A numerical scheme for continuous population models for single and interacting species

Abstract

In this article, the dynamic of models such as logistic growth model, prey-predator model and 2-species Lotka-Volterra competition model is approximately solved by the Chebyshev collocation method.  These nonlinear mathematical models are transformed into the matrix form by Chebyshev expansion method and converted nonlinear algebraic equation system. Chebyshev coefficients are obtained by solving nonlinear equation system. Results are compared with Homotopy perturbation and Adomian decomposition method and then comparision numerical result and exact solution are presented by graphics for logistic growth model. Plots are showed the numbers of prey and predator versus time for various N values on predaor prey model. In the 2 spices Lotka Volterra competition model numerical results are presented by graphics. Matlab R2010a and Mapple14 are used for all calculations and graphs. In the conclusion part, the CPU times of the programs are given and the models are compared

Keywords

• Logistic growth model,prey and predator model,Lotka-Volterra model,2-species Lotka-Volterra model,system of nonlinear differential equations

Tek ve etkileşimli türlerin sürekli populasyon modelleri için bir sayısal yöntem

Abstract

Bu makalede, lojistik büyüme modeli, av avcı modeli ve 2-tür Lotka-Volterra yaşama mücadelesi modeli gibi modeller Chebyshev sıralama metodu ile çözülmüştür. Bu lineer olmayan matematiksel modeller Chebyshev açılımı metodu ile matris formuna dönüştürülmüş ve lineer olmayan cebirsel denklem sistemine indirgenmiştir. Lineer olmayan denklem sistemi çözülerek Chebyshev katsayıları elde edilmiştir. Lojistik büyüme modeli için sonuçlar homotopy perturbation metodu ve Adomian decomposition metodu ile karşılaştırılmış ve elde edilen nümerik sonuçlar ile tam çözümün karşılaştırılması grafiklerle sunulmuştur. Av-avcı modelinde grafikler yardımı ile av ve avcı sayılarının zamana karşı olan durumları farklı N değerleri için gösterilmiştir. 2 tür Lotka Volterra yaşama mücadelesi modelinde nümerik sonuçlar grafik ile ifade edilmiştir. Yapılan tüm hesaplamalar ve grafik çizimlerinde Matlab R2010a ve Maple14 kullanılmıştır. Ayrıca sonuç kısmında programların CPU zamanları verilerek modeller arası karşılaştırmalar yapılmıştır. 

Keywords

• Lojistik büyüme modeli,av-avcı modeli,Lotka-Volterra modeli,2-tür Lotka-Volterra modeli,linear olmayan diferansiyel denklem sistemi

References

1. Murray, J.D., Mathematical Biology, Springer, Berlin, (1993).
2. Simmons, G.F., Differential Equations with Applications and Historical Notes, McGraw-Hill, (1972).
3. Biazar, J. ve Montazeri, R., A computational method for solution of the prey and predator problem, Applied Mathematics and Computation, 163,2,841–847, (2005).
4. Biazar, J., Ilie, M. ve Khoshkenar,A., A new approach to the solution of the prey and predator problem and comparison of the results with the Adomian method, Applied Mathematics and Computation, 171,1,486–491, (2005).
5. Rafei, M., Daniali, H., Ganji, D.D. ve Pashaedi, H., Solution of the prey and predator problem by homotopy perturbation method, Applied Mathematics and Computation, 188, 1419–1425, (2007).
6. Pamuk, S., The decomposition method for continuous population models for single and interacting species, Applied Mathematics and Computation, 163, 79–88, (2005).
7. Pamuk, S. ve Pamuk, N., He’s homotopy perturbation method for continuous population models for single and interacting species, Computational Mathematics and Applications, 59, 612–621, (2010).
8. Pamuk, S., A review of some recent results for the approximate analytical solutions of non-linear differential equations, Mathematical Problems in Engineering, 34, (2009).

9. Pamuk, S. ve Pamuk, N., He’s homotopy perturbation method for continuous population models for single and interacting species, Computational Mathematics and Applications, 59, 612-621, (2010).
10. Hu, X., Liu, G. ve Yan, J., Existence of multiple positive periodic solutions of delayed predator–prey models with functional responses, Computational Mathematics and Applications, 52, 1453–1462, (2006).
11. Edelstein-Keshet, L., Mathematical Models in Biology, Random House, New York, (1988).
12. Takeuchi, Y., Du, N.H., Hieu, N.T. ve Sato, K., Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment, Journal of Mathematical Analysis and Applications, 323, 938–957, (2006).
13. Akyüz, A. ve Sezer, M., Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coeficients, Applied Mathematics and Computation, 144,237-247, (2003).
14. Gülsu, M., Öztürk, Y. ve Sezer, M., A new collocation method for solution of mixed linear integro-differential-difference equations, Applied Mathematics and Computation, 216, 2183-2198, (2010).
15. Sezer, M. ve Dogan, S., Chebyshev series solutions of Fredholm integral equations, International Journal of Mathematical Education in Science and Technology, 27, 5, 649-657, (1996).
16. Gülsu, M., Öztürk, Y. ve Sezer, M., On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials, Applied Mathematics and Computation, 217, 4827-4833, (2011).
17. Daşçıoğlu, A. ve Yaslan, H., The solution of high-order nonlinear ordinary differential equations by Chebyshev polynomials, Applied Mathematics and Computation, 217, 2, 5658-5666,(2011).
18. Daşcıoglu, A., Chebyshev solutions of systems of linear integral equations, Applied Mathematics and Computation, 151, 221-232, (2004).
19. Dascioglu, A., ve Sezer, M., Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro-differential equations, Journal of The Franklin Institute, 342, 688-701, (2005).
20. Öztürk, Y., Gülsu, A., ve Gülsu, M., On solution of a modified epidemiological model for drug release systems, Scholars Journal of Physics, Mathematics and Statistics, 3,1, 1-5, (2016).
21. Öztürk, Y., Gülsu, A., ve Gülsu, M., A numerical approach for solving modified epidemiological model for drug release systems, Nevşehir Bilim ve Teknoloji Dergisi, 2 ,2, 56-64, (2013).
22. Öztürk, Y., Gülsu, A.,ve Gülsu, M., A numerical method for solving the mathematical model of controlled drug release, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, 2, 2, 169-175, (2013).
23. Mason, J.C. ve Handscomb, D.C., Chebyshev polynomials, Chapman and Hall/CRC, New York,(2003).
24. Body, J.P., Chebyshev and fourier spectral methods, University of Michigan, New York, (2000).
25. Rivlin, T. J., Introduction to the approximation of functions, London, (1969).