Science World Journal Vol. 15(No 3) 2020 www.scienceworldjournal.org ISSN: 1597-6343 Published by Faculty of Science, Kaduna State University https://doi.org/10.47514/swj/15.03.2020.001 Application of the −homotopy analysis transform method (−HATM) to the solution of a fractional attraction Keller-Segel chemotaxis model APPLICATION OF THE −HOMOTOPY ANALYSIS TRANSFORM METHOD (−HATM) TO THE SOLUTION OF A FRACTIONAL ATTRACTION KELLER-SEGEL CHEMOTAXIS MODEL ∗ Newton I. Okposo 1 , Abel M. Jonathan 2 1 Department of Mathematics, Delta State University, Abraka, Nigeria 2 Department of Mathematics, Delta State University, Abraka, Nigeria *Corresponding Author’s Email Address: newstar4sure@gmail.com ABSTRACT The main purpose of this paper is to construct approximate analytic solutions for a time-fractional attraction Keller-Segel (TF- AKS) chemotaxis model using the −homotopy analysis transform method (−HATM). The obtained results and numerical simulations for three sets of initial data describe the behavior of the system. This further assert the convenience, computational efficiency and wide applicability of the proposed method even to more complex coupled systems of partial differential equations arising from mathematical biology. Keywords: Keller-Segel chemotaxis model, Caputo derivative, Laplace transform, −homotopy analysis method INTRODUCTION Obtaining exact solutions of fractional differential equations appear to be more difficult than their classical integer-order counterparts. Hence, a lot of attention have been devoted to develop very effective semi-analytical and numerical techniques for finding approximate solutions to this class of problems. Some of these techniques include the Adomian decomposition method (ADM) (Momani, 2005; Momani et al., 2006), Laplace decomposition method (LDM) (Jafari et al., 2011; Khan et al., 2011), Homotopy analysis method (HAM) (Liao, 1992; Liao, 2003; Zurigat et al., 2010), Homotopy perturbation method (HPM) (Momani et al., 2008) and Variational iteration method (VIM) (Jafari et al., 2012). Another very powerful technique is the −homotopy analysis transform method ( − HATM) (Kumar et al., 2017; Prakasha et al., 2017; Singh et al., 2019). It combines the traditional −homotopy analysis method (−HAM) due to El-Tawil and Huseen (El-Tawil et al., 2012; El- Tawil et al., 2013) with the Laplace transform method (LTM) to simplify computational procedures without any need for discretization or restrictive assumptions. The −HAM extends the classical homotopy analysis method (HAM) by incorporating a parameter ∈ [0, 1 n ] , n ≥ 1. The presence of the term ( 1 ) in the −HATM solution ensures faster convergence than the classical HAM. The central focus of this paper is to employ the −HATM to construct approximate series solutions for the following one-dimensional time-fractional Keller-Segel chemotaxis model (TF-AKS): { (, ) = 2 (, ) 2 − ((, ) ((, )) ) (, ) = 2 (, ) 2 − (, ) + (, ) (1) with associated initial conditions (, ) = 0 (), (, ) = 0 (), ∈ = (, ) (2) where 0<≤1 is the fractional differential parameter, , , and are various positive constants of biological importance (see Table 1 for their definitions and values), = (, ) and = (, ) are unknown state variables denoting the density of amoebae and concentration of chemoattractive substance, respectively and ((, )) represents the signal- dependent chemotatic sensitivity function. The chemotactic ( ((,)) ) term appearing in the first equation of (1) measures sensitivity of the amoebae cells to the chemical substance. If, for instance,=1 and () = with (. < 0), the system (1) reduces to the classical one- dimensional attraction (resp. repulsion) Keller-Segel chemotaxis model (Keller et al., 1970) which describes the aggregation dynamics of the amoeba Dictyostelium discoideum in response to cyclic Adenosine Monophosphate (cAMP) which mediate their aggregation. Generally, chemotaxis refers to the oriented motion of cellular species either in the direction of an attraction-type or away from a repulsion-type chemical signal. In biological processes, it accounts for cellular communication among motile marine organisms in their quest for mates, nutrients and survival. Among higher organisms, it dictates the processes of wound healing, pattern formation, cell-organization and positioning, embryogenesis, tumor cell invasion and cancer metastisis of living tissues. The classical Keller-Segel chemotaxis model (i.e., equation (1) with =1) as well as several of its variant formulations have been extensively studied from different mathematical perspectives. For instance, it has been shown that the classical model admits globally bounded solutions in the one- dimensional settings (Hillen et al., 2004; Yagi, 1997) whereas in higher dimensions a more complex dynamics arise in the sense that the solutions may blow up either in finite or infinite time (Blanchet et al., 2006; Horstmann et al., 2001; Senba et al., 2001). Specifically, in the two-dimensional settings, it was conjectured that there exists a threshold value > 0 for which the model admits global solution in time if ∫ 0 () < and for which blow up occurs if ∫ 0 () > (Childress et al., Full Length Research Article 1