In this paper, an analytic approximation for Volterra's model for population growth of a species in a closed system is presented. The nonlinear integro-differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation. The decomposition algorithm are implemented independently to a related ODE. The Pade approximants, that often show superior performance over series approximations, are effectively used in the analysis to capture the essential behavior of the population u(t) of identical individuals.
References
1.
AdomianG., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 19.
2.
AndrianovI.V. and BulanovaN.S., Using Pade approximants to estimate the domain of applicability of small-parameter method, J. Sov. Math., Vol. 63, (5) (1993) 532–534.
3.
BakerG. A. and Graves-MorrisP., Essentials of Pade Approximants, Cambridge University Press, Cambridge, 1996.
4.
BoydJ., Pade approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Comp. Phys., 11 (3) (1997) 299–303.
5.
KhuriS.A., A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Appl. Math.1 (4) (2001) 141–155.
6.
ScudoF.M., Vito volterra and theoretical ecology, Theoret. Population Biol.2 (1971) 1–23.
7.
SmallR.D., Population growth in a closed model, Mathematical Modelling: Classroom Notes in Applied Mathematics, SIAM, Philadelphia, PA, 1989.
8.
TeBeestK.G., Numerical and analytical solutions of Volterra's population method, SIAM Rev., 39 (3) (1997) 484–493.
9.
AgadjanovYusufoglu E., Numerical Solution of Duffing Equation by the Laplace Decomposition algorithm, Applied Mathematics and Computation, 177, 572–580, (2006).
10.
[W] WazwazA.M., Analytical approximations and Pade Approximants for Volterra's population model, Appl. Math. Comput., 100 (1999) 13–25.