Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics
We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fract...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2017/5716015 |
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| author | J. E. Macías-Díaz |
| author_facet | J. E. Macías-Díaz |
| author_sort | J. E. Macías-Díaz |
| collection | DOAJ |
| description | We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks. |
| format | Article |
| id | doaj-art-57c238cc0bff44deafea0bf2e6f3e00f |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-57c238cc0bff44deafea0bf2e6f3e00f2025-02-03T05:51:53ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2017-01-01201710.1155/2017/57160155716015Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional DynamicsJ. E. Macías-Díaz0Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, 20131 Aguascalientes, AGS, MexicoWe depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.http://dx.doi.org/10.1155/2017/5716015 |
| spellingShingle | J. E. Macías-Díaz Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics Discrete Dynamics in Nature and Society |
| title | Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics |
| title_full | Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics |
| title_fullStr | Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics |
| title_full_unstemmed | Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics |
| title_short | Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics |
| title_sort | existence and uniqueness of positive and bounded solutions of a discrete population model with fractional dynamics |
| url | http://dx.doi.org/10.1155/2017/5716015 |
| work_keys_str_mv | AT jemaciasdiaz existenceanduniquenessofpositiveandboundedsolutionsofadiscretepopulationmodelwithfractionaldynamics |