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Journals / Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi / 2019 / Cilt: 19 - Sayı: 3

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

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

Ali ŞENDUR
Pages
653–661
Publication Date
2022-10-12
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The numerical results depict that the algorithm is efficient and feasible.

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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.
📊
Key Findings & Data:We consider the Pseudo Residual-free Bubble (PRFB) method which is based onaugmenting the finite element space by appropriate functions for the space discretization.
💡
Conclusion & Impact:The numerical results depict that the algorithm is efficient and feasible.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

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.

Özet

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.

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Cite This Article

ŞENDUR, A. (2022). A Note on Stabilized Finite Element Methods for Predator-Prey Systems. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, Cilt: 19 - Sayı: 3, 653–661. https://ajindex.com/articles/429876

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References (13)

  1. 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.
  2. 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.
  3. Brezzi, F. and Russo, A., 1994. Choosing bubbles for advection-diffusion problems. Mathematical Models and Methods in Applied Sciences, 4, 571–587.
  4. 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.
  5. 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.
  6. 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.
  7. Cosner, C., 2014. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 34, 1701–1745.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. Garzon-Alvarado, D.A., Galeano, C.H. and Mantilla, J.M., 2012. Computational examples