A Note on Stabilized Finite Element Methods for Predator-Prey Systems

Ali ŞENDUR (Alanya Alaaddin Keykubat Üniversitesi)
Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi
1 1
Yıl: 2019 Cilt: 19 Sayı: 3 Sayfa Aralığı: 653 - 661 Metin Dili: İngilizce DOI: 10.35414/ akufemubid.597506 İndeks Tarihi: 01-04-2020

A Note on Stabilized Finite Element Methods for Predator-Prey Systems

Öz:
A numerical method that will improve and produce effective results for solving mathematical model forthe system of predator-prey interactions which is defined by convection-diffusion-reaction problem isstudied herein. We consider the Pseudo Residual-free Bubble (PRFB) method which is based onaugmenting the finite element space by appropriate functions for the space discretization. The methodis applied on different test problems and the numerical solutions are in good agreement with the resultavailable in literature. The numerical results depict that the algorithm is efficient and feasible.
Anahtar Kelime:

Av-Avcı Problemleri için Kararlı Sonlu Eleman Yöntemleri Üzerine Bir Not

Öz:
Bu çalışmada, konveksiyon-difüzyon-reaksiyon problemleri ile modellenebilen av-avcı denklem sistemlerinin simülasyonunda kullanılan sayısal çözüm tekniklerini iyileştirecek ve daha etkin sonuçlar üretecek sayısal bir yöntem önerilmiştir. Uzay ayrıklaştırması için, sonlu elemanlar metodunu uygularken seçilen polinom baz fonksiyonlarına ilaveten fonksiyon uzayının özel tip fonksiyonlarla (residual-free bubbles) zenginleştirilmesine dayanan Pseudo Residual-free Bubble (PRFB) yöntemi kullanılmıştır. Söz konusu yöntem, çeşitli test örneklerine uygulanmış olup elde edilen sayısal çözümlerin, literatürde mevcut olan sonuçlar ile iyi bir uyum içinde olduğu gözlemlenmiştir. Sayısal sonuçlar, önerilen yöntemin verimli ve uygulanabilir olduğunu göstermektedir.
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • Allen, L. J. S., 2007. An Introduction to Mathematical Biology. Marcia J. Horton, Pearson/Prentice Hall, New Jersey, 365.
  • Brezzi, F., Bristeau, M. O., Franca, L. P., Mallet, M. and Roge, G., 1992. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg. 96, 117–129.
  • Brezzi, F., Franca, L. P., Hughes, T.J.R. and Russo, A., 1997. 𝑏 = ∫ 𝑔. Computer Methods in Applied Mechanics and Engineering, 145, 329–339.
  • Brezzi, F. and Russo, A., 1994. Choosing bubbles for advection-diffusion problems. Mathematical Models and Methods in Applied Sciences, 4, 571–587.
  • Brezzi, F., Marini, D. and Russo, A., 1998. Applications of pseudo residual-free bubbles to the stabilization of convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 166, 51–63.
  • Brezzi, F., Marini, D. and Russo, A., 2005. On the choice of a stabilizing sub-grid for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 194, 127–148.
  • Chong, O. A., Diniz, G. L. and Villatoro, F. R., 2005. Dispersal of fish populations in dams: modelling and simulation. Ecological modelling, 186, 290–298.
  • Cosner, C., 2014. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 34, 1701–1745.
  • Dimitrov, T.D. and Kojouharov, H.V., 2006. Positive and elementary stable nonstandard numerical methods with applications to predator - prey models. Journal of Computational and Applied Mathematics, 189, 98–108.
  • Dimitrov, D.T. and Kojouharov, H.V., 2007. Stabilitypreserving finite-difference methods for general multidimensional autonomous dynamical systems. International Journal of Numerical Analysis and Modeling, 4, 282–292.
  • Franca, L. P., Nesliturk, A. and Stynes, M., 1998. On the stability of residual-free bubbles for convectiondiffusion problems and their approximation by a twolevel finite element method. Computer Methods in Applied Mechanics and Engineering, 166, 35–49.
  • Garvie, M. R., 2007. Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator-Prey Interactions in MATLAB. Bulletin of mathematical biology, 69 (3), 931-956.
  • Garvie, M. R., Burkardt, J. and Morgan, J., 2015. Simple Finite Element Methods for Approximating PredatorPrey Dynamics in Two Dimensions Using Matlab. Bulletin of mathematical biology, 77 (3), 548-578.
  • Garzon-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M., 2012. Computational examples of reaction- convectiondiffusion equations solution under the influence of fluid flow: First example. Applied Mathematical Modelling, 36, 5029–5045.
