Reaction-diffusion equations modeling Predator-Prey interaction are of current interest. Standard approaches such as first-order (in time) finite difference schemes for approximating the solution are widely spread. Though, this paper shows that recent advance methods can be more favored. In this work, we have incorporated, throughout numerical comparison experiments, spectral methods, for the space discretization, in conjunction with second and fourth-order time integrating methods for approximating the solution of the reaction-diffusion differential equations. The results have revealed that these methods have advantages over the conventional methods , some of which to mention are: the ease of implementation, accuracy and CPU time.
KeywordsFinite Difference MethodsExponential IntegratorExponential Time Differencing MethodReaction-Diffusion System
Murray, J.D. (1993) Mathematical Biology. 2nd Edition, Springer-Verlag, Berlin Heidelberg.
Turing, A. (1952) The Chemical Basis of Morphogenesis. Philosophical Transactions of the Royal Society B, 237, 37-72. https://doi.org/10.1098/rstb.1952.0012
Neville, A.A., Matthews, P.C. and Byrne, H.M. (2006) Interactions between Pattern Formation and Domain Growth. Bulletin of Mathematical Biology, 68, 1975-2003. https://doi.org/10.1007/s11538-006-9060-5
Haque, M. (2009) Ratio-Dependent Predator-Prey Models of Interacting Populations. Bulletin of Mathematical Biology, 71, 430-452. https://doi.org/10.1007/s11538-008-9368-4
Garvie, M.R. (2007) Finite-Difference Schemes for Reaction-Diffusion Equations Modelling Predator-Prey Interactions in MATLAB. Bulletin of Mathematical Biology, 69, 931-956. https://doi.org/10.1007/s11538-006-9062-3
Apreutesei, N. and Dimitriu, G. (2010) On a Preypredator Reactiondiffusion System with Holling Type III Functional Response. Journal of Computational and Applied Mathematics, 235, 366-379. https://doi.org/10.1016/j.cam.2010.05.040
Garfinkel, D., Marbach, C.B. and Shapiro, N.Z. (1977) Stiff Differential Equations. Annual Review of Biophysics and Bioengineering, 6, 525-542. https://doi.org/10.1146/annurev.bb.06.060177.002521
Hairer, E. and Wanner, G. (1996) Solving Ordinary Differential Equations II. 2nd Edition, Springer-Verlag, Berlin. https://doi.org/10.1007/978-3-642-05221-7
Shampine, L.F. and Gear, C.W. (1979) A User’s View of Solving Stiff Ordinary Differential Equations. SIAM Review, 21, 1-17. https://doi.org/10.1137/1021001
Gear, C.W. (1971) Automatic Integration of Ordinary Differential Equations. Communications of the ACM, 14, 176-179. https://doi.org/10.1145/362566.362571
Curtiss, C.F. and Hirschfelder, J.O. (1952) Integration of Stiff Equations. Proceedings of the National Academy of Sciences, 38, 235-243. https://doi.org/10.1073/pnas.38.3.235
Cox, S.M. and Matthews, P.C. (2002) Exponential Time Differencing for Stiff Systems. Journal of Computational Physics, 176, 430-455. https://doi.org/10.1006/jcph.2002.6995
Boyd, J.P. (2001) Chebyshev and Fourier Spectral Methods. 2nd Edition, Dover, New York.
Fornberg, B. (1996) A Practical Guide to Pseudo-Spectral Methods. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511626357
Fornberg, B. and Driscoll, T.A. (1999) A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion. Journal of Computational Physics, 155, 456-467. https://doi.org/10.1006/jcph.1999.6351
Craster, R.V. and Sassi, R. (2006) Spectral Algorithm for Reaction-Diffusion Equations. Note del Polo 99, Universita degli Studi di Milano, Polp Didattico e di Ricerca di Crema.
Dimitriu, G. and Tefanescu, R.S. (2009) Numerical Experiments for Reaction-Diffusion Equations Using Exponential Integrators. N. A. A., 5434, 249-256. https://doi.org/10.1007/978-3-642-00464-3_26
Kassam, A.K. (2003) Solving Reaction-Diffusion Equations 10 Times Faster. Oxford University, Oxford, Numerical Analysis Group Research Report No. 16.
Khaliq, A.Q.M., Martin-Vaquero, J., Wade, B.A. and Yousuf, M. (2009) Smoothing Schemes for Reaction-Diffusion Systems with Nonsmooth Data. Journal of Computational and Applied Mathematics, 223, 374-386. https://doi.org/10.1016/j.cam.2008.01.017
Trefethen, L.N. (2000) Spectral Methods in MATLAB. SIAM, Philadelphia. https://doi.org/10.1137/1.9780898719598
Minchev, B.V. and Wright, W.M. (2005) A Review of Exponential Integrators for First Order Semi-Linear Problems. Tech. Rep. NTNU.
Ashi, H.A., Cummings, L.J. and Matthews, P.C. (2009) Comparison of Methods for Evaluating Functions of a Matrix Exponential. Applied Numerical Mathematics, 59, 468-486. https://doi.org/10.1016/j.apnum.2008.03.039
Ashi, H.A. (2008) Numerical Methods for Stiff Systems. PhD Thesis, University of Nottingham, Nottingham.
Skaflestad, B. and Wright, W.M. (2009) The Scaling and Modified Squaring Method for Matrix Functions Related to the Exponential. Applied Numerical Mathematics, 59, 783-799. https://doi.org/10.1016/j.apnum.2008.03.035
Kassam, A.K. (2004) High Order Time Stepping for Stiff Semi-Linear Partial Differential Equations. PhD Thesis, Oxford University, Oxford.
Schmelzer, T. and Trefethen, L.N. (2007) Evaluating Matrix Functions for Exponential Integrators via Carathéodory-Fejér Approximation and Contour Integrals. Electronic Transactions on Numerical Analysis, 29, 1-18.
Higham, N.J. (2005) The Scaling and Squaring Method for the Matrix Exponentials Revisited. SIAM Journal on Matrix Analysis and Applications, 26, 1179-1193. https://doi.org/10.1137/04061101X
Koikari, S. (2007) An Error Analysis of the Modified Scaling and Squaring Method. Computers & Mathematics with Applications, 53, 1293-1305. https://doi.org/10.1016/j.camwa.2006.04.032
Beylkin, G., Keiser, J.M. and Vozovoi, L. (1998) A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs. Journal of Computational Physics, 147, 362-387. https://doi.org/10.1006/jcph.1998.6093
Berland, H., Skeflestad, B. and Wright, W.M. (2007) EXPINT—A Matlab Package for Exponential Integrators. ACM Transactions on Mathematical Software, 33, Article No. 4.
Kassam, A.K. and Trefethen, L.N. (2005) Fourth-Order Time Stepping for Stiff PDEs. SIAM: SIAM Journal on Scientific Computing, 26, 1214-1233. https://doi.org/10.1137/S1064827502410633
Ruuth, S.J. (1995) Implicit-Explicit Methods for reaction-Diffusion Problems in Pattern Formation. Journal of Mathematical Biology