Research Article Dynamics, Chaos Control, and Synchronization in a Fractional-Order Samardzija-Greller Population System with Order Lying in (0, 2) A. Al-khedhairi, 1 S. S. Askar, 1,2 A. E. Matouk , 3 A. Elsadany, 4 and M. Ghazel 5 1 Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 3 Department of Basic Engineering Sciences, College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia 4 Mathematics Department, College of Sciences and Humanities Studies Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia 5 Mathematics Department, Faculty of Science, Hail University, Hail 2440, Saudi Arabia Correspondence should be addressed to A. E. Matouk; aematouk@hotmail.com Received 28 January 2018; Revised 3 July 2018; Accepted 16 July 2018; Published 10 September 2018 Academic Editor: Matilde Santos Copyright © 2018 A. Al-khedhairi 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. This paper demonstrates dynamics, chaos control, and synchronization in Samardzija-Greller population model with fractional order between zero and two. The fractional-order case is shown to exhibit rich variety of nonlinear dynamics. Lyapunov exponents are calculated to confirm the existence of wide range of chaotic dynamics in this system. Chaos control in this model is achieved via a novel linear control technique with the fractional order lying in (1, 2). Moreover, a linear feedback control method is used to control the fractional-order model to its steady states when 0< α <2. In addition, the obtained results illustrate the role of fractional parameter on controlling chaos in this model. Furthermore, nonlinear feedback synchronization scheme is also employed to illustrate that the fractional parameter has a stabilizing role on the synchronization process in this system. The analytical results are confirmed by numerical simulations. 1. Introduction Dynamic analysis of engineering and biological models has become an important issue for research [1–10]. One of the most fascinating dynamical phenomena is the existence of chaotic attractors. Due to the importance of chaos, it has been investigated in various academic disciplines [11–15]. The sensitivity to initial conditions which characterizes the exis- tence of chaotic attractors was first noticed by Poincaré [16]. According to their potential applications in a wide variety of settings, fractional-order differential equations (FODEs) have received increasing attention in engineering [17–23], physics [24], mathematical biology [25-26], and encryption algorithms [27]. Moreover, FODEs play an important role in the description of memory which is essential in most biological models. Chaos synchronization and control in dynamical systems are essential applications of chaos theory. They have become focal topics for research since the elegant work of Ott et al. in chaos control [28] and the pioneering work of Pecora and Carroll in chaos synchronization [29]. Chaos control is sometimes needed to refine the behavior of a chaotic model and to remove unexpected performance of power electronics. Synchronization of chaos has also useful applications to biological, chemical, physical systems and secure communi- cations. Furthermore, synchronization and control in chaotic fractional-order dynamical systems have been investigated by authors [30–32]. The integer-order Samardzija-Greller population model is a system of ODEs that generalizes the Lotka-Volterra equa- tions and expresses the behaviors of two species predator- prey population dynamical system. This model was proposed Hindawi Complexity Volume 2018, Article ID 6719341, 14 pages https://doi.org/10.1155/2018/6719341