High order accuracy scheme for modeling the dynamics of predator and prey in heterogeneous environment

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Abstract

The aim of this work is to develop a compact finite-difference approach for modeling the dynamics of predator and prey based on reaction-diffusion-advection equations with variable coefficients. Methods. To discretize a spatially inhomogeneous problem with nonlinear terms of taxis and local interaction, the balance method is used. Species densities are determined on the main grid whereas fluxes are computed at the nodes of the staggered grid. Integration over time is carried out using the high-order Runge-Kutta method. Results. For the case of one-dimensional annular interval, the finite-difference scheme on the three-point stencil has been constructed that makes it possible to increase the order of accuracy compared to the standard second-order approximation scheme. The results of computational experiment are presented and comparison of schemes for stationary and non-stationary solutions is carried out. We conduct the calculation of accuracy order basing on the Aitken process for sequences of spatial grids. The calculated values of the effective order accuracy for the proposed scheme were greater than the standard two: for the diffusion problem, values of at least four were obtained. Decrease was obtained when directional migration was taken into account. This conclusion was also confirmed for non-stationary oscillatory regimes. Conclusion. The results demonstrate the effectiveness of the derived scheme for dynamics of predator and prey system in a heterogeneous environment.

About the authors

Buu Hoang Nguyen

Southern Federal University

ORCID iD: 0009-0001-1644-5800
Scopus Author ID: 58109765900
ul. Bol`shaya Sadovaya 105/42, Rostov-on-Don, 344006, Russia

Vyacheslav Georgievich Tsybulin

Southern Federal University

ORCID iD: 0000-0003-4812-278X
Scopus Author ID: 6507974728
ResearcherId: S-7753-2016
ul. Bol`shaya Sadovaya 105/42, Rostov-on-Don, 344006, Russia

References

  1. Толстых А. И. Компактные разностные схемы и их применение в задачах аэрогидродинамики. М.: Наука, 1990. 232 с.
  2. Толстых А. И. Компактные и мультиоператорные аппроксимации высокой точности для уравнений в частных производных. М.: Наука, 2015. 350 с.
  3. Zhang L., Ge Y. Numerical solution of nonlinear advection diffusion reaction equation using high-order compact difference method // Applied Numerical Mathematics. 2021. Vol. 166. P. 127–145. doi: 10.1016/j.apnum.2021.04.004.
  4. Deka D., Sen S. Compact higher order discretization of 3D generalized convection diffusion equation with variable coefficients in nonuniform grids // Applied Mathematics and Computation. 2022. Vol. 413, no. 5. P. 126652. doi: 10.1016/j.amc.2021.126652.
  5. Матус П. П., Утебаев Б. Д. Компактные и монотонные разностные схемы для обобщенного уравнения Фишера // Дифференциальные уравнения. 2022. T. 58, № 7. C. 947–961. doi: 10.31857/S037406412207007X.
  6. He M., Liao W. A compact ADI finite difference method for 2D reaction–diffusion equations with variable diffusion coefficients // Journal of Computational and Applied Mathematics. 2024. Vol 436. P. 115400. doi: 10.1016/j.cam.2023.115400.
  7. Xu P., Ge Y., Zhang L. High-order finite difference approximation of the Keller-Segel model with additional self-and cross-diffusion terms and a logistic source // Networks & Heterogeneous Media. 2022. Vol. 18, no. 4. P. 1471–1492. doi: 10.3934/nhm.2023065.
  8. Самарский А. А. Теория разностных схем. М.: Наука, 1989. 616 с.
  9. Калиткин Н. Н. Численные методы. СПб.: БХВ-Петербург, 2011. 592 с.
  10. Хайрер Э., Нерсетт С., Ваннер Г. Решение обыкновенных дифференциальных уравнений. Нежесткие задачи. М.: Мир, 1990. 512 с.
  11. Мюррей Дж. Математическая биология. Т. 2 Пространственные модели и их приложения в биомедицине. М.-Ижевск: НИЦ «Регулярная и хаотическая динамика», Институт компьютерных исследований, 2011. 1104 с.
  12. Rubin A., Riznichenko G. Mathematical biophysics. New York: Springer, 2014. 273 p. doi: 10.1007/978-1-4614-8702-9.
  13. Cantrell R. S., Cosner C. Spatial Ecology Via Reaction–Diffusion Equations. Chichester: John Wiley and Sons Ltd, 2003. 428 p. doi: 10.1002/0470871296.
  14. Malchow H., Petrovskii S. V., Venturino E. Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation. New York: Chapman and Hall, 2008. 469 p.
  15. Budyansky A. V., Frischmuth K., Tsybulin V. G. Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat // Discrete & Continuous Dynamical Systems – B. 2019. Vol. 24, no. 2. P. 547–561. doi: 10.3934/dcdsb.2018196.
  16. Будянский А. В., Цибулин В. Г. Моделирование многофакторного таксиса в системе «хищник– жертва» // Биофизика. 2019. Т. 64, № 2, С. 343–349. doi: 10.1134/S0006302919020133.
  17. Цибулин В. Г., Ха Т. Д., Зеленчук П. А. Нелинейная динамика системы хищник-жертва на неоднородном ареале и сценарии локального взаимодействия видов // Известия высших учебных заведений. Прикладная нелинейная динамика. 2021. Т. 29, № 5. С. 751–764. doi: 10.18500/0869-6632-2021-29-5-751-764.

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