  • Hilker, F.M. and Lewis, M.A., 2010. Predator-prey systems in streams and rivers. Theoretical Ecology, 3, 175–193.
  • Hughes, T. J. R., 1995. Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods. Computer Methods in Applied Mechanics and Engineering, 127, 387–401.
  • Medvinsky, A. B., Petrovskii, S. V., Tikhonova, I. A., Malchow, H. and Li, B. L., 2002. Spatiotemporal complexity of plankton and fish dynamics. SIAM review, 44, 311–370.
  • Meyer, J. F. C. A., and Diniz, G. L., 1997. Changes of habitat of fish populations: a mathematical model. International Journal of Mathematical Education in Science and Technology, 28, 519–529.
  • Mickens, R. E., 1994. Nonstandard finite difference model of differential equations. World Scientific, Singapore. Moghadas, S. M., Alexander, M. E. and Corbett, B. D. A., 2004. Non-standard numerical scheme for a generalized Gause-type predator-prey model. Journal of Physics D, 188, 134–151.
  • Murray, J. D., 2003. Mathematical Biology II: Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, 18, Springer, New York.
  • Sendur, A. and Nesliturk, A. I., 2012. Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems. Calcolo, 49, 1–19.
  • Sendur, A., Nesliturk, A. I. and Kaya, A., 2014. Applications of the pseudo residual-free bubbles to the stabilization of the convection-diffusion-reaction problems in 2D. Computer Methods in Applied Mechanics and Engineering, 277, 154–179.
  • Stefano, M., Perotto, S. and David, F., 2013. Model adaptation enriched with an anisotropic mesh spacing for nonlinear equations: application to environmental and CFD problems. Numerical Mathematics: Theory, Methods and Applications, 6, 447–478.
  • Zhang, T. and Jin, Y., 2017. Traveling waves for a reactiondiffusion-advection predator-prey model. Nonlinear Analysis: Real World Applications, 36, 203–232.
APA Sendur A (2019). A Note on Stabilized Finite Element Methods for Predator-Prey Systems. , 653 - 661. 10.35414/ akufemubid.597506
Chicago Sendur Ali A Note on Stabilized Finite Element Methods for Predator-Prey Systems. (2019): 653 - 661. 10.35414/ akufemubid.597506
MLA Sendur Ali A Note on Stabilized Finite Element Methods for Predator-Prey Systems. , 2019, ss.653 - 661. 10.35414/ akufemubid.597506
AMA Sendur A A Note on Stabilized Finite Element Methods for Predator-Prey Systems. . 2019; 653 - 661. 10.35414/ akufemubid.597506
Vancouver Sendur A A Note on Stabilized Finite Element Methods for Predator-Prey Systems. . 2019; 653 - 661. 10.35414/ akufemubid.597506
IEEE Sendur A "A Note on Stabilized Finite Element Methods for Predator-Prey Systems." , ss.653 - 661, 2019. 10.35414/ akufemubid.597506
ISNAD Sendur, Ali. "A Note on Stabilized Finite Element Methods for Predator-Prey Systems". (2019), 653-661. https://doi.org/10.35414/ akufemubid.597506
APA Sendur A (2019). A Note on Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 19(3), 653 - 661. 10.35414/ akufemubid.597506
Chicago Sendur Ali A Note on Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi 19, no.3 (2019): 653 - 661. 10.35414/ akufemubid.597506
MLA Sendur Ali A Note on Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, vol.19, no.3, 2019, ss.653 - 661. 10.35414/ akufemubid.597506
AMA Sendur A A Note on Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi. 2019; 19(3): 653 - 661. 10.35414/ akufemubid.597506
Vancouver Sendur A A Note on Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi. 2019; 19(3): 653 - 661. 10.35414/ akufemubid.597506
IEEE Sendur A "A Note on Stabilized Finite Element Methods for Predator-Prey Systems." Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 19, ss.653 - 661, 2019. 10.35414/ akufemubid.597506
ISNAD Sendur, Ali. "A Note on Stabilized Finite Element Methods for Predator-Prey Systems". Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi 19/3 (2019), 653-661. https://doi.org/10.35414/ akufemubid.597506
Sayfa Oluşturulma Tarihi
09-05-2024 5:15

İLETİŞİM

TÜBİTAK ULAKBİM TR Dizin

Yüzüncüyıl, İşçi Blokları Mahallesi

Muhsin Yazıcıoğlu Caddesi No:51/C

06530 Çankaya / ANKARA +90 (312) 298 92 00

trdizin@tubitak.gov.tr

TÜBİTAK ULAKBİM Ulusal Akademik Ağ ve Bilgi Merkezi Cahit Arf Bilgi Merkezi © Tüm Hakları Saklıdır. Gizlilik Politikası Kullanım Şartları