Restarted Adomian Decomposition Method for Solving Volterra ’ s Population Model

In this paper, we used an efficient algorithm to obtain an analytic approximation for Volterra’s model for population growth of a species within a closed system, called the Restarted Adomian decomposition method (RADM) to solve the model. The numerical results illustrate that RADM has the good accuracy.


Introduction
Integro-differential equations arise in many areas of mathematics and sciences, such as biology, ecology, medicine, physics and technology.This class of equations arises while modeling various engineering and natural science problems, and hence it attracts much attention in numerical computation and analysis.
racterizing the population dynamics of an isolated species, is to consider an asexually reproducing organism for which age is irrelevant and behavior does not change with time or with the number of the organisms.The number, u, must be sufficiently large so that the process can be well approximated by a deterministic treatment and by real, rather than integer, numbers.It may mean number of individuals, in which case it is an integer, but it may also mean total weight, weight of certain parts, total metabolism, or some other measure of quantity of life.
Under these assumptions the change in u is given by the Malthusian equation in [8]: where a is birth rate, by integration, a geometrical law of increase (or decrease, if 0 a < ) is obtained: Volterra's practice discussed deeply the restrictive assumptions under which his formulas were derived.Then, he proceeded to remove one or two of the assumptions at a time, in the best Baconian fashion.The above assumptions imply, for example, that unlimited environmental resources are available to the species.
One can easily allow for a finite environmental capacity by taking for a decreasing function of u.By assuming that a decreases linearly with u, one obtains the Verhlust-Pearl equation where b is crowding coefficient, the integral, often called the "logistic curve," is widely used even outside ecology.By taking into account specific mechanisms affecting reproduction or mortality lead to a much more complex functional relationship.For example, by taking a population living in a completely closed environment, such as some microorganisms confined to a test tube, the amount of nutrients available decreases with time in proportion to the total amount of "metabolism" that takes place in the tube from the beginning of the experiment.
Total metabolism also determines the concentration of toxic waste in the medium.For simplicity, it is assumed that the metabolic activity of the population is directly proportional to the number of individuals and that its total amount affects linearly the coefficient of self-increase.Hence, the system can be represented by the integro-differential equation If the integral term is missing, the well-known logistic equation with birth rate 0 a > and crowding coefficient 0 b > is obtained.The last term, containing the integral, indicates the "total metabolism" or total amount of toxins produced since time zero.The individual death rate is proportional to this integral, so the population death rate due to toxicity must include a factor u. The presence of the toxic term, by considering the system being always closed, causes the population level to fall to zero in the long run.The relative size of the sensitivity to toxins, c, determines the manner in which the population evolves before its fated decay.
By introducing the non-dimensional variables the time and population scales are obtained as: And the non-dimensional problem takes the form:

t kP T P T P T P T P
where ( ) is the scaled population of identical individuals at a time T, and the non-dimensional parameter c K ab = is a prescribed parameter.The nondimensional parameter K plays a great role in the behavior of ( ) P T concerning the rapid rise to a certain amplitude followed by an exponential decay to extinction.It is important to point out that for K small, the population is not sensitive to toxins, whereas the population is strongly sensitive to toxins for large K.

General Description of the Adomian Decomposition Method
Equation ( 4) may be written as With initial condition: ( ) , applying 1 L − of both sides in Equation (7), and using the initial conditions, we obtain The unknown solution function u assumed to be given by a series of the form ( ) ( ) The nonlinear term usually represented by an infinite series of the so-called Adomian polynomials ( ) ( ) , n B t τ , respectively, i.e., we set Substituting ( 9) and ( 10) in (8) given the recursive relation ( ) From this recursive relation, we can compute 0 1 2 , , , .u u u  The solution of Equation ( 4) is now determined.However, in practice series

Restarted Adomian Decomposition Method (RADM)
In 2003, E. Babolian, et al. [10] introduced a new algorithm called "Restarted Adomian Method", to improve the accuracy dramatically.This new method depends on adding a term to both sides of Equation (8).Let G be the proper term, which is determined next; then By applying the "Modified Adomian Method" on Equation (12), we obtain: ( ) Hence, the following algorithm is presented.

• The algorithm
Choose small natural numbers m, n.
2) The Adomian Decomposition Method' usually gives the sum of the first few terms, and, consequently, gives an approximation of u.In the new algorithm (RADM), 0 u are updated, while the terms with large index are not calculated.
Therefore, m and n are considered to be small, say, m = 3 and n = 2. Applying ( ) ( )

Computation Results and Analysis
0.09 0.0005 0.036 0.00058333 0.00000167 The series solutions are By applying the new algorithm (RADM) with 2 n = and 3 m = , we obtain: Step 1 0.0068999999 0.001935 0.00028258 0.00137999 0.00057532 0.00006516 0.00034499 0.00046423 0.00007783 0.00034499 0.00011923 0.00005931 0.1 0.09 0.0355 0.00631667 0.00055375 0.00051289 0.00083165 The results produced by the present method with only few components (m = 5) are in a very good agreement with the best of the results of the methods listed in Table 1.By applying the new algorithm (RADM) with 2 n = and 3 m = , we obtain: Step 1 ( ) ( ) ( ) 0.09 0.005 0.036 0.00583333 0.00016667   The results produced by the present method with only few components (m = 5) are in a very good agreement with the best of the results of the methods listed in Table 2.

Conclusion
In this paper, we have applied Restarted Adomian method in solving nonlinear integro-differential equations.The numerical results show that RADM is more accurate than Adomian decomposition method of the solution Volterra's population model in Table 1 and Table 2.

Table 1 .
Error values obtained by the Adomian decomposition method and RestartedAdomian decomposition method for u(t).

Table 2 .
Error values obtained by the Adomian decomposition method and RestartedAdomian decomposition method for u(t).