IJNSNS 2019; aop A. M. Yousef, S. Z. Rida, Y. Gh. Gouda and A. S. Zaki ∗ Dynamical Behaviors of a Fractional-Order Predator–Prey Model with Holling Type IV Functional Response and Its Discretization https://doi.org/10.1515/ijnsns-2017-0152 received July 09, 2017; accepted January 12, 2019 Abstract: In this paper, we investigate the dynam- ical behaviors of a fractional-order predator–prey with Holling type IV functional response and its discretized counterpart. First, we seek the local stability of equilib- ria for the fractional-order model. Also, the necessary and sufficient conditions of the stability of the discretized model are achieved. Bifurcation types (include transcrit- ical, flip and Neimark–Sacker) and chaos are discussed in the discretized system. Finally, numerical simulations are executed to assure the validity of the obtained theoretical results. Keywords: predator–prey system, fractional calculus, dis- cretization, functional response, bifurcations, chaos MSC ® (2010). 26A33, 34C23, 37D45 1 Introduction In the last decades, enormous studies for population mod- els appearing in different branches of mathematical bio- logy have been examined by scientists because of their wide existence and importance [1]. Indeed, there are vari- ous mathematical approximations to study population models, e.g. ordinary differential equations, difference equations, partial differential equations and fractional- order differential equations. Fractional-order differential equations (FOD) are very convenient tools for modeling the memory and genetical effects of various processes and substances. Numerous models in interdisciplinary fields can be modeled by FOD, such as diffusion waves [2], non- linear oscillation of earthquakes [3], viscoelastic material *Corresponding author: A. S. Zaki, Faculty of Science, Aswan University, Aswan, Egypt, E-mail: zasmaasabry@yahoo.com A. M. Yousef: E-mail: ahmedyousef1981@gmail.com, S. Z. Rida: E-mail: szagloul@yahoo.com, Department of Mathematics, Faculty of Science, South valley University, Qena, Egypt Y. Gh. Gouda, Faculty of Science, Aswan University, Aswan, Egypt, E-mail: yas.gouda@yahoo.com models [4], hydrologic models [5], wave propagation in nonlocal elastic continua [6], world economies models [7], gyros systems [8], energy supply–demand equations [9] and muscular blood vessel model [10]. Furthermore, FOD are closely related to fractals which may be applied in mathematical biology widely. Moreover, the fractional- order equations are naturally related to systems with memory, which exist in most biological populations. So, fractional-order equations are more convenient than sys- tems with integer-order in economic, biological and social systems where the memory has an effective and import- ant role [11]. Also, approaching many phenomena by FOD is more suitable than integer-order differential equations [12–14]. Therefore, fractional-order equations are widely utilized in mathematical biology [15–18] and some other interdisciplinary fields [2, 3, 19, 20]. There are many definitions for the fractional derivat- ive. More famous definition is Caputo definition of frac- tional derivative [21] given as follows: D ! f (t)= I n–! D n f (t), D ≡ d dt , (1) where ! ∈ (n – 1, n), n ∈ N and I ( is the Riemann–Liouville integral operator of order ( which is given by I ( h(t)= 1 A(()  t 0 (t – 4) (–1 h(4)d4, ( > 0, (2) where A(() represents Euler’s Gamma function. For the physical and geometric meaning of fractional derivatives, one can see [22, 23]. Actually, chaos is an extremely complex nonlin- ear phenomenon that has comprehensive studies in the last four decades due to its wide utilization in multiple fields such as technology and science [24–34]. Chaos in fractional-order population systems is a motivating topic that has been studied and reported in [35–42]. In fact, using difference equations, for approximation population dynamics of some mathematical models, is more convenient and genuine especially when the popu- lation contains nonoverlapping species such as plant that completes its life cycle within one year and an univolt- Authenticated | ahmedyousef1981@gmail.com author's copy Download Date | 1/28/19 11:48 AM