A numerical method for solving continuous population models for single and interacting species

Year 2019, Volume: 23 Issue: 3, 403 - 412, 01.06.2019

Elçin Gökmen , Elçin Çelik

https://doi.org/10.16984/saufenbilder.410641

Abstract

In this
study, a numerical approach is presented to obtain the approximate solutions of
continuous population models for single and interacting species. This method is
essentially based on the truncated Taylor series and its matrix representations
with collocation points. By using Taylor polynomials and collocation points,
this method transforms population models into a matrix equation. The matrix
equation corresponds to a system of nonlinear equations with the unknown Taylor
coefficients. To illustrate reliability and efficiency of the method, numerical
examples are presented and results are compared with the other numerical
methods. Additionally, residual correction procedure is applied to estimate the
absolute errors. All numerical computations have been performed on the computer
algebraic system Maple 15.

Keywords

Continuous population models, Single and interacting species, Nonlinear differential equations and their systems, Taylor polynomials and series, Collocation points

References

• Referans1 A. Bellen, M. Zennaro, “Numerical Methods for Delay Differential Equations,” Clarendon Press, London, 2003.
• Referans2 F. Brauer, C. C. Chavez, “Mathematical Models in Population Biology and Epidemiology,” Springer, New York, 2010.
• Referans3 M. Dehghan, R. Salehi, “Solution of a nonlinear time-delay model in biology via semi-analytical approaches,” Computer Physics Communications, vol. 181, pp. 1255–1265, 2010.
• Referans4 J. D. Murray, “Mathematical Biology,” Springer, Berlin, 1993.
• Referans5 V. Volterra, “Variazioni e fluttazioni del numero d’individui in specie animali conviventi,” Mem. Acad. Sci. Lincei, vol. 2, pp. 31-13, 1926.
• Referans6 S. Pamuk, “The decomposition method for continuous population models for single and interacting species,” Applied Mathematics and Computation, vol 163, pp. 79–88, 2005.
• Referans7 S. Pamuk, N. Pamuk, “He’s homotopy perturbation method for continuous population models for single and interacting species,” Comput. Math. Appl, vol 59, pp. 612–621, (2010).
• Referans8 S. Yuzbasi, “Bessel collocation approach for solving continuous population models for single and interacting species,” Appl. Math. Modelling, vol. 36, pp. 3787–3802, 2012.
• Referans9 J. Biazar, Montazeri RA, “Computational method for solution of prey and predator problem,” Applied Mathematics and Computation, vol. 163, pp. 841-847, 2005.
• Referans10 J. Biazar, “Solution of the epidemic model by Adomian decomposition method,” Applied Mathematics and Computation, vol. 173, pp. 1101-1106, 2006.
• Referans11 S. Yuzbasi, “Bessel collocation approach for solving continuous population models for single and interacting species,” Appl. Math. Modelling, vol. 36, pp. 3787–3802, 2012.
• Referans12 M. Rafei, H. Daniali, D. D. Ganji, H. Pashaei, “Solution of the prey and predator problem by homotopy perturbation method,” Applied Mathematics and Computation, vol. 188, pp. 1419, 2007.
• Referans13 M. Rafei, H. Daniali, D. D. Ganji, “Variational iteration method for solving the epidemic model and the prey and predator problem,” Applied Mathematics and Computation, vol. 186, pp. 1701-1709, 2007.
• Referans14 E. Yusufoglu, B. Erbas, “He’s variational iteration method applied to the solution of the prey and predator problem with variable coefficients,” Physiscs Letter A., vol. 372, pp. 3829-3835, 2008.
• Referans15 F. A. Oliveira, “Collocation and residual correction,” Numer Math., vol. 36, pp. 27–31, 1980.
• Referans16 I. Çelik, “Approximate calculation of eigenvalues with the method of weighted residual collocation method,” Applied Mathematics and Computation, vol. 160, no. 2, pp. 401-410, 2005.
• Referans17 I. Çelik, “Collocation method and residual correction using Chebyshev series,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 910-920, 2006.
• Referans18 M. Gulsu, M. Sezer, “A Taylor polynomial approach for solving differential-difference equations,” Journal of Computational and Applied Mathematics, vol.186, pp. 349-364, 2006.
• Referans19 M. Sezer, A. A. Dascioglu, “A Taylor method for numerical solution of generalized pantograph equations with linear functional argument,” Journal of Computational and Applied Mathematics, vol. 200, pp. 217-225, 2007.
• Referans20 E. Gokmen, M. Sezer, “Taylor collocation method for systems of high order linear differential–difference equations with variable coefficients,” Ain Shams Engineering Journal, vol.4, no. 1, pp. 117-125, 2013.
• Referans21 E Gokmen, M. Sezer, “Approximate solution of a model describing biological species living together by Taylor collocation method,” New Trends in Math. Sci,. vol 3, no. 2, pp. 147-158, 2015.
• Referans22 E. Gokmen, O. R. Isik, M. Sezer, “Taylor collocation approach for delayed Lotka–Volterra predator–prey system,” Appl. Math. Comput., vol 268, pp. 671-684, 2015.