A Galerkin-like approach to solve continuous population models for single and interacting species

Authors

  • Suayip Yuzbasi Akdeniz University
  • Murat Karacayir

Keywords:

Continuous population models, Galerkin method, logistic equation, predator-prey equation, residual error correction.

Abstract

In this paper, we present a Galerkin-like approach to numerically solve continuous population models for single and interacting species. After taking inner product of a set of monomials with a vector obtained from the problem under consideration, the problem is transformed to a nonlinear system of algebraic equations. The solution of this system gives the coefficients of the approximate solutions. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed in some detail. The method and the residual correction technique areillustrated with two examples. The results are also compared with numerous existing methods from the literature.

Author Biography

Suayip Yuzbasi, Akdeniz University

Mathermatics

References

Adomian, G. (1988). A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135.2:501-544.

Batiha, B., Noorani, M.S.M. & Hashim, I. (2007). Variational iteration method for solving multispecies Lotka-Volterra equations, Computers and Mathematics with Applications, 54.7: 903-909.

Bayat, M., Pakar, I. & Bayat, M. (2015). Nonlinear vibration of mechanical systems by means of homotopy perturbation method, Kuwait Journal of Science, 42.3:64-85.

Bayat, M., Pakar, I. & Domairry, G. (2012). Recent developments of some asymptotic methods and theirapplications for nonlinear vibration equations in engineering problems: A review, Latin American Journalof Solids and Structures, 9(2):145-234.

Chowdhury, M.S.H. & Rahman, M.M. (2012). An accurate solution to the Lotka-Volterra equations by modified homotopy perturbation method, International Journal of Modern Physics: Conference Series, Vol. 9, World Scientific Publishing Company.

Gander, M.J. (1994). A nonspiraling integrator for the Lotka Volterra equation, Il Volterriano, 4:21-28.

Grozdanovski, T. & Shepherd, J. (2008). Approximating the periodic solutions of the Lotka-Volterra system, ANZIAM Journal, 49:243-257.

He, J.H. (1999). Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178.3:257-262.

He, J.H. (2007). Variational approach for nonlinear oscillators, Chaos, Solitons & Fractals, 34(5):1430-1439.

Hirsch, M. W. & Smale, S. (1974). Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press.

Jamshidi, N. & Ganji, D.D. (2010). Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire, Current Applied Physics, 10:484-486.

Kim, Y. H. & Choo, S. (2015). A new approach to global stability of discrete Lotka-Volterra predator-prey models, Discrete Dynamics in Nature and Society, 11 pages.

Lotka, A. J. (1910). Contribution to the theory of periodic reactions, The Journal of Physical Chemistry, 14.3:271-274.

Lotka, A. J. (1920). Analytical note on certain rhythmic relations in organic systems, Proceedings of the National Academy of Sciences of the United States of America, 6.7:410.

Lotka, A. J. (1925). Elements of physical biology, Williams and Wilkins Company.

McKendrick, A.G. & Kesava Pai, M. (1911). The rate of multiplication of microorganisms: a mathematical study, Proceedings of the Royal Society of Edinburgh, Vol. 31.

Mickens, R.E. (2003). A nonstandard finite-difference scheme for the Lotka-Volterra system, Applied NumericalMathematics, 45.2:309-314.

Murty, K.N. & Rao, D.V.G. (1987). Approximate analytical solutions of general Lotka-Volterra equations, Journal of Mathematical Analysis and Applications, 122.2:582-588.

Olek, S. (1994). An accurate solution to the multispecies Lotka-Volterra equations, SIAM Review, 36.3:480-488.

Pamuk, S. (2005). The decomposition method for continuous population models for single and interactingspecies, Applied Mathematics and Computation, 163.1:79-88.

Pamuk, S. & Pamuk, N. (2010). Hes homotopy perturbation method for continuous population models for single and interacting species, Computers and Mathematics with Applications, 59.2:612-621.

Pearl, R. & Reed, L.J. (1922). A further note on the mathematical theory of population growth, Proceedings of the National Academy of Sciences of the United States of America, 8.12:365.

Trky?lmazo?lu, M. (2014). An effective approach for numerical solutions of high-order Fredholm integro-differential equations, Applied Mathematics and Computation, 227:384398.

Verhulst, P.F. (1845). La loi daccroissement de la population, Nouveaux Memories de lAcadmie Royale des Sciences et Belles-Lettres de Bruxelles, 18:14-54.

Verhulst, P.F. (1847) Deuxime memoire sur la loi daccroissement de la population, Memoires de lAcadmie Royale des Sciences, des lettres et des beaux-arts de Belgique, 20:1-32.

Volterra, V. (1927). Variazioni e fluttuazioni del numero dindividui in specie animali conviventi, C. Ferrari.

Volterra, V. & Brelot, M. (1931). Leons sur la thorie mathmatique de la lutte pour la vie, Vol. 1, Paris: Gauthier-Villars.

Wu, G. (2011). Adomian decomposition method for non-smooth initial value problems, Mathematical and Computer Modelling, 54(9-10):2104-2108.

Xu, N. & Zhang, A. (2009). Variational approach to analyzing catalytic reactions in short monoliths, Computers & Mathematics with Applications, 58(11-12):2460-2463.

Yzba??, ?. (2012a). Bessel collocation approach for solving continuous population models for single and interacting species, Applied Mathematical Modelling, 36.8:3787-3802.

Yzba??, ?. (2012b). A collocation approach to solve the Riccati-type differential equation systems, International Journal of Computer Mathematics, 89.16:2180-2197.

Downloads

Published

03-05-2017