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 A Stochastic Differential Equation SIS Epidemic Model

2011 ◽  
Vol 71 (3) ◽  
pp. 876-902 ◽  
Author(s):  
A. Gray ◽  
D. Greenhalgh ◽  
L. Hu ◽  
X. Mao ◽  
J. Pan
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Epidemic Model ◽  
Sis Epidemic Model ◽  
Sis Epidemic

scholarly journals TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350007 ◽  
Author(s):  
HUIJIE QIAO ◽  
JINQIAO DUAN
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Topological Equivalence ◽  
Random Differential Equation ◽  
Jump Discontinuities ◽  
Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

scholarly journals Mean reflected stochastic differential equations with jumps

2020 ◽  
Vol 52 (2) ◽  
pp. 523-562
Author(s):  
Phillippe Briand ◽  
Abir Ghannoum ◽  
Céline Labart
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
The Law

AbstractIn this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

scholarly journals Lie-Point Symmetries and Backward Stochastic Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1153
Author(s):  
Na Zhang ◽  
Guangyan Jia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equation ◽  
Backward Stochastic Differential Equations ◽  
Point Symmetry ◽  
Lie Point Symmetries ◽  
Lie Point Symmetry ◽  
Symmetry Method

In this paper, we introduce the Lie-point symmetry method into backward stochastic differential equation and forward–backward stochastic differential equations, and get the corresponding deterministic equations.

scholarly journals An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

2020 ◽  
Vol 28 (1) ◽  
pp. 1-18
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Risk Sensitive ◽  
Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

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