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Does assortative mating lead to a polymorphic population? A toy model justification
January
2018, 23(1): 473-485.
doi: 10.3934/dcdsb.2018032
Solution to a stochastic pursuit model using moment equations
| 1. | University of Białystok, Faculty of Mathematics and Informatics, K. Ciołkowskiego 1M, 15-245 Białystok, Poland |
| 2. | Kiev National Economics V. Hetman University Faculty of Information System and Technology, Kyiv 03068, Peremogy 54/1, Ukraine |
| 3. | Palacký University Olomouc, Faculty of Education, Žižkovo nám. 5, Olomouc, Czech Republic |
| 4. | Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95,602 00 Brno, Czech Republic |
The paper investigates the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters. The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. Simulations are presented to test effectiveness of the approach.
Keywords:
Stochastic pursuit model,
differential equations with random parameters,
Markov process,
moment equations,
simulation of stochastic process.
Mathematics Subject Classification:
Primary: 34F05; Secondary: 60J28.
Citation:
Miroslava Růžičková, Irada Dzhalladova, Jitka Laitochová, Josef Diblík. Solution to a stochastic pursuit model using moment equations. Discrete and Continuous Dynamical Systems - B,
2018, 23
(1)
: 473-485.
doi: 10.3934/dcdsb.2018032
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References:
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A. G. Aleksandrov, Optimal and Adaptive Systems, Nauka, Moscow, 1989.
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J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková,
Modeling of applied problems by stochastic systems and their analysis using the moment equations, Adv. Difference Equ., 2013 (2013), 12 pp.
doi: 10.1186/1687-1847-2013-152. |
| [3] |
J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková, Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty Abstr. Appl. Anal. 2013 (2013), Art. ID 172847, 11 pp.
doi: 10.1155/2013/172847. |
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I. A. Dzhalladova, Optimization of Stochastic System, KNEU, Kiev, 2005.
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| [5] |
I. Dzhalladova, M. Růžičková, M. Štoudková and M. Růžičková,
Stability of the zero solution of nonlinear differential equations under the influence of white noise, Adv. Difference Equ., 2015 (2015), 11 pp.
doi: 10.1186/s13662-015-0482-y. |
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A. I. Egorov, Optimal Control with Linear Systems, KNEU, Kiev, 1988.
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| [7] |
H. Gao and C. Sun,
Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Continuous Dynam. Systems -B, 21 (2016), 3053-3073.
doi: 10.3934/dcdsb.2016087. |
| [8] |
C. G. Guo, C. X. Guo, S. Ahmed and F. X. Liu,
Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays, Discrete Continuous Dynam. Systems -B, 21 (2016), 2473-2489.
doi: 10.3934/dcdsb.2016056. |
| [9] |
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, New York, 1987.
doi: 10.1007/978-3-662-02514-7.
|
| [10] |
M. Růžičková and I. Dzhalladova, The optimization of solutions of the dynamic systems with random structure Abstr. Appl. Anal. 2011 (2011), Art. ID 486714, 18 pp.
doi: 10.1155/2011/486714. |
| [11] |
M. Šagát,
Stochastic Differential Equations and their Applications Mgr. thesis University of Žilina, Žilina -2814420132009,2013. |
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K. G. Valeev and I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006.
|
show all references
References:
| [1] |
A. G. Aleksandrov, Optimal and Adaptive Systems, Nauka, Moscow, 1989.
|
| [2] |
J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková,
Modeling of applied problems by stochastic systems and their analysis using the moment equations, Adv. Difference Equ., 2013 (2013), 12 pp.
doi: 10.1186/1687-1847-2013-152. |
| [3] |
J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková, Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty Abstr. Appl. Anal. 2013 (2013), Art. ID 172847, 11 pp.
doi: 10.1155/2013/172847. |
| [4] |
I. A. Dzhalladova, Optimization of Stochastic System, KNEU, Kiev, 2005.
|
| [5] |
I. Dzhalladova, M. Růžičková, M. Štoudková and M. Růžičková,
Stability of the zero solution of nonlinear differential equations under the influence of white noise, Adv. Difference Equ., 2015 (2015), 11 pp.
doi: 10.1186/s13662-015-0482-y. |
| [6] |
A. I. Egorov, Optimal Control with Linear Systems, KNEU, Kiev, 1988.
|
| [7] |
H. Gao and C. Sun,
Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Continuous Dynam. Systems -B, 21 (2016), 3053-3073.
doi: 10.3934/dcdsb.2016087. |
| [8] |
C. G. Guo, C. X. Guo, S. Ahmed and F. X. Liu,
Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays, Discrete Continuous Dynam. Systems -B, 21 (2016), 2473-2489.
doi: 10.3934/dcdsb.2016056. |
| [9] |
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, New York, 1987.
doi: 10.1007/978-3-662-02514-7.
|
| [10] |
M. Růžičková and I. Dzhalladova, The optimization of solutions of the dynamic systems with random structure Abstr. Appl. Anal. 2011 (2011), Art. ID 486714, 18 pp.
doi: 10.1155/2011/486714. |
| [11] |
M. Šagát,
Stochastic Differential Equations and their Applications Mgr. thesis University of Žilina, Žilina -2814420132009,2013. |
| [12] |
K. G. Valeev and I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006.
|
Figure 2.
The mean value of the process s(t) with parameters λ and p: λ = 0.01; p = 0.1, 0.2, ..., 1; s(0) = 0, 4, 8, ..., 200
Figure 3.
The mean value of the process s(t) with parameters λ and p: λ = 0.01, 0.02, ..., 0.2; p = 0.1, 0.2, ..., 1; s(0) = 0, 4, 8, ..., 200
Figure 4.
The mean value of the process $s(t)$ with parameters $\lambda$ and $p$ : $\lambda=0.01, 0.011, 0.012, \ldots, 0.21$ ; $p=0, 0.01, \ldots, 1$ ; $s(0)=0$
Figure 5.
The mean value of the process $s(t)$ with parameters $\lambda$ and $p$ : $\lambda=0.01, 0.011, 0.012, \ldots, 0.21$ ; $p=0, 0.01, \ldots, 1$ ; $s(0)=0, 20, \ldots, 100$
Figure 6.
The mean value $ E_{1}^{(1)}\{s( t )\}$ of the stochastic process as the solution to (12) and the mean value of two simulations of the stochastic process
Figure 7.
The mean value $ E_{1}^{(1)}\{s( t )\}$ of the stochastic process as the solution to(12) and the mean value of one hundred simulations of the stochastic process
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