On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property
In this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a popula...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2021/4445496 |
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| author | Cutberto Romero-Meléndez David Castillo-Fernández Leopoldo González-Santos |
| author_facet | Cutberto Romero-Meléndez David Castillo-Fernández Leopoldo González-Santos |
| author_sort | Cutberto Romero-Meléndez |
| collection | DOAJ |
| description | In this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a population modelled by a Lotka–Volterra system with stochastic environmental fluctuations. Even though there are numerous results in the deterministic case, there are few results for the behavior of numerical solutions in a population dynamic with random fluctuations. First, we show, using the Euler–Maruyama scheme, that the boundedness of numerical solutions and the convergence of the scheme are preserved in the stochastic case. Second, we analyze a property of the long-term behavior of a Lotka–Volterra system with stochastic environmental fluctuations known as turnpike property. In optimal control theory, the optimal solutions dwell mostly in the neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution. Our study shows, by means of the Stochastic Maximum Principle, that this turnpike property is preserved, when the noise in the system is small. Numerical simulations are implemented to support our results. |
| format | Article |
| id | doaj-art-ec141b8555af469a8eb1fce294906924 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-ec141b8555af469a8eb1fce2949069242025-02-03T06:12:51ZengWileyComplexity1076-27871099-05262021-01-01202110.1155/2021/44454964445496On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike PropertyCutberto Romero-Meléndez0David Castillo-Fernández1Leopoldo González-Santos2Basic Sciences Department, Metropolitan Autonomous University, Av. San Pablo 180, col. Reynosa, alcaldía Azcapotzalco 02220, Ciudad de Mexico, MexicoDepartment of Applied Mathematics and Systems, Metropolitan Autonomous University, Ciudad de Mexico, MexicoNeurobiology Institute, National Autonomous University of Mexico, Ciudad de Mexico, MexicoIn this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a population modelled by a Lotka–Volterra system with stochastic environmental fluctuations. Even though there are numerous results in the deterministic case, there are few results for the behavior of numerical solutions in a population dynamic with random fluctuations. First, we show, using the Euler–Maruyama scheme, that the boundedness of numerical solutions and the convergence of the scheme are preserved in the stochastic case. Second, we analyze a property of the long-term behavior of a Lotka–Volterra system with stochastic environmental fluctuations known as turnpike property. In optimal control theory, the optimal solutions dwell mostly in the neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution. Our study shows, by means of the Stochastic Maximum Principle, that this turnpike property is preserved, when the noise in the system is small. Numerical simulations are implemented to support our results.http://dx.doi.org/10.1155/2021/4445496 |
| spellingShingle | Cutberto Romero-Meléndez David Castillo-Fernández Leopoldo González-Santos On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property Complexity |
| title | On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property |
| title_full | On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property |
| title_fullStr | On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property |
| title_full_unstemmed | On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property |
| title_short | On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property |
| title_sort | on the boundedness of the numerical solutions mean value in a stochastic lotka volterra model and the turnpike property |
| url | http://dx.doi.org/10.1155/2021/4445496 |
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