Constant Steady-state Solution Research Articles

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

A Passivity-Based PI Control of Quasi-Resonant Buck Converter

In order to guarantee the stability of quasi-resonant converters, the zero-current switching quasi-resonant buck topology is supplied with a PI passivity-based controller in this study. An adequate and full-order model with constant steady-state solutions is generated, via the interconnection of port-Hamiltonian systems based on the principal structure of the converter. The passivity-based approach is employed to complete the control design, and numerical validation results are provided to ensure performance estimations.

Spatial pattern formation driven by the cross-diffusion in a predator–prey model with Holling type functional response

In this paper, we consider a cross-diffusion predator–prey system with Holling type functional response. We study the local stability, Turing instability, spatial pattern formation, Hopf and Turing–Hopf bifurcation of the equilibrium. Numerical simulation with zero-flux boundary conditions discloses that the system under consideration experiences the occurrence of cross-diffusion-driven instability. The dynamical system in Turing space emerges spots, stripe-spot mixtures and labyrinthine patterns, which reveals that the interaction of both self- and cross-diffusions play a significant role on the pattern formation of the present system in a way to enrich the pattern. We obtain the normal form of the Turing–Hopf bifurcation and observe that the system has stably spatially homogeneous periodic solutions, stable constant and nonconstant steady-state solutions, which indicates that the intrinsic growth rate coefficient and the environmental carrying capacity coefficient are two important factors for predator–prey system, and affect the stability of predator–prey system.

Dynamics of an aquatic diffusive predator–prey model with double Allee effect and [formula omitted]-dependent capture rate

To investigate ocean acidification and Allee effects on the dynamics of a marine predator–prey system, an aquatic diffusive predator–prey model with double Allee effect on prey and pH-dependent capture rate is considered. First, we study the stability of constant steady state solutions using linearized theory. Second, the nonexistence of nonconstant positive steady state solutions is shown for appropriate ranges of parameters. Furthermore, we show the existence of a Hopf bifurcation and derive the direction and stability of the bifurcating periodic solutions. Both theoretical analysis and numerical simulation show that changing predator–prey interaction strengths, due to changing environmental conditions, can fundamentally change the system dynamics, even for apparently small changes in interaction strength. As the interaction strength decreases due to decreasing ocean pH, the system dynamics transition from persistent fluctuations in species abundances (periodic solutions), to stable coexistence, to predator extinct (with stable non-zero prey abundance), suggesting the potential for ocean acidification to decrease the abundance and diversity of marine species by weakening predation rates. Moreover, double Allee effect parameters together determine the stability of periodic solutions when the spatially homogeneous bifurcating periodic solutions exist, and the wavelength becomes longer as the Allee effect increases.

Dynamics analysis of a diffusional immunosuppressive infection model with Beddington-DeAngelis functional response

<abstract><p>This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations.</p></abstract>

Dynamics of a predator–prey system with nonlinear prey-taxis

In this paper, we investigate a predator–prey system with nonlinear prey-taxis under Neumann boundary condition. For a class of chemotactic sensitive functions, we obtain the existence and boundedness of global classical solutions for initial boundary value problems in arbitrary dimensional space. In addition, we also study the local stability of the constant steady state solution, and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions. Furthermore, the steady state bifurcation, Hopf bifurcation and fold-Hopf Singularity are analysed in detail by using Lyapunov–Schmidt reduction method.

Spatiotemporal patterns and bifurcations with degeneration in a symmetry glycolysis model

In this paper, we consider the glycolysis system with symmetry and degeneration. Based on the detailed stability of constant steady state solution and symmetry of nonlinear glycolysis system, for the Hopf bifurcations, we obtain the existence, stability and bifurcation direction not only for the homogeneous and inhomogeneous periodic solutions, but also for the degenerate bifurcating periodic solutions. The approach we adopted is the Lyapunov–Schmidt reduction method instead of the center manifold theory. On the other hand, we derive the nonexistence of the steady state solutions for small input flux of substrate. Additionally, for the steady state bifurcations, main object is to discuss degenerate bifurcations by means of Lyapunov–Schmidt procedure and singularity theory because the classical Crandall–Rabinowitz theorem cannot be applied. Some numerical simulations are achieved to illustrate the analytical results, especially the degenerate bifurcation results. It is interesting to noticed that the degenerate bifurcations can lead to richer spatiotemporal patterns, including the meeting of two Hopf bifurcation branches and the coupling of two eigenfunction models.

Stabilization in a two-species chemotaxis system with two chemicals under small initial conditions

This paper deals the two-species chemotaxis system in a smoothly bounded domain Ω⊂Rn(n≥1). It is proved that the solution of the system is globally bounded and exponentially convergent to the constant steady state solution under small initial conditions ‖u0‖Ln+β(Ω) and ‖ω0‖Ln+β(Ω) for any β>0, and the convergence rates are calculated accurately. The present work improves the results under small initial conditions ‖u0‖L∞(Ω) and ‖ω0‖L∞(Ω) in Tao and Winkler (2015).

Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme

<p style='text-indent:20px;'>A depletion-type reaction-diffusion Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme and the homogeneous Neumann boundary conditions is introduced and investigated in this paper. Firstly, the boundedness of positive solution of the parabolic system is given, and the constant steady state solutions of the model are exhibited by the Shengjin formulas. Through rigorous theoretical analysis, the stability of the corresponding positive constant steady state solution is explored. Next, a priori estimates, the properties of the nonconstant steady states, non-existence and existence of the nonconstant steady state solution for the corresponding elliptic system are investigated by some estimates and the Leray-Schauder degree theory, respectively. Then, some existence conditions are established and some properties of the Hopf bifurcation and the steady state bifurcation are presented, respectively. It is showed that the temporal and spatial bifurcation structures will appear in the reaction-diffusion model. Theoretical results are confirmed and complemented by numerical simulations.</p>

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

Boundedness and stabilization in a two-species chemotaxis system with two chemicals

Concerned is a two-species chemotaxis system with cross-diffusion and logistic source in a smooth bounded domain. When the system is parabolic-elliptic-parabolic-elliptic, if the logistic damping effect is strong enough then the solutions of the system are global, bounded and exponentially convergent to the constant steady-state solution. This is also true for the fully parabolic system. It is also shown that the damping effect is conducive to the global existence of solutions, which prevents the occurrence of blow-up in the presence of chemotaxis.

Spatiotemporal patterns of a diffusive plant–herbivore model with toxin-determined functional responses: Multiple bifurcations

In this paper, a homogeneous diffusive plant–herbivore model with toxin-determined functional response subject to the homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. By using Hopf bifurcation theorem and steady state bifurcation theorem due to Yi et al. (2009), we are able to show the existence of Hopf bifurcating periodic solutions (spatially homogeneous and non-homogeneous) and bifurcating non-constant steady state solutions. In particular, under certain conditions, the globally asymptotic stability of the positive constant steady state solutions and the non-existence of non-constant positive steady state solutions are investigated. These results allow the clear understanding of the mechanisms of the spatiotemporal pattern formations of this ecology model. In particular, our numerical results authenticate that toxicant parameter will play important roles in the stability and instability of the periodic solutions.

Global Stabilization of Two Dimensional Viscous Burgers\u2019 Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying C^0-conforming finite element method in spatial direction, optimal error estimates in L^infty (L^2) and in L^infty (H^1)-norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Patterns in a Modified Leslie–Gower Model with Beddington–DeAngelis Functional Response and Nonlocal Prey Competition

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions.

Bifurcations and pattern formation in a generalized Lengyel–Epstein reaction–diffusion model

Abstract This paper investigates the formation of spatial patterns in a general reaction–diffusion system based on the Lengyel–Epstein CIMA model. By analyzing the properties of the system’s unique positive equilibrium in the ODE and PDE cases, we establish the existence of non–constant steady state solutions thereby confirming the existence of Turing instability. Hopf–bifurcation analysis of the system show the existence of periodic solutions in the absence and presence of diffusion. Numerical simulations are presented to validate the theoretical results of the paper.

Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting.

Based on ecological significance, a delayed diffusive predator-prey system with food-limited and nonlinear harvesting subject to the Neumann boundary conditions is investigated in this paper. Firstly, the sufficient conditions of the stability of nonnegative constant steady state solutions of system are derived. The existence of Hopf bifurcation is obtained by analyzing the associated characteristic equation and the conditions of Turing instability are derived when the system has no delay. Furthermore, the occurrence conditions the Hopf bifurcation are discussed by regarding delay expressing the gestation time of the predator as the bifurcation parameter. Secondly, by using upper-lower solution method, the global asymptotical stability of a unique positive constant steady state solution of system is investigated. Moreover, we also give the detailed formulas to determine the direction, stability of Hopf bifurcation by applying the normal form theory and center manifold reduction. Finally, numerical simulations are carried out to demonstrate our theoretical results.

Dynamics of a Leslie-Gower predator-prey system with cross-diffusion

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well.

PI Passivity-Based Control and Performance Analysis of MMC Multiterminal HVDC Systems

In this paper, a decentralized PI passivity-based controller is applied to modular multilevel converters to ensure global stability of a multiterminal high-voltage direct current system. For the derivation of the controller, an appropriate model with constant steady-state solutions is obtained via a multifrequency orthogonal coordinates transformation. The control design is then completed using passivity arguments, and performance guarantees are established by a small-signal analysis. The obtained results are validated by means of detailed time-domain simulations both on a single-terminal and a four-terminal benchmarks.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.