Yıl 2019, Cilt 19 , Sayı 3, Sayfalar 653 - 661 2019-12-31

Av-Avcı Problemleri için Kararlı Sonlu Eleman Yöntemleri Üzerine Bir Not
Stabilized Finite Element Methods for Predator-Prey Systems

Ali Şendur [1]


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.

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

  • [1] Allen, L. J. S. An Introduction to Mathematical Biology. Prentice Hall, New Jersey, 2007.
  • [2] Brezzi, F., Bristeau, M. O., Franca, L. P., Mallet, M. & Roge, G. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg. 96, (1992), pp.117–129.
  • [3] Brezzi, F., Franca, L. P., Hughes, T.J.R. & Russo, A. b=∫ g Computer Methods in Applied Mechanics and Engineering, 145, (1997), pp.329–339.
  • [4] Brezzi, F. & Russo, A. Choosing bubbles for advection-diffusion problems. Mathematical Models and Methods in Applied Sciences, 4, (1994), pp.571–587.
  • [5] Brezzi, F., Marini, D. & Russo, A. Applications of pseudo residual-free bubbles to the stabilization of convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 166, (1998), pp.51–63.
  • [6] Brezzi, F., Marini, D. & Russo, A. On the choice of a stabilizing sub-grid for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 194, (2005), pp.127–148.
  • [7] Chong, O. A., Diniz, G. L. and Villatoro, F. R. Dispersal of fish populations in dams: modelling and simulation. Ecological modelling, 186, (2005), pp.290–298.
  • [8] Cosner, C. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 34, (2014), pp.1701–1745.
  • [9] Dimitrov, T.D. and Kojouharov, H.V. Positive and elementary stable nonstandard numerical methods with applications to predator - prey models. Journal of Computational and Applied Mathematics, 189, (2006), pp.98–108.
  • [10] Dimitrov, D.T. and Kojouharov, H.V. Stability-preserving finite-difference methods for general multi-dimensional autonomous dynamical systems. International Journal of Numerical Analysis and Modeling, 4, (2007), pp.282–292.
  • [11] Franca, L. P., Nesliturk, A. , & Stynes, M. On the stability of residual-free bubbles for convection- diffusion problems and their approximation by a two-level finite element method. Computer Methods in Applied Mechanics and Engineering, 166, (1998), pp.35–49.
  • [12] Garvie, M. R. Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator-Prey Interactions in MATLAB. Bulletin of mathematical biology, 69, 3, (2007), pp.931-956.
  • [13] Garvie, M. R., Burkardt, J. and Morgan, J. Simple Finite Element Methods for Approximating Predator-Prey Dynamics in Two Dimensions Using Matlab. Bulletin of mathematical biology, 77, 3, (2015), pp.548-578.
  • [14] Garzon-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M. Computational examples of reaction- convection-diffusion equations solution under the influence of fluid flow: First example. Applied Mathematical Modelling, 36, (2012), pp.5029–5045.
  • [15] Hilker, F.M. and Lewis, M.A. Predator-prey systems in streams and rivers. Theoretical Ecology, 3, 3, (2010), pp.175–193.
  • [16] Hughes, T. J. R. Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127, (1995), pp.387–401.
  • [17] Medvinsky, A. B., Petrovskii, S. V., Tikhonova, I. A., Malchow, H. and Li, B. L. Spatiotemporal complexity of plankton and fish dynamics. SIAM review, 44, (2002), pp.311–370.
  • [18] Meyer, J. F. C. A., and Diniz, G. L. Changes of habitat of fish populations: a mathematical model. International Journal of Mathematical Education in Science and Technology, 28, (1997), pp.519–529.
  • [19] Mickens, R. E. Nonstandard finite difference model of differential equations. World Scientific, Singapore , (1994).
  • [20] Moghadas S. M., Alexander M. E. and Corbett B. D. A non-standard numerical scheme for a generalized Gause-type predator-prey model. Journal of Physics D, 188, (2004), pp.134–151.
  • [21] Murray, J. D. Mathematical Biology II: Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, 18, Springer, New York, 2003.
  • [22] Sendur, A. and Nesliturk, A. I. Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems. Calcolo, 49, (2012), pp.1–19.
  • [23] Sendur, A., Nesliturk, A. I. & Kaya, A. 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, (2014), pp.154–179.
  • [24] Stefano, M., Perotto, S. and David, F. Model adaptation enriched with an anisotropic mesh spacing for nonlinear equations: application to environmental and CFD problems. Numerical Mathematics: Theory, Methods and Applications, 6, (2013), pp.447–478.
  • [25] Zhang, T. and Jin, Y. Traveling waves for a reaction-diffusion-advection predator-prey model. Nonlinear Analysis: Real World Applications, 36, (2017), pp.203–232.
Birincil Dil en
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Orcid: 0000-0001-8628-5497
Yazar: Ali Şendur (Sorumlu Yazar)
Kurum: ALANYA ALAADDIN KEYKUBAT UNIVERSITY
Ülke: Turkey


Tarihler

Başvuru Tarihi : 27 Temmuz 2019
Yayımlanma Tarihi : 31 Aralık 2019

Bibtex @araştırma makalesi { akufemubid597506, journal = {Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi}, issn = {}, eissn = {2149-3367}, address = {}, publisher = {Afyon Kocatepe Üniversitesi}, year = {2019}, volume = {19}, pages = {653 - 661}, doi = {}, title = {Stabilized Finite Element Methods for Predator-Prey Systems}, key = {cite}, author = {Şendur, Ali} }
APA Şendur, A . (2019). Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi , 19 (3) , 653-661 . Retrieved from https://dergipark.org.tr/tr/pub/akufemubid/issue/51083/597506
MLA Şendur, A . "Stabilized Finite Element Methods for Predator-Prey Systems". Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 19 (2019 ): 653-661 <https://dergipark.org.tr/tr/pub/akufemubid/issue/51083/597506>
Chicago Şendur, A . "Stabilized Finite Element Methods for Predator-Prey Systems". Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 19 (2019 ): 653-661
RIS TY - JOUR T1 - Stabilized Finite Element Methods for Predator-Prey Systems AU - Ali Şendur Y1 - 2019 PY - 2019 N1 - DO - T2 - Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi JF - Journal JO - JOR SP - 653 EP - 661 VL - 19 IS - 3 SN - -2149-3367 M3 - UR - Y2 - 2019 ER -
EndNote %0 Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi Stabilized Finite Element Methods for Predator-Prey Systems %A Ali Şendur %T Stabilized Finite Element Methods for Predator-Prey Systems %D 2019 %J Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi %P -2149-3367 %V 19 %N 3 %R %U
ISNAD Şendur, Ali . "Stabilized Finite Element Methods for Predator-Prey Systems". Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 19 / 3 (Aralık 2020): 653-661 .
AMA Şendur A . Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2019; 19(3): 653-661.
Vancouver Şendur A . Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2019; 19(3): 661-653.