Research Article Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics J. E. Macías-Díaz Departamento de Matem´ aticas y F´ ısica, Universidad Aut´ onoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, 20131 Aguascalientes, AGS, Mexico Correspondence should be addressed to J. E. Mac´ ıas-D´ ıaz; jemacias@correo.uaa.mx Received 8 March 2017; Accepted 19 April 2017; Published 8 May 2017 Academic Editor: Douglas R. Anderson Copyright © 2017 J. E. Mac´ ıas-D´ ıaz. his 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. 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 diferences. he 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. 1. Introduction he development in recent decades of fractional calculus has led to important discoveries in many scientiic areas [1, 2]. For example, research in the physical sciences has developed toward the construction of a physically meaningful calculus of variations for fractional systems [3, 4]. Here, the problem lies in the fact that the notions of “energy” and “Hamiltonian” still lack a concrete physical connotation, so the determina- tion of the physical signiicance of those concepts is still a fruitful area of study [5]. In the ield of diferential/diference equations, the determination of theorems on the existence and the uniqueness of solutions of fractional systems is nowadays an area of analytic importance [6]. In this context, the determination of the properties of the relevant solutions of fractional systems is also a transited avenue of research, with the conditions of positivity and boundedness being of particular interest [7]. From a numerical perspective, the design of computational techniques with desirable numer- ical properties (convergence, stability, consistency, etc.) is also a scientiic problem that has attracted the attention of researchers in the area [8]. However, the development of numerical methods that are capable of preserving structural properties of their continuous counterparts (the positivity, the boundedness, the monotonicity, or the preservation of energy, mass, momentum, etc.) is still scarce. One of the most studied partial diferential equations in the literature is the well-known difusion-reaction model investigated simultaneously and independently in 1937 by Fisher [9] and Kolmogorov et al. [10]. hat model possesses positive, bounded, and traveling-wave solutions, and it was used initially to describe the propagation of mutant genes that are advantageous to the survival of populations distributed in linear habitats [11]. Fisher’s equation has also found applications in the investigation of the neutron lux and temperature in prompt feedback nuclear reactors [12], where the governing law is the Pearl–Verhulst equation. As many other equations from the physical sciences, that model has also been extended to the fractional scenario in order to describe more accurately the phenomena of interest (see [13] and the references therein). A vast number of criteria for fractional diferentiation have been employed to that end, but some of these extensions are capable of resembling the properties of the solutions of the model investigated in 1937, like the existence and uniqueness of traveling-wave solutions that are positive and bounded [14]. On the other hand, the exact determination of solutions of fractional extensions of Fisher’s equation is a diicult task. Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 5716015, 7 pages https://doi.org/10.1155/2017/5716015