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International Journal of Mathematics and Mathematical Sciences
Volume 2017 (2017), Article ID 8273430, 9 pages
https://doi.org/10.1155/2017/8273430
Research Article

Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect

Department of Mathematics, Brawijaya University, Jl. Veteran, Malang 65145, Indonesia

Correspondence should be addressed to Agus Suryanto

Received 1 March 2017; Accepted 3 May 2017; Published 28 May 2017

Academic Editor: Shyam L. Kalla

Copyright © 2017 Agus Suryanto et al. 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 analyze the dynamics of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect by means of local stability. In this respect, all possible equilibria and their existence conditions are determined and their stability properties are established. We also construct nonstandard numerical schemes based on Grünwald-Letnikov approximation. The constructed scheme is explicit and maintains the positivity of solutions. Using this scheme, we perform some numerical simulations to illustrate the dynamical behavior of the model. It is noticed that the nonstandard Grünwald-Letnikov scheme preserves the dynamical properties of the continuous model, while the classical scheme may fail to maintain those dynamical properties.