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

The main purpose of this article is to suggest an approximation method for solving continuous population models. Using the collocation method and matrix operations, the problems are reduced into a system of nonlinear algebraic equations. The desired approximate solutions are obtained by solving this system via Maple 15. The error analysis for the proposed method is also introduced using the residual function. Numerical experiments are given to demonstrate the efficiency of the method. The results obtained from the proposed method are compared with the known results.

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A. Bellen, M. Zennaro, “Numerical Methods for Delay Differential Equations,” Clarendon Press, London, 2003.

F. Brauer, C. C. Chavez, “Mathematical Models in Population Biology and Epidemiology,” Springer, New York, 2010.

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.

J. D. Murray, “Mathematical Biology,” Springer, Berlin, 1993.

V. Volterra, “Variazioni e fluttazioni del numero d’individui in specie animali conviventi,” Mem. Acad. Sci. Lincei, vol. 2, pp. 31-13, 1926.

S. Pamuk, “The decomposition method for continuous population models for single and interacting species,” Applied Mathematics and Computation, vol 163, pp. 79–88, 2005.

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).

S. Yuzbasi, “Bessel collocation approach for solving continuous population models for single and interacting species,” Appl. Math. Modelling, vol. 36, pp. 3787–3802, 2012.

J. Biazar, Montazeri RA, “Computational method for solution of prey and predator problem,” Applied Mathematics and Computation, vol. 163, pp. 841-847, 2005.

J. Biazar, “Solution of the epidemic model by Adomian decomposition method,” Applied Mathematics and Computation, vol. 173, pp. 1101-1106, 2006.

S. Yuzbasi, “Bessel collocation approach for solving continuous population models for single and interacting species,” Appl. Math. Modelling, vol. 36, pp. 3787–3802, 2012.

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.

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.

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.

F. A. Oliveira, “Collocation and residual correction,” Numer Math., vol. 36, pp. 27– 31, 1980.

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.

I. Çelik, “Collocation method and residual correction using Chebyshev series,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 910-920, 2006.

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.

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.

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.

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.