Skip to main content

Advertisement

SpringerLink
Log in

Fractional-step θ-method for solving singularly perturbed problem in ecology

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrus, J.F.: Stability of a multi-rate method for numerical integration of ODE’s. Computers & Mathematics with Applications 25(2), 3–14 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Armstrong, R.A., McGehee, R.: Competitive exclusion. Am. Nat. 115(2), 151–170 (1980)

    Article  MathSciNet  Google Scholar 

  3. Chrispell, J.C., Ervin, V.J., Jenkins, E.W.: A fractional step 𝜃-method for convection-diffusion problems. J. Math. Anal. Appl. 333(1), 204–218 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Constantinescu, M., Sandu, A.: Extrapolated multirate methods for differential equations with multiple time scales. J. Sci. Comput. 56(1), 28–44 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Constantinescu, M., Sandu, A.: Extrapolated implicit-explicit time stepping. SIAM J. Sci. Comput. 31(6), 4452–4477 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cortez, M.H., Ellner, S.P.: Understanding rapid evolution in predator-prey interactions using the theory of fast-slow dynamical systems. Am. Nat. 176(5), E109–E127 (2010)

    Article  Google Scholar 

  7. Duminil, S., Sadok, H.: Reduced rank extrapolation applied to electronic structure computations. Electron. Trans. Numer. Anal. 38, 347–362 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Fang, H., Saad, Y.: Two classes of multisecant methods for nonlinear acceleration. Numerical Linear Algebra with Applications 16(3), 197–221 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gear, C.W., Wells, D.R.: Multirate linear multistep methods. BIT Numer. Math. 24(4), 484–502 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ginoux, J.-M., Rossetto, B., Jamet, J.-L.: Chaos in a three dimensional Volterra-Gause model of predator-prey type. International Journal of Bifurcation and Chaos 15(5), 1689–1708 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hek, G.: Geometric singular perturbation theory in biological practice. J. Math. Biol. 60(3), 347–386 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Higham, N.J., Strabić, N.: Anderson acceleration of the alternating projections method for computing the nearest correlation matrix. Numerical Algorithms 72, 1021–1042 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hundsdorfel, W., Savcenco, V.: Analysis of a multirate theta-method for stiff ODEs. Appl. Numer. Math. 59(3), 693–706 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jopp, F., Breckling, B., Reuter, H.: Modelling complex ecological dynamics. Springer, Berlin (2011)

    Book  Google Scholar 

  15. Kelley, C.: Solving nonlinear equations with Newton’s method. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  16. Kim, J., Parviz, M.: Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59(2), 308–323 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and application. J. Comput. Phys. 193(2), 357–397 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lott, P.A., Walker, H.F., Woodward, C.S., Yang, U.M.: An accelerated Picard method for nonlinear systems related to variably saturated flow. Adv. Water Resour. 38, 92–101 (2012)

    Article  Google Scholar 

  19. Kuehn, C.: Multiple time scale dynamics. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  20. Lenbury, Y.: Singular perturbation analysis of a model for a predator-prey system invaded by a parasite. Biosystems 39(3), 251–262 (1996)

    Article  Google Scholar 

  21. Lue, W., Xiao, D., Yi, Y.: Relaxation oscillations in a class of predator-prey systems. J. Diff. Equat. 188(1), 306–331 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341–434 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Melis, T.M., Khatiwala, S.: Fast dynamical spin-up of ocean general circulation models using Newton-Krylov methods. Ocean Model. 21(3), 97–105 (2008)

    Article  Google Scholar 

  24. Mergia, W.D., Patidar, K.C.: Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes. Quaest. Math. 39(5), 689–714 (2016)

    Article  MathSciNet  Google Scholar 

  25. Obaid, H.A., Ouifki, R., Patidar, K.C.: An unconditionally stable non standard finite difference method applied to a mathematical model of HIV infection. Int. J. Appl. Math. Comput. Sci. 23(2), 357–372 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Perot, J.B.: An analysis of the fractional step method. J. Comput. Phys. 108 (1), 51–58 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rice, J.R.: Split Runge-Kutta methods for simultaneous equations. J. Res. Natl. Bur. Stand.-B. Math. Math. Phys. 64B(3), 151–170 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  28. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vijalapura, P.K., Stain, J., Govindjee, S.: Fractional step methods for index-1 differential-algebraic equations. J. Comput. Phys. 203(1), 305–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, 7535, South Africa

    Woinshet D. Mergia & Kailash C. Patidar

Authors
  1. Woinshet D. Mergia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kailash C. Patidar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kailash C. Patidar.

Additional information

Communicated by: Silas Alben

The research contained in this chapter is also supported by the South African National Research Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-017-9554-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation