Analytic approximation of Volterra’s population model

Biazar, J.; Hosseini, K.

doi:10.1515/jamsi-2017-0001

Abstract

In this paper, the Volterra’s population model is studied for population growth of a species within a closed system. Modified Adomian decomposition method (MADM) in conjunction with Pade technique is formally proposed to obtain an analytic approximation for the solution of the model, which is a nonlinear intgro-differential equation. The results of the method are compared with the existing exact results, confirming the accuracy and the efficiency of the proposed approach.

References

1. Scudo F (1971) Vito Volterra and theoretical ecology. Theor Popul Biol 2:1-2310.1016/0040-5809(71)90002-5
Search in Google Scholar Back to article
2. TeBeest K (1997) Numerical and analytical solutions of Volterra’s population model. SIAM Rev 39:484-49310.1137/S0036144595294850
Search in Google Scholar Back to article
3. Small R (1983) Population growth in a closed system. SIAM Rev 25:93-9510.1137/1025005
Search in Google Scholar Back to article
4. Dehghan M, Shahini M (2015) Rational pseudospectral approximation to the solution of a nonlinear integro-differential equation arising in modeling of the population growth. Appl Math Modelling 39:5521-553010.1016/j.apm.2015.01.001
Search in Google Scholar Back to article
5. Mohyud-Din ST, Yıldırım A, Gulkanat Y (2010) Analytical solution of Volterra’s population model. J King Saud Univ 22:247-25010.1016/j.jksus.2010.05.005
Search in Google Scholar Back to article
6. Wazwaz AM (1999) Analytical approximations and Pade approximants for Volterra’s population model. Appl Math Comput 100:13-2510.1016/S0096-3003(98)00018-6
Search in Google Scholar Back to article
7. Al-Khaled K (2005) Numerical approximations for population growth models. Appl Math Comput 160:865-87310.1016/j.amc.2003.12.005
Search in Google Scholar Back to article
8. Ramezani M, Razzaghi M, Dehghan M (2007) Composite spectral functions for solving Volterra’s population model. Chaos Solitons Fract 34:588-59310.1016/j.chaos.2006.03.067
Search in Google Scholar Back to article
9. Parand K, Abbasbandy S, Kazem S, Rad JA (2011) A novel application of radial basis functions for solving a model of first-order integro-ordinary differential equation. Commun Nonlinear Sci Numer Simulat 16:4250-425810.1016/j.cnsns.2011.02.020
Search in Google Scholar Back to article
10. Parand K, Delafkar Z, Pakniat N, Pirkhedri A, Kazemnasab Haji M (2011) Collocation method using sinc and rational Legendre functions for solving Volterra’s population model. Commun Nonlinear Sci Numer Simulat 16:1811-181910.1016/j.cnsns.2010.08.018
Search in Google Scholar Back to article
11. El-Nahhas A (2007) Analytic approximations for Volterra’s population equation. Proc Pakistan Acad Sci 44:255-261
Search in Google Scholar Back to article
12. Tabatabaei K, Gunerhan E (2015) The differential transform method for solving Volterra’s population model. AASCIT Commun 2:262-267
Search in Google Scholar Back to article
13. Xu H (2009) Analytical approximations for a population growth model with fractional order. Commun Nonlinear Sci Numer Simulat 14:1978-198310.1016/j.cnsns.2008.07.006
Search in Google Scholar Back to article
14. Babolian E, Biazar J (2003) Solving concrete examples by Adomian method. Appl Math Comput 135:161-16710.1016/S0096-3003(01)00321-6
Search in Google Scholar Back to article
15. Parand K, Hossayni SA, Rad JA (2016) An operation matrix method based on Bernstein polynomials for Riccati differential equation and Volterra population model. Appl Math Modelling 40:993-101110.1016/j.apm.2015.07.002
Search in Google Scholar Back to article
16. Parand K, Razzaghi M (2004) Rational Chebyshev tau method for solving Volterra’s population model. Appl Math Comput 149:893-90010.1016/j.amc.2003.09.006
Search in Google Scholar Back to article
17. Yuzbasi S (2013) A numerical approximation for Volterra’s population growth model with fractional order. Appl Math Modelling 37:3216-322710.1016/j.apm.2012.07.041
Search in Google Scholar Back to article
18. Duan S (2011) Convenient analytic recurrence algorithms for the Adomian polynomials. Appl Math Comput 217:6337-634810.1016/j.amc.2011.01.007
Search in Google Scholar Back to article
19. Aminikhah H, Biazar J (2009) A new HPM for ordinary differential equations. Numer Methods Partial Differ Equ 26:480-48910.1002/num.20413
Search in Google Scholar Back to article
20. Hosseini K, Biazar J, Ansari R, Gholamin P (2012) A new algorithm for solving differential equations. Math Methods Appl Sci 35:993-99910.1002/mma.1601
Search in Google Scholar Back to article
21. Biazar J, Hosseini K (2016) A modified Adomian decomposition method for singular initial value Emden-Fowler type equations. Int J Appl Math Research 5:69-7210.14419/ijamr.v5i1.5666
Search in Google Scholar Back to article
22. Pedersen S (2015) From calculus to analysis. Springer, Switzerland10.1007/978-3-319-13641-7
Search in Google Scholar Back to article

Analytic approximation of Volterra’s population model

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