World Applied Sciences Journal 11 (12): 1567-1572, 2010 ISSN 1818-4952 © IDOSI Publications, 2010 Corresponding Author: Dr. Ahmet Yildirim, Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey 1567 HPM-Padé Technique for Solving a Fractional Population Growth Model Ahmet Yildirim and Yagmur Gülkanat Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey Abstract: The aim of this paper is to obtain approximate solutions of a fractional population growth model in a closed system by using HPM-Padé technique. The time-fractional derivative is considered in the Caputo sense. The solution procedure is based on Homotopy Perturbation Method (HPM) and the solutions are calculated in the form of a convergent series with easily computable components. Then the Padé technique is effectively used in the analysis to capture the essential behavior of the population u(t) of identical individuals. Key words : Homotopy perturbation method • population dynamics • fractional derivative • padé technique • volterra integral equation INTRODUCTION In recent years, it has been found that derivatives of non-integer order are very effective for the description of many physical phenomena such as rheology, damping laws, diffusion process. These findings invoked the growing interest of studies of the fractal calculus in various fields such as physics, chemistry and engineering [1-4]. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [1] and the fluid-dynamic traffic model with fractional derivatives [2] can eliminate the deficiency arising from the assumption of continuum traffic flow. A review of some applications of fractional derivatives in continuum and statistical mechanics is given by Mainardi [5]. The solution of a fractional differential equation is much involved. In general, there exists no method that yields an exact solution for a fractional differential equation. Only approximate solutions can be derived using the linearization or perturbation methods. No analytical method was available before 1998 for such equations even for linear fractional differential equations. In 1998, the variational iteration method was first proposed to solve fractional differential equations with greatest success [3]. Many authors found VIM as an effective way to solving fractional equations, both linear and nonlinear [6, 7]. Momani [8] Ganji [9] and Yildirim [10, 11] applied the homotopy perturbation method (HPM) to fractional differential equations and revealed that HPM is an alternative analytical method for solving fractional differential equations. Momani [12] and Odibat [13] compared solution procedure between VIM and HPM. This paper outlines reliable numerical strategies for solving the fractional population growth model of a species within a closed system. The model is characterized by the nonlinear fractional Volterra integro-differential equation () () () ( ) () t 2 0 du au t bu t cutuxdx,u0 ,0 1 dt α α = − − =β <α≤ ∫ (1) where u = u(t) is the population of identical individuals at time t which exhibits crowding and sensitivity to the amount of toxins produced [14], α is a parameter describing the order of the time-fractional derivative, a > 0 is the birth rate coefficient, b > 0 is the crowding coefficient and c > 0 is the toxicity coefficient. The coefficient c indicates the essential behavior of the population evolution before its level falls to zero in the long run. If c = 0,we have the well-know logistic equation [14, 15]. The last term contains the integral that indicates the ‘‘total metabolism’’ or total amount of toxins produced since time zero. The individual death rate is proportional to this integral and so the population death rate due to toxicity must include a factor u. Since the system is closed, the presence of the toxic term always causes the population level to fall to zero in the long run, as will be seen later. The relative size of the sensitivity to toxins, c, determines the manner in which the population evolves before its extinction. The time- fractional derivative in Eq. (1) is considered in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses.