Periodic Solution and Stationary Distribution of Stochastic Predator–Prey Models with Higher-Order Perturbation

In this paper, two stochastic predator–prey models with general functional response and higher-order perturbation are proposed and investigated. For the nonautonomous periodic case of the system, by using Khasminskii’s theory of periodic solution, we show that the system admits a nontrivial positive T-periodic solution. For the system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent.     

On the two-dimensional tidal dynamics system: stationary solution and stability

ABSTRACT

In this work, we consider the two-dimensional stationary and non-stationary tidal dynamic equations and examine the asymptotic behavior of the stationary solution. We prove the existence and uniqueness of weak and strong solutions of the stationary tidal dynamic equations in bounded domains using compactness arguments. Using maximal monotonicity property of the linear and nonlinear operators, we also establish that the solvability results are even valid in unbounded domains. Later, we obtain a uniform Lyapunov stability of the steady state solution. Finally, we remark that the stationary solution is exponentially stable if we add a suitable dissipative term in the equation corresponding to the deviations of free surface with respect to the ocean bottom. This exponential stability helps us to ensure the mass conservation of the modified system, if we choose the initial data of the modified system as stationary solution.     

Stability of stationary solutions for the non-isentropic Euler-Maxwell system in the whole space

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R³. It is known in the authors’ previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.     

Study on a Cross Diffusion Parabolic System

This paper considers a kind of strongly coupled cross diffusion parabolic system,which can be usedas the multi-dimensional Lyumkis energy transport model in semiconductor science.The global existence andlarge time behavior are obtained for smooth solution to the initial boundary value problem.When the initialdata are a small perturbation of an isothermal stationary solution,the smooth solution of the problem under theinsulating boundary condition,converges to that stationary solution exponentially fast as time goes to infinity.     

THE STABILITY OF STATIONARY SOLUTION FOR OUTFLOW PROBLEM ON THE NAVIER-STOKES-POISSON SYSTEM

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.     

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.     

Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case

We prove the existence of solutions for a Navier-Stokes model in two dimensions with an external force containing infinite delay effects in the weighted space C_γ(H). Then, under additional suitable assumptions, we prove the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions.     

Periodic solutions of affine stochastic differential equations

We discuss the problem of the existence of periodic and stationary solutions of affine stochastic differential equations. We prove that under a controllability condition the system has a periodic solution if and only if the linear part is eyponentially stable in mean square.

It is also shown that the controllability assumption is necessary for the existence of a “unique” weakly periodic solution with nondegenerate covariance.     

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities.     

Stationary solutions to the Falk system on shape memory alloys

We study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physically closed stationary state is formulated as a nonlinear eigenvalue problem with a non‐local term. Then, some results on existence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamically stable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits

The present paper proves the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a quantum hydrodynamic model of semiconductors over a one-dimensional bounded domain. We also discuss on a singular limit from this model to a classical hydrodynamic model without quantum effects. Precisely, we prove that a solution for the quantum model converges to that for the hydrodynamic model as the Planck constant tends to zero. Here we adopt a non-linear boundary condition which means quantum effect vanishes on the boundary. In the previous researches, the existence and the asymptotic stability of a stationary solution are proved under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, the typical doping profile in actual devices does not satisfy this assumption. Thus, we prove the above theorems without this flatness assumption. Firstly, the existence of the stationary solution is proved by the Leray-Schauder fixed-point theorem. Secondly, we show the asymptotic stability theorem by using an elementary energy method, where the equation for an energy form plays an essential role. Finally, the classical limit is considered by using the energy method again.     

On a singularly perturbed system of two second-order equations in the case of intersecting roots of the degenerate equation

For a singularly perturbed system of two second-order differential equations (one rapid and one slow), we prove the existence of a solution and obtain its asymptotics for the case in which the degenerate equation has two intersecting roots. In addition, we show that this solution is an asymptotically stable stationary solution of the corresponding parabolic problem and find its local attraction domain.     

Existence of a stationary solution for the modified Ward–King tumor growth model

This paper studies existence of a stationary solution to a tumor growth model proposed by Ward and King in 1997 and 1998, with biologically reasonable modifications. Mathematical formulation of this problem is a two-point free boundary problem of a system of ordinary differential equations, one of which is singular at boundary points. By using the Schauder fixed point theorem we prove existence of a solution for this problem.     

Boundary layer solution to system of viscous conservation laws in half line

In the present paper, we show the existence and asymptotic stability of a stationary solution for a system of viscous conservation laws in a one-dimensional half space. We also discus the application to the model system of compressible viscous gases.     

Optimal exploitation of two competing size-structured populations

For a model describing the dynamics of two competing size-structured populations under chosen intensities of their exploitation, the existence and uniqueness of a stationary solution is proved. It is shown that there exist exploitation intensities that maximize a given profit functional on the stationary solution corresponding to them.     

On the dynamics of radially symmetric granulomas

A granuloma is a collection of macrophages that contains bacteria or other foreign substances that the body?s immune response is unable to eliminate. In this paper we present a simple mathematical model of radially symmetric granuloma dynamics. The model consists of a coupled system of two semi-linear parabolic equations for the macrophage density, and the bacterial density. The boundary of the granuloma is free. This simple framework makes it possible to conduct a mathematical analysis of the system dynamics. In particular, we show that the model system has a unique solution, and that, depending on the biological parameters; the bacterial load either disappears over time or persists. We use numerical methods to establish the existence of stationary solutions and examine how a stationary solution changes with the reproductive rate of the bacteria. These simulations show that the structure of the granuloma breaks down as the reproductive rate of the bacteria increases.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

Turing instability for a ratio-dependent predator-prey model with diffusion

Ratio-dependent predator-prey models have been increasingly favored by field ecologists where predator-prey interactions have to be taken into account the process of predation search. In this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction-diffusion model in which predator mortality is neither a constant nor an unbounded function, but it is increasing with the predator abundance. We show that analytically at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion). We also show that the stationary solution becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises. A numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented. Numerical examples are also included.     

Qualitative Analysis of a Predator-prey System with Ratio-dependent and Modified Leslie-Gower Functional Response

In this paper, a predator-prey model with ratio-dependent and modified Leslie-Gower functional response subject to homogeneous Neumann boundary condition is considered. First, properties of the constant positive sta- tionary solution are shown, including the existence, nonexistence, multiplicity and stability. In addition, a comparatively characterization of the stability is obtained. Moreover, the existing result of global stability is improved. Finally, properties of nonconstant positive stationary solutions are further studied. By a priori estimate and the theory of Leray-Schauder degree, it is shown that nonconstant positive stationary solutions may exist when the system has two constant positive stationary solutions.