Abstract
We consider a predator-prey model arising in ecology that describes a slow-fast dynamical system. The dynamics of the model is expressed by a system of nonlinear differential equations having different time scales. Designing numerical methods for solving problems exhibiting multiple time scales within a system, such as those considered in this paper, has always been a challenging task. To solve such complicated systems, we therefore use an efficient time-stepping algorithm based on fractional-step methods. To develop our algorithm, we first decouple the original system into fast and slow sub-systems, and then apply suitable sub-algorithms based on a class of θ-methods, to discretize each sub-system independently using different time-steps. Then the algorithm for the full problem is obtained by utilizing a higher-order product method by merging the sub-algorithms at each time-step. The nonlinear system resulting from the use of implicit schemes is solved by two different nonlinear solvers, namely, the Jacobian-free Newton-Krylov method and the well-known Anderson’s acceleration technique. The fractional-step θ-methods give us flexibility to use a variety of methods for each sub-system and they are able to preserve qualitative properties of the solution. We analyze these methods for stability and convergence. Several numerical results indicating the efficiency of the proposed method are presented. We also provide numerical results that confirm our theoretical investigations.
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Acknowledgments
Authors wish to acknowledge the anonyms referees for their valuable suggestions that have improved the presentation of the work in this paper. WDM acknowledges the African Institute for Mathematical Sciences, South Africa as well as University of the Western Cape for the financial support. KCP’s research was also supported by the South African National Research Foundation.
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Communicated by: Silas Alben
The research contained in this chapter is also supported by the South African National Research Foundation.
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Mergia, W.D., Patidar, K.C. Fractional-step θ-method for solving singularly perturbed problem in ecology. Adv Comput Math 44, 645–671 (2018). https://doi.org/10.1007/s10444-017-9554-8
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DOI: https://doi.org/10.1007/s10444-017-9554-8
Keywords
- Mathematical ecology
- Slow-fast dynamics
- Fractional-step θ-method
- Jacobian-free Newton-Krylov method
- Anderson’s acceleration method