusive prey–predator model, in which predator is assumed to undergo exploitation. By using the theory of partial functional differential equations, the local stability of an interior equilibrium is established and the existence of Hopf bifurcations at the interior equilibrium is also discussed. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Finally, the complex dynamics are obtai

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Xiao, Zaowang, Zhong Li, Zhenliang Zhu, and Fengde Chen. "Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge." Open Mathematics 17, no. 1 (March 26, 2019): 141–59. http://dx.doi.org/10.1515/math-2019-0014.

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Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on dens

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Bracken, Paul. "Shape equations for two-dimensional manifolds with nonempty boundary based on a variational method." International Journal of Geometric Methods in Modern Physics 17, no. 06 (April 30, 2020): 2050082. http://dx.doi.org/10.1142/s0219887820500826.

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A smooth surface is considered which has a curved boundary. A system of exterior differential forms is introduced which describes the surface and boundary curves completely in the moving frame approach. A total free energy functional is defined based on these forms for which an equilibrium equation and boundary conditions of the surface are derived by calculating the variation of the total free energy. These results can be applied to a surface with several freely exposed edges.

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BERETTA, E., P. FERGOLA, and C. TENNERIELLO. "CHEMOSTAT EQUATIONS FOR A PREDATOR-PREY CHAIN WITH DELAYED NUTRIENT RECYCLING." Journal of Biological Systems 03, no. 02 (June 1995): 483–94. http://dx.doi.org/10.1142/s0218339095000459.

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The problem of the equilibrium stability for the chemostat equation is considered in the special case of a biotic species feeding on one limiting nutrient and predated by another biotic species. Both the biotic species through the decomposition process can return with delay to the chemostat fraction of dead biomass as new nutrient. The delay kernels of nutrient recycling are assumed to be general L2(0, +∞) non-negative functions which admit up to second order finite moments. Two approaches are adopted: the first one applies when both the biotic species have a self-regulating term in their evol

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ZHANG, CUN-HUA, and XIANG-PING YAN. "STABILITY AND HOPF BIFURCATIONS IN A DELAYED PREDATOR–PREY SYSTEM WITH A DISTRIBUTED DELAY." International Journal of Bifurcation and Chaos 19, no. 07 (July 2009): 2283–94. http://dx.doi.org/10.1142/s0218127409024062.

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This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bi

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SAMANTA, G. P., A. Mondal, D. Sahoo, and P. Dolai. "A prey-predator system with herd behaviour of prey in a rapidly fluctuating environment." Mathematics in Applied Sciences and Engineering 1, no. 1 (December 6, 2019): 16–26. http://dx.doi.org/10.5206/mase/8196.

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A statistical theory of non-equilibrium fluctuation in damped Volterra-Lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. The method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. The characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response.

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Di Francesco, Marco, Klemens Fellner, and Peter A. Markowich. "The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2100 (August 21, 2008): 3273–300. http://dx.doi.org/10.1098/rspa.2008.0214.

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We study the long-time asymptotics of reaction–diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functiona

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