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Nonstandard finite-difference methods for predator–prey models with general functional response

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  • Dimitrov, Dobromir T.
  • Kojouharov, Hristo V.

Abstract

Predator–prey systems with linear and logistic intrinsic growth rate of the prey are analyzed. The models incorporate the mutual interference between predators into the functional response which stabilizes predator–prey interactions in the system. Positive and elementary stable nonstandard (PESN) finite-difference methods, having the same qualitative features as the corresponding continuous predator–prey models, are formulated and analyzed. The proposed numerical techniques are based on a nonlocal modeling of the growth-rate function and a nonstandard discretization of the time derivative. This discretization approach leads to significant qualitative improvements in the behavior of the numerical solution. In addition, it allows for the use of an essentially implicit method for the cost of an explicit method. Applications of the PESN methods to specific predator–prey systems are also presented.

Suggested Citation

  • Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2008. "Nonstandard finite-difference methods for predator–prey models with general functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(1), pages 1-11.
    • Handle: RePEc:eee:matcom:v:78:y:2008:i:1:p:1-11
    DOI: 10.1016/j.matcom.2007.05.001
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    References listed on IDEAS

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    1. Jansen, H. & Twizell, E.H., 2002. "An unconditionally convergent discretization of the SEIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(2), pages 147-158.
    2. Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2005. "Analysis and numerical simulation of phytoplankton–nutrient systems with nutrient loss," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(1), pages 33-43.
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    Cited by:

    1. Korkut, Sıla Ö. & Erdoğan, Utku, 2018. "Positivity preserving scheme based on exponential integrators," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 731-739.
    2. Wood, Daniel T. & Kojouharov, Hristo V. & Dimitrov, Dobromir T., 2017. "Universal approaches to approximate biological systems with nonstandard finite difference methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 133(C), pages 337-350.
    3. Abraham J. Arenas & Gilberto González-Parra & Jhon J. Naranjo & Myladis Cogollo & Nicolás De La Espriella, 2021. "Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    4. Jódar, Lucas & Villanueva, Rafael J. & Arenas, Abraham J. & González, Gilberto C., 2008. "Nonstandard numerical methods for a mathematical model for influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 622-633.
    5. Tuan Hoang, Manh & Nagy, A.M., 2019. "Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 24-34.
    6. Joel Alba-Pérez & Jorge E. Macías-Díaz, 2019. "Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion," Mathematics, MDPI, vol. 7(12), pages 1-31, December.
    7. Pasha, Syed Ahmed & Nawaz, Yasir & Arif, Muhammad Shoaib, 2023. "On the nonstandard finite difference method for reaction–diffusion models," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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