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Stabilized Finite Element Methods for Predator-Prey Systems

Yıl 2019, Cilt: 19 Sayı: 3, 653 - 661, 31.12.2019
https://doi.org/10.35414/akufemubid.597506

Öz

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

Kaynakça

  • [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.

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

Yıl 2019, Cilt: 19 Sayı: 3, 653 - 661, 31.12.2019
https://doi.org/10.35414/akufemubid.597506

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.

Kaynakça

  • [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.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Ali Şendur 0000-0001-8628-5497

Yayımlanma Tarihi 31 Aralık 2019
Gönderilme Tarihi 27 Temmuz 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 19 Sayı: 3

Kaynak Göster

APA Şendur, A. (2019). Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 19(3), 653-661. https://doi.org/10.35414/akufemubid.597506
AMA Şendur A. Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. Aralık 2019;19(3):653-661. doi:10.35414/akufemubid.597506
Chicago Şendur, Ali. “Stabilized Finite Element Methods for Predator-Prey Systems”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 19, sy. 3 (Aralık 2019): 653-61. https://doi.org/10.35414/akufemubid.597506.
EndNote Şendur A (01 Aralık 2019) Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 19 3 653–661.
IEEE A. Şendur, “Stabilized Finite Element Methods for Predator-Prey Systems”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, c. 19, sy. 3, ss. 653–661, 2019, doi: 10.35414/akufemubid.597506.
ISNAD Şendur, Ali. “Stabilized Finite Element Methods for Predator-Prey Systems”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 19/3 (Aralık 2019), 653-661. https://doi.org/10.35414/akufemubid.597506.
JAMA Şendur A. Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2019;19:653–661.
MLA Şendur, Ali. “Stabilized Finite Element Methods for Predator-Prey Systems”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, c. 19, sy. 3, 2019, ss. 653-61, doi:10.35414/akufemubid.597506.
Vancouver Şendur A. Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2019;19(3):653-61.


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