Solution Of Above System Research Articles

A new approach toward stabilization in a two-species chemotaxis model with logistic source

This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem ut=d1Δu−χ1∇⋅(u∇w)+μ1u(1−u−a1v),x∈Ω,t>0,vt=d2Δv−χ2∇⋅(v∇w)+μ2v(1−a2u−v),x∈Ω,t>0,0=d3Δw−w+u+v,x∈Ω,t>0,in a bounded domain Ω⊂Rn,n≥1, with smooth boundary, where d1,d2,d3,χ1,χ2,μ1,μ2,a1,a2 are positive constants.When a1∈(0,1) and a2∈(0,1), it is shown that under some explicit largeness assumptions on the logistic growth coefficients μ1 and μ2, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state (u∗,v∗,w∗) of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential.When a1≥1 and a2∈(0,1), if μ2 is suitable large, for all sufficiently regular nonnegative initial data u0 and v0 with u0≢0 and v0≢0, the globally bounded solution of above system will stabilize toward (0,1,1) as t→∞ in algebraic.

Global asymptotic stability of periodic solutions for inertial delayed BAM neural networks via novel computing method of degree and inequality techniques

Firstly, by combining Mawhin’s continuation theorem of coincidence degree theory with Lyapunov functional method and by using inequality techniques, a sufficient condition on the existence of periodic solutions for inertial BAM neural networks is obtained. Secondly, a novel sufficient condition which can ensure the global asymptotic stability of periodic solutions of the system is obtained by using Lyapunov functional method and by using inequality techniques. In our paper, the assumption for boundedness on the activation functions in existing paper is removed, the conditions in inequality form in existing papers are replaced with novel conditions, and the prior estimate method of periodic solutions is replaced with Lyapunov functional method. Hence, our result on global asymptotic stability of periodic solutions for above system is less conservative than those obtained in existing paper and more novel than those obtained in existing papers.

HOMOCLINIC ORBITS OF NONPERIODIC SUPERQUADRATIC HAMILTONIAN SYSTEM

In this paper, we study the following first-order nonperiodic Hamiltonian system $$\dot{z} = \mathcal {J}H_{z}(t,z),$$ where $H \in C^{1}(\mathbb{R} \times \mathbb{R}^{2N}, \mathbb{R})$ is the form $H(t,z) = \frac{1}{2} L(t)z \cdot z + R(t,z)$. Under weak superquadratic condition on the nonlinearitiy. By applying the generalized Nehari manifold method developed recently by Szulkin and Weth, we prove the existence of homoclinic orbits, which are ground state solutions for above system.

Homoclinic orbits for an infinite dimensional Hamiltonian system with periodic potential

In this paper, we study the following diffusion system Open image in new window where \(z:(u,v):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}\), \(V(x)\in C(\mathbb{R}^{N},\mathbb{R})\) is a general periodic function, g(t,x,v), f(t,x,u) are periodic in t,x and superquadratic in v,u at infinity. By using much more direct methods to prove all Cerami sequences for the energy functional are bounded and establish the existence of homoclinic orbits, which are ground state solutions for above system.

Positive Almost Periodic Solution on a Nonlinear Differential Equation

We study the following nonlinear equation dx(t)/dt = x(t)[a(t) − b(t)xα(t) − f(t, x(t))] + g(t), by using fixed point theorem, the sufficient conditions of the existence of a unique positive almost periodic solution for above system are obtained, by using the theories of stability, the sufficient conditions which guarantee the stability of the unique positive almost periodic solution are derived.

Existence and global attractivity of a periodic solution for a prey–predator model with sex-structure

In this paper, we investigate the following periodic sex-structure model: x 1 ′ ( t ) = x 1 ( t ) b 1 ( t ) x 2 ( t ) x 1 ( t ) - d 1 ( t ) - k ( t ) x 1 ( t ) - k ( t ) x 2 ( t ) - c 1 ( t ) x 3 ( t ) , x 2 ′ ( t ) = x 2 ( t ) β ( t ) - k ( t ) x 1 ( t ) - k ( t ) x 2 ( t ) - c 1 ( t ) x 3 ( t ) , x 3 ′ ( t ) = x 3 ( t ) - a ( t ) - b ( t ) x 3 ( t ) + c 2 ( t ) x 1 ( t ) + c 2 ( t ) x 2 ( t ) . The sufficient and realistic conditions are obtained for the existence of positive periodic solution of above system. Further, by constructing a V functional and using the result of the existence of positive periodic solutions, the global attractivity of a positive periodic solution for above system is obtained.

Global attractivity of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion

By employing the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion is established. Further, by constructing a Lyapunov functional and using the result of the existence of positive periodic solution, the attractivity of a positive periodic solution for above system is obtained.

SUPERSYMMETRIC YANG-MILLS QUANTUM MECHANICS IN VARIOUS DIMENSIONS

Recent analytical and numerical solutions of above systems are reviewed. Discussed results include: a) exact construction of the supersymmetric vacua in two space-time dimensions, and b) precise numerical calculations of the coexisting, continuous and discrete, spectra in the four-dimensional system, together with the identification of dynamical supermultiplets and SUSY vacua. New construction of the gluinoless SO (9) singlet state, which is vastly different from the empty state, in the ten-dimensional model is also briefly summarized.

Determination by enhanced luminescence technique of liver antioxidant capacity.

A simple approach to quantitative determination of antioxidant capacity of rat liver homogenate is proposed. It consists of measuring chemiluminescence generated by a suitable system "detector" for .OH radicals produced from sodium perborate. The system generating the light signal contained luminol and compounds producing enhancement of light emission, such as sodium benzoate and indophenol. Two different methods, utilizing the same technique of enhanced luminescence, were set up. In a previous work, a parameter b, contained in the equation, which best describes the dependence of the intensity of light emission (E) on liver homogenate concentration (C) (E = a.C/exp(b.C), was found to be related to the level of antioxidants in the homogenate. Therefore, in the first method, the light emission from several dilutions of both liver homogenates, and homogenate and antioxidant mixtures, stressed with sodium perborate, was detected by a luminometer. The best fitting of data to theoretical equation provided b values, which were introduced in a system of equations relating such values to the antioxidant concentration. The solution of above system supplied the antioxidant concentration in the homogenate in terms of the equivalent concentration of the antioxidant used. In the other method, evaluations of the antioxidant capacity of liver homogenates were obtained by the determination of the ability of 10% homogenates to quench the light emission induced by either peroxidase or cytochrome c in comparison to the ability of antioxidant solutions. Both methods are able to evidence the decrease of the antioxidant concentration of liver homogenates after oxidative stress with ter-butylhydroperoxide. The value of both concentration changes and standard errors indicates that the method using a standard curve obtained with peroxidase, such as catalyst of radical reaction, and deferoxamine, such as antioxidant, is to be preferred.