scholarly journals Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Dan Li ◽  
Jing’an Cui ◽  
Guohua Song
Keyword(s):  
Differential Equation ◽  
Initial Data ◽  
Stochastic Differential Equation ◽  
Asymptotic Behaviour ◽  
Average Rate ◽  
Sufficient Conditions ◽  
Jump Diffusion ◽  
Volterra Model ◽  
Environmental Perturbations ◽  
Stochastically Ultimate Boundedness

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

scholarly journals Exponential Stability of Stochastic Differential Equation with Mixed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Xinfeng Ruan ◽  
Zhao Zhao
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Stochastic Differential Equation ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Martingale Inequality ◽  
Mixed Delay ◽  
Exponentially Stable ◽  
Theoretical Results

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

scholarly journals Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
R. Rath ◽  
N. Misra ◽  
L. Padhy
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
T Tau

AbstractIn this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE) $$\left[ {r(t)(y(t) - p(t)y(t - \tau ))'} \right]^\prime + q(t)G(y(h(t))) = 0$$ are obtained, where q, h ∈ C([0, ∞), ℝ) such that q(t) ≥ 0, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C ([0, ∞), ℝ), G ∈ C (ℝ, ℝ) and τ ∈ ℝ+. Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.

Analysis of nonautonomous two species system in a polluted environment

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
G. Samanta
Keyword(s):  
Periodic Solution ◽  
Population Growth ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Volterra Model ◽  
Asymptotically Stable ◽  
Polluted Environment ◽  
Globally Asymptotically Stable ◽  
Bounded Functions

AbstractIn this paper, a two-species nonautonomous Lotka-Volterra model of population growth in a polluted environment is proposed. Global asymptotic behaviour of this model by constructing suitable bounded functions has been investigated. It is proved that each population for competition, predation and cooperation systems respectively is uniformly persistent (permanent) under appropriate conditions. Sufficient conditions are derived to confirm that if each of competition, predation and cooperation systems respectively admits a positive periodic solution, then it is globally asymptotically stable.

scholarly journals On a non-standard two-species stochastic competing system and a related degenerate parabolic equation

2020 ◽  
Vol 61 ◽  
pp. C1-C14
Author(s):  
Hidekazu Yoshioka ◽  
Yumi Yoshioka
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Growth Rates ◽  
Epidemic Model ◽  
Volterra Model ◽  
Sis Epidemic Model ◽  
Sis Epidemic

We propose and analyse a new stochastic competing two-species population dynamics model. Competing algae population dynamics in river environments, an important engineering problem, motivates this model. The algae dynamics are described by a system of stochastic differential equations with the characteristic that the two populations are competing with each other through the environmental capacities. Unique existence of the uniformly bounded strong solution is proven and an attractor is identified. The Kolmogorov backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. Our mathematical analysis results can be effectively utilized for a foundation of modelling, analysis, and control of the competing algae population dynamics. References S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two correlated brownian motions. Nonlin. Dyn., 97(4):2175–2187, 2019. doi:10.1007/s11071-019-05114-2. S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two independent brownian motions. J. Math. Anal. App., 474(2):1536–1550, 2019. doi:10.1016/j.jmaa.2019.02.039. U. Callies, M. Scharfe, and M. Ratto. Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe river. Ecol. Mod., 213(2):229–244, 2008. doi:10.1016/j.ecolmodel.2007.12.015. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):229–244, 1992. doi:10.1090/S0273-0979-1992-00266-5. N. H. Du and V. H. Sam. Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. App., 324(1):82–97, 2006. doi:10.1016/j.jmaa.2005.11.064. P. Grandits, R. M. Kovacevic, and V. M. Veliov. Optimal control and the value of information for a stochastic epidemiological SIS model. J. Math. Anal. App., 476(2):665–695, 2019. doi:10.1016/j.jmaa.2019.04.005. B. Horvath and O. Reichmann. Dirichlet forms and finite element methods for the SABR model. SIAM J. Fin. Math., 9(2):716–754, 2018. doi:10.1137/16M1066117. J. Hozman and T. Tichy. DG framework for pricing european options under one-factor stochastic volatility models. J. Comput. Appl. Math., 344:585–600, 2018. doi:10.1016/j.cam.2018.05.064. G. Lan, Y. Huang, C. Wei, and S. Zhang. A stochastic SIS epidemic model with saturating contact rate. Physica A, 529(121504):1–14, 2019. doi:10.1016/j.physa.2019.121504. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Berlin Heidelberg, 1972. doi:10.1007/978-3-642-65161-8. J. Lv, X. Zou, and L. Tian. A geometric method for asymptotic properties of the stochastic Lotka–Volterra model. Commun. Nonlin. Sci. Numer. Sim., 67:449–459, 2019. doi:10.1016/j.cnsns.2018.06.031. S. Morin, M. Coste, and F. Delmas. A comparison of specific growth rates of periphytic diatoms of varying cell size under laboratory and field conditions. Hydrobiologia, 614(1):285–297, 2008. doi:10.1007/s10750-008-9513-y. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6. O. Oleinik and E. V. Radkevic. Second-order Equations with Nonnegative Characteristic Form. Springer Boston, 1973. doi:10.1007/978-1-4684-8965-1. S. Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion. Springer-Verlag Berlin Heidelberg, 2019. doi:10.1007/978-3-662-59903-7. T. S. Schmidt, C. P. Konrad, J. L. Miller, S. D. Whitlock, and C. A. Stricker. Benthic algal (periphyton) growth rates in response to nitrogen and phosphorus: parameter estimation for water quality models. J. Am. Water Res. Ass., 2019. doi:10.1111/1752-1688.12797. Y. Toda and T. Tsujimoto. Numerical modeling of interspecific competition between filamentous and nonfilamentous periphyton on a flat channel bed. Landscape Ecol. Eng., 6(1):81–88, 2010. doi:10.1007/s11355-009-0093-4. H. Yoshioka, Y. Yaegashi, Y. Yoshioka, and K. Tsugihashi. Optimal harvesting policy of an inland fishery resource under incomplete information. Appl. Stoch. Models Bus. Ind., 35(4):939–962, 2019. doi:10.1002/asmb.2428.

