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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 5716015, 7 pages
https://doi.org/10.1155/2017/5716015
Research Article

Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics

Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, 20131 Aguascalientes, AGS, Mexico

Correspondence should be addressed to J. E. Macías-Díaz; xm.aau.oerroc@saicamej

Received 8 March 2017; Accepted 19 April 2017; Published 8 May 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 J. E. Macías-Díaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.