In this paper, we consider the evolution of a system composed of two predator-prey deterministic systems described by Lotka-Volterra equations in random environment. It is proved that under the influence of telegraph noise, all positive trajectories of such a system always go out from any compact set of int R 2 + with probability one if two rest points of the two systems do not coincide. In case where they have the rest point in common, the trajectory either leaves from any compact set of int R 2 + or converges to the rest point. The escape of the trajectories from any compact set means that the system is neither permanent nor dissipative.

In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with a Beddington-DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator-prey models are also given.

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This paper presents a survey of recent results on the robust stability analysis and the distance to instability for linear time-invariant and time-varying differential-algebraic equations (DAEs). Different stability concepts such as exponential and asymptotic stability are studied and their robustness is analyzed under general as well as restricted sets of real or complex perturbations. Formulas for the distances are presented whenever these are available and the continuity of the distances in terms of the data is discussed. Some open problems and challenges are indicated.

This paper investigates asymptotic behavior of a stochastic SIR epidemic model, which is a system with degenerate diffusion. It gives sufficient conditions that are very close to the necessary conditions for the permanence. In addition, this paper develops ergodicity of the underlying system. It is proved that the transition probabilities converge in total variation norm to the invariant measure. Our result gives a precise characterization of the support of the invariant measure. Rates of convergence are also ascertained. It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree.

This paper continues the study of Mao et al. investigating two aspects of the equationThe first of these is to slightly improve results in [X. Mao, S. Sabais, E. Renshaw, Asymptotic behavior of stochastic Lotka-Volterra model, J. Math. Anal. 287 (2003) 141-156] concerning with the upper-growth rate of the total quantity n i=1 x i (t) of species by weakening hypotheses posed on the coefficients of the equation. The second aspect is to investigate the lower-growth rate of the positive solutions. By using Lyapunov function technique and using a changing time method, we prove that the total quantity n i=1 x i (t) always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower-growth rate.

This paper is addressed to some questions concerning the exponential stability and its robustness measure for linear timevarying differential-algebraic systems of index 1. First, the Bohl exponent theory that is well known for ordinary differential equations is extended to differential-algebraic equations. Then, it is investigated that how the Bohl exponent and the stability radii with respect to dynamic perturbations for a differentialalgebraic system depend on the system data. The paper can be considered as a continued and complementary part to a recent paper on stability radii for time-varying differential-algebraic equations [N.H. Du, V.H. Linh, Stability radii for linear timevarying differential-algebraic equations with respect to dynamic perturbations, J. Differential Equations 230 (2006) 579-599].

Necessary and sufficient conditions for exponential stability of linear time invariant delay differential-algebraic equations (DDAEs) are presented. The robustness of this property is studied when the equation is subjected to structured perturbations and a computable formula for the structured stability radius is derived. The results are illustrated by several examples.