scholarly journals Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

2011 ◽  
Vol 43 (3) ◽  
pp. 688-711 ◽  
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

scholarly journals Nonexplosion and Pathwise Uniqueness of Stochastic Differential Equation Driven by Continuous Semimartingale with Non-Lipschitz Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Jinxia Wang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Pathwise Uniqueness ◽  
Strong Uniqueness ◽  
Continuous Semimartingale

We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients. New sufficient conditions on the strong uniqueness and the nonexplosion are derived ford-dimensional stochastic differential equations onRd(d>2)with non-Lipschitz coefficients, which extend and improve Fei’s results.

Third-order leader-following consensus with circumstance noise under impulsive and switching control

2014 ◽  
Vol 25 (08) ◽  
pp. 1450026 ◽  
Author(s):  
Mei Sun ◽  
Dun Han ◽  
Dandan Li ◽  
Qiang Jia ◽  
Yaqi Wang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Hybrid Control ◽  
Sufficient Conditions ◽  
Switching Control ◽  
Theoretical Research ◽  
Third Order ◽  
Noise Perturbation ◽  
Leader Following ◽  
Multi Agent

This research is aimed at investigating the leader–follower problem of third-order multi-agent with noise perturbation over fixed network under impulsive and switching control. Based on stochastic differential equation theory and hybrid control theory, effective impulsive and switching controllers are proposed, and the sufficient conditions for reaching multi-agent leader-following consensus are acquired. Numerical simulations verify the validity of the theoretical research results.

scholarly journals Portfolio problems based on jump-diffusion models

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 573-583 ◽  
Author(s):  
Tiantian Liu ◽  
Jun Zhao ◽  
Peibiao Zhao
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Asset Price ◽  
Efficient Frontier ◽  
Jump Diffusion ◽  
Natural Generalization ◽  
Optimal Portfolio ◽  
Linear Quadratic ◽  
Price Process ◽  
Portfolio Problem

Zhou and Li [49] by virtue of stochastic linear-quadratic control theory studied the optimal portfolio problems with the asset price process satisfying a diffusion stochastic differential equation, and proposed the celebrated LQ framework and the efficient frontier for the given portfolio problem. In this paper, we consider the optimal portfolio problems based on the asset price process satisfying a jump-diffusion stochastic differential equation. Similarly, we also arrive at the efficient frontier of the optimal portfolio selection problem. The conclusions obtained here can be regarded as a natural generalization of the work by Zhou and Li [49].

A Class of Jump-Diffusion Stochastic Differential System Under Markovian Switching and Analytical Properties of Solutions

2020 ◽  
Vol 8 (1) ◽  
pp. 17-32
Author(s):  
Xiangdong Liu ◽  
Zeyu Mi ◽  
Huida Chen
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Stochastic Differential Equation ◽  
Differential System ◽  
Numerical Solutions ◽  
Jump Diffusion ◽  
Markovian Switching ◽  
Analytical Properties ◽  
Global Nature ◽  
First Moment

AbstractOur article discusses a class of Jump-diffusion stochastic differential system under Markovian switching (JD-SDS-MS). This model is generated by introducing Poisson process and Markovian switching based on a normal stochastic differential equation. Our work dedicates to analytical properties of solutions to this model. First, we give some properties of the solution, including existence, uniqueness, non-negative and global nature. Next, boundedness of first moment of the solution to this model is considered. Third, properties about coefficients of JD-SDS-MS is proved by using a right continuous markov chain. Last, we study the convergence of Euler-Maruyama numerical solutions and apply it to pricing bonds.

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