Chaos, Solitons and Fractals 133 (2020) 109624 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Frontiers Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system Shatha Hasan a , Ahmad El-Ajou b,c,∗ , Samir Hadid d , Mohammed Al-Smadi a , Shaher Momani d,e a Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan b Department of Mathematics, Faculty of Science, Taibah University, Madina, Saudi Arabia c Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan d Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, United Arab Emirates e Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan a r t i c l e i n f o Article history: Received 5 October 2019 Revised 5 January 2020 Accepted 8 January 2020 Keywords: Atangana-Baleanu derivative Generalized Mittag-Leffler function Nonlinear fractional logistic equation Schmidt orthogonalization process Reproducing kernel approach a b s t r a c t In this article, a class of population growth model, the fractional nonlinear logistic system, is studied an- alytically and numerically. This model is investigated by means of Atangana-Baleanu fractional derivative with a non-local smooth kernel in Sobolev space. Existence and uniqueness theorem for the fractional logistic equation is provided based on the fixed-point theory. In this orientation, two numerical tech- niques are implemented to obtain the approximate solutions; the reproducing-kernel algorithm is based on the Schmidt orthogonalization process to construct a complete normal basis, while the successive sub- stitution algorithm is based on an appropriate iterative scheme. Convergence analysis associated with the suggested approaches is provided to demonstrate the applicability theoretically. The impact of the frac- tional derivative on population growth is discussed by a class of nonlinear logistical models using the derivatives of Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Using specific examples, numerical simu- lations are presented in tables and graphs to show the effect of the fractional operator on the population curve as. The present results confirm the theoretical predictions and depict that the suggested schemes are highly convenient, quite effective and practically simplify computational time. © 2020 Elsevier Ltd. All rights reserved. 1. Introduction The fractional logistic equation is one of the well-known non- linear fractional differential equations that appear in ecology, biol- ogy and social studies. The population growth model, in particu- lar, can be considered as one of the famous typical applications of the fractional logistic equations where the rate of system change depends on previous memory of entire historical states [1–2]. In point of fact, Malthus was the first economist who proposed a sys- tematic theory of population [2]. He gathered experimental data and assumed that human population grows exponentially. After that, Verhulst presented a model to population dynamics in 1838 to illustrate the periodic doubling and chaotic behavior of the dy- namical system [3]. This standard model was closely related to the exponential growth model suggested by Malthus. More specif- ically, Verhulst presented a nonlinear first-order ordinary differ- ential equation of population growth which subsequently became ∗ Corresponding author. E-mail addresses: ajou42@yahoo.com, ajou43@gmail.com (A. El-Ajou). known as the logistic equation as follows: dM dt = ρ M  1 − M k  , t ≥ 0, where M(t) indicates the size of population growth at a time t, ρ > 0 is Malthusian parameter related with the maximum growth rate, and k describes the carrying capacity. Reciprocally, if we put N(t ) = M/k, then the standard logistic differential equation (LDE) can be expressed as: dN dt = ρ N(1 − N), (1) in which the exact solution of such problem can be given by N(t ) = N 0 N 0 + (1 − N 0 )e −ρt , (2) along with initial population data N 0 = N(0). Indeed, this solution explains the rate of population growth which doesn’t include the reducing food supplies or spreading diseases [4]. Further, the LDE has been introduced in the fractional sense through many applica- tions on different fractional operators for standard logistic issues, https://doi.org/10.1016/j.chaos.2020.109624 0960-0779/© 2020 Elsevier Ltd. All rights reserved.