%0 Journal Article
%T Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra's Population Model
%A Najeeb Alam Khan
%A Amir Mahmood
%A Nadeem Alam Khan
%A Asmat Ara
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/845945
%X This paper suggests two component homotopy method to solve nonlinear fractional integrodifferential equations, namely, Volterra's population model. Pad¨¦ approximation was effectively used in this method to capture the essential behavior of solutions for the mathematical model of accumulated effect of toxins on a population living in a closed system. The behavior of the solutions and the effects of different values of fractional-order are indicated graphically. The study outlines significant features of this method as well as sheds some light on advantages of the method over the other. The results show that this method is very efficient, convenient, and can be adapted to fit a larger class of problems. 1. Introduction Ecology is the study of different species in relation to their surroundings, competition for resources within and among the species, and predator-prey [1] relations among them. At times the surroundings may be infected by metabolic actions of the crowd [2, 3]. In all these situations, since time rates of changes of population sizes are concerned, it is natural that the mathematical modeling be given by differential equations or integrodifferential equations. Integrodifferential equations are usually difficult to solve especially analytically, so an effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. Also, fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a ¡°memory¡± term in the model. This memory term insures the history and its impact to the present and future. Problems of this type have gained escalating importance in recent years and many interesting outcomes have been accumulated. In the living thing population, the accumulation of metabolic products may cause inconvenience to the whole population and may ultimately result in a fall of the birth rate while the death rate is increased. We assumed that the total toxic effect on birth and death rates be expressed by the following nonlinear fractional-order integrodifferential equation [4]: where is the birth rate coefficient, is the crowding coefficient, and is the toxicity coefficient. The coefficient indicates the essential behavior of the population evolution before its level falls to zero in the long term, is the initial population, denotes the
%U http://www.hindawi.com/journals/ijde/2012/845945/