Constant Steady-state Solution Research Articles

Global asymptotic stability of 3-species Lotka–Volterra models with diffusion and time delays

This paper is concerned with three 3-species time-delayed Lotka–Volterra reaction–diffusion models with homogeneous Neumann boundary condition. Some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady-state solutions. These conditions are explicit and easily verifiable, and they involve only the reaction rate constants and are independent of the diffusion and time delays. The result of global asymptotic stability not only implies the nonexistence of positive steady-state solution but also gives some extinction results of the models in the ecological sense. The instability of some nonnegative semitrivial constant steady-state solutions is also shown. The conclusions for the reaction–diffusion systems are directly applicable to the corresponding ordinary differential systems.

Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion

In the paper, we investigate a three-species food chain model with diffusion and ratio-dependent predation functional response. We mainly focus on the coexistence of the three species. For this coupled reaction-diffusion system, we study the persistent property of the solution, the stability of the constant positive steady state solution, and the existence and nonexistence of nonconstant positive steady state solutions. Both the general stationary pattern and Turing pattern are observed as a result of diffusion. Our results also exhibit some interesting effects of diffusion and functional responses on pattern formation. (An erratum to this article has been appended at the end of the pdf file.)

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.

Asymptotic behavior of solutions for a cooperation-diffusion model with a saturating interaction

This paper is concerned with a Lotka-Volterra cooperation-diffusion model with a saturating interaction term for one species. The goal of the paper is to investigate the asymptotic behavior of the time- dependent solution in relation to the corresponding steady-state solutions under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants so that for every nontrivial nonnegative initial function the corresponding time-dependent solution converges to one of the nonnegative constant steady-state solutions as time tends to infinity. This convergence result leads to the existence and uniqueness of a positive (or nonnegative) steady-state solution and the global asymptotic stability of a given nonnegative constant steady-state solution. In terms of ecological dynamics, it also gives some coexistence, permanence and extinction results for the model.

Asymptotic behavior of solutions for a class of predator–prey reaction–diffusion systems with time delays

The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator–prey reaction–diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays.

Stability analysis for Allen–Cahn type equation associated with the total variation energy

In this paper, we shall discuss about the large-time behavior of solutions of an Allen–Cahn type equation generated by the total variation functional with constraints. In the one-dimensional case, the large time behavior of solutions has been studied in (Nonlinear Anal. 46 (2001) 435; Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195). According to the results, all steady-state patterns are represented as piecewise constant solutions of the equation, and any stable (steady-state) solution takes only values corresponding to pure phases. In the argument, the authors in (Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195) introduced an original concept, named as “local stability”, and discussed the stability of steady-state solutions by means of this concept. The main objective of this paper is to investigate the situation of multi-dimensional solutions. Referring to the results in the one-dimensional case, we target piecewise constant (steady-state) solutions as the object of consideration, and try to extend the theory of local stability to multi-dimensional cases. Consequently, some geometric conditions concerned with the structure of steady-state solutions and the stability will be shown.

Stability analysis for Allen–Cahn type equation associated with the total variation energy

In this paper, we shall discuss about the large-time behavior of solutions of an Allen–Cahn type equation generated by the total variation functional with constraints. In the one-dimensional case, the large time behavior of solutions has been studied in (Nonlinear Anal. 46 (2001) 435; Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195). According to the results, all steady-state patterns are represented as piecewise constant solutions of the equation, and any stable (steady-state) solution takes only values corresponding to pure phases. In the argument, the authors in (Funkcial. Ekvac. 44 (2001) 119; J. Math. Anal. 47 (2001) 3195) introduced an original concept, named as “local stability”, and discussed the stability of steady-state solutions by means of this concept. The main objective of this paper is to investigate the situation of multi-dimensional solutions. Referring to the results in the one-dimensional case, we target piecewise constant (steady-state) solutions as the object of consideration, and try to extend the theory of local stability to multi-dimensional cases. Consequently, some geometric conditions concerned with the structure of steady-state solutions and the stability will be shown.

Diffusion-Driven Instability in Reaction–Diffusion Systems

For a stable matrix A with real entries, sufficient and necessary conditions for A−D to be stable for all non-negative diagonal matrices D are obtained. Implications of these conditions for the stability and instability of constant steady-state solutions to reaction–diffusion systems are discussed and an example is given to show applications.

Sufficient conditions for stability of linear differential equations with distributed delay

We develop conditions for the stability of the constant (steady state) solutions oflinear delay differential equations with distributed delay when only information about the moments of the density of delays is available. We use Laplace transforms to investigate the properties of different distributions of delay. We give a method to parametrically determine the boundary of the region of stability, and sufficient conditions for stability based on the expectation of the distribution of the delay. We also obtain a result based on the skewness of the distribution. These results are illustrated on a recent model of peripheral neutrophil regulatory system which include a distribution of delays. The goal of this paper is to give a simple criterion for the stability when little is known about the distribution of the delay.

CHAOTIC PHENOMENA OF SUPERCONDUCTIVITY

One-dimensional steady state and evolutionary Ginzburg-Landau equations for superconductivity is disussed. Instability of constant steady state solutions and the existence of limit sets of infinite-dimensional dynamic system are proved. Using the author's definition of chaos for finite-and infinite-dimensional dynamic systems, we conclude that superconductors governed by G-L equations possess chaotic phenomena. Therefore strange phenomena may occur in conducting. The theoretical results indicate that it is advisable to improve the design of experiments or try to find new structures of superconductors in future research to suppress the chaotic behavior.

Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions

This paper is concerned with some dynamical property of a reaction-diffusion equation with nonlocal boundary condition. Under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function, the asymptotic behavior of the time-dependent solution is characterized in relation to a finite or an infinite set of constant steady-state solutions. This characterization is determined solely by the initial function and it leads to the stability and instability of the various steady-state solutions. In the case of finite constant steady-state solutions, the time-dependent solution blows up in finite time when the initial function in greater than the largest constant solution. Also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain.

On the dynamics of rings rotating with variable spin speed

This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.

On a semilinear parabolic equation arising in optical bistability

Optical bistable systems are at present of interest to physicists as such systems seem to have the potential to serve as local elements in an all optical computer. These systems consist essentially of a passive optical resonator containing a medium with a non-linear optical response to an external pump laser beam; when the output beam intensity is monitored against the input beam intensity, hysteresis effects are observed, i.e., there are two possible stable outputs. The ability of such a device to switch very rapidly between the two stable transmission states provides a fast optical switch. Detailed expositions on optical bistable systems can be found in the physics literature (see [3, 91). In this paper we discuss a reaction-diffusion equation arising in an optical bistable system in which diffusion of the excitation dominates the diffraction effects. We consider the equation ;-ddu+u=kZo(x) g(u(x)) for (x, C)ER” x [0, co), (1.1) where the unknown function u denotes the phase shift in the transmitted beam. Z,,(x) corresponds to the input beam, g is a smooth bounded function defined by g(u) = l/( 1 + m sin’(u - 6)), and d, k, m, and 6 are positive constants. Hysteresis corresponds to the existence of multiple non- negative steady-state solutions of (1.1). In Section 2 we consider the case where Z,(x) = 1, i.e., the input beam is a plane wave. In this case for appropriate values of k, m, and S, Eq. (1.1) has three constant steady-state solutions. By using the results of Fife and

Optimal Periodic Control: A Scenario for Local Properness

A fundamental problem in optimal periodic control is to decide whether proper periodic controls and trajectories yield better average performance than constant steady-state solutions. The present paper describes a situation where this holds true, because “nearby” the linearized system equation has a pair of eigenvalues on the imaginary axis. An example involving a retarded Lieneard equation is discussed in detail.

Bifurcation studies in reaction-diffusion

and estimated the critical lengths of the domain at which bifurcation occurs in the cases b = 0, a and 1. Also, we were able to characterise some of the non constant steady state solutions which appear beyond the points of instability by a steady state solution of the form u(x)=

On a Certain Semilinear Parabolic System Related to the Lotka-Volterra Ecological Model

A mathematical model proposed to explain the horizontal structure of prey and predator populations is represented by a semilinear parabolic system of equations. In this paper some mixed problems for this kind of system, Lotka-Volterra system, are considered and asymptotic behaviors of the solutions are investigated by use of Energy Method. A remarkable fact is that, in opposition to Steele's conjecture (see [3]), the solution in the case of the Neumann boundary condition is asymptotically spatially homogeneous but does not tend to any constant steady state solution.