Paradox of Enrichment in a Stochastic Predator-Prey Model

We propose a stochastic predator-prey model to study a novel idea that involves investigating random noises effects on the enrichment paradox phenomenon. Existence and stochastic boundedness of a unique positive solution with positive initial conditions are proved. The global asymptotic stability is studied to determine the occurrence of the enrichment paradox phenomenon. We show theoretically that intensive noises play an important role in the occurrence of the phenomenon, where increasing intensive noises lead to occurrence of the paradox of enrichment. We perform numerical simulations to verify and demonstrate the theoretical results. The new results in this study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.

Analysis of a mutualism model with time-related coefficients in a stochastic environment

In this paper, a mutualism model with stochastic perturbations is considered and some of its coefficients are related to time. Under some assumptions, we make efforts to prove the existence and uniqueness of a positive solution, and the asymptotic behavior to the problem is discussed. Furthermore, we also prove the properties of stochastic boundedness, uniform continuity and stochastic permanence of this system. At last, some numerical simulations are introduced to illustrate our main results.

Stochastic boundedness of state trajectories of stable LTI systems in the presence of non-vanishing stochastic perturbation

This paper studies stochastic boundedness of trajectories of a non-vanishing stochastically perturbed stable linear time-invariant system. First, two definitions on stochastic boundedness are presented, then, the boundedness is analyzed via Lyapunov theory. A theorem is proposed, which shows that under a condition on the Lipchitz constant of the perturbation kernel, the trajectories remain stochastically bounded, and the bounds are calculated. Also, the limiting behaviour of the trajectories is studied. At the end, an illustrative example is presented, which shows the effectiveness of the proposed theory.

Analysis of a stochastic predator–prey model with prey subject to disease and Lévy noise

We consider a non-autonomous predator–prey model, with prey subject to the disease and Lévy noise. We show the existence of global positive solution and stochastic boundedness. Then, we examine the asymptotic properties of the solution. Finally, we offer sufficient conditions for persistence and extinction.

Asymptotic Behavior of a Stochastic Two-Species Competition System with Impulsive Effects

In this paper, we consider a stochastic two-species competition system with impulsive effects. Some dynamical properties are investigated and sufficient conditions for the stochastic boundedness, stochastic permanence and global attractivity are established. Under some conditions, we conclude that the stochastic model is persistent in mean and extinction. An example is given to illustrate the main result.

Stochastic boundedness filter design for Markovian jump linear systems with guaranteed H∞ filtering performance

This paper concerns the filtering problem for a class of continuous-time Markovian jump linear systems, where the Markovian jump is supposed to frequently occur in some short time intervals. For this class of Markovian jump system, the boundedness of estimation error deserves our investigation. By introducing the concepts of stochastic boundedness with respect to a finite-time interval, an observer ensuring the estimation error bounded in a prescribed boundary is constructed and the result is extended to the [Formula: see text] filtering problem with norm bounded disturbances. By formulating an optimization algorithm, we derive the optimal [Formula: see text] stochastic boundedness filter with an an optimized convex combination of estimation error boundary and [Formula: see text] performance index. We propose a design algorithm for when parameter optimization is involved. Numerical design examples are given to illustrate the effectiveness of our results.

Dynamical Behaviors of Stochastic Delayed One-Predator and Two-Competing-Prey Systems with Holling Type IV and Crowley-Martin Type Functional Responses

This paper is devoted to stochastic delayed one-predator and two-competing-prey systems with two kinds of different functional responses. By establishing appropriate Lyapunov functions, the globally positive solution and stochastic boundedness are investigated. In some case, the stochastic permanence and extinction are also obtained. Moreover, sufficient conditions of the global asymptotic stability of the system are established. Finally, some numerical examples are provided to explain our conclusions.

On the Stochastic Stability and Boundedness of Solutions for Stochastic Delay Differential Equation of the Second Order

We present two qualitative results concerning the solutions of the following equation: x¨(t)+g(x˙(t))+bx(t-h)+σx(t)ω˙(t)=p(t,x(t),x˙(t),x(t-h)); the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case p≡0, while the second one discusses the uniform stochastic boundedness of all solutions in case p≢0. Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.