scholarly journals Dynamics of Stochastic Coral Reefs Model with Multiplicative Nonlinear Noise

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Zaitang Huang
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Probability Measure ◽  
Coral Reefs ◽  
Stationary Distribution ◽  
Markov Semigroup ◽  
Support Theorem ◽  
Stationary Density ◽  
Behavior Dynamics ◽  
Nonlinear Noise

Little seems to be known about the ergodicity of random dynamical systems with multiplicative nonlinear noise. This paper is devoted to discern asymptotic behavior dynamics through the stochastic coral reefs model with multiplicative nonlinear noise. By support theorem and Hörmander theorem, the Markov semigroup corresponding to the solutions is to prove the Foguel alternative. Based on boundary distributions theory, the required conservative operators related to the solutions are further established to ensure the existence a stationary distribution. Meanwhile, the density of the distribution of the solutions either converges to a stationary density or weakly converges to some probability measure.

The convex hull of a spherically symmetric sample

1981 ◽  
Vol 13 (4) ◽  
pp. 751-763 ◽  
Author(s):  
William F. Eddy ◽  
James D. Gale
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Probability Measure ◽  
Convex Hull ◽  
Random Sample ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Extreme Value ◽  
Spherically Symmetric ◽  
Extreme Value Distributions

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.

Stationary distributions for dams with additive input and content-dependent release rate

1977 ◽  
Vol 9 (03) ◽  
pp. 645-663 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Stationary Distribution ◽  
Release Rate ◽  
Zero Set ◽  
Stationary Distributions ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Limiting Case ◽  
Finite Dam

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt } of an infinite dam whose cumulative input {At } is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = x α and {A t } is stable with index β ∊ (0, 1). In general if EAt , < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA 1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA 1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt } as t → ∞. For {At } stable with index β and r(x) = x α , α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt } in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

On perturbed stochastic discrete systems

1974 ◽  
Vol 11 (3) ◽  
pp. 385-393 ◽  
Author(s):  
B.G. Pachpatte
Keyword(s):  
Asymptotic Behavior ◽  
Probability Measure ◽  
Measure Space ◽  
Discrete System ◽  
Discrete Systems ◽  
Image Position ◽  
Probability Measure Space

The object of this paper is to study a stochastic discrete system, including an operator T, of the formas a perturbation of the linear stochastic discrete systemwhere ω ∈ Ω, the supporting set of probability measure space (Ω, A, P) and n ∈ N, the set of nonnegative integers. We are concerned vith the existence, uniqueness, boundedness, and asymptotic behavior of random solutions of the above equation.

The asymptotic behavior of a stochastic multigroup SIS model

2018 ◽  
Vol 11 (03) ◽  
pp. 1850037 ◽  
Author(s):  
Chunyan Ji ◽  
Daqing Jiang
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Endemic Equilibrium ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Sis Model ◽  
Stochastic Perturbations ◽  
Long Time ◽  
Theoretical Results

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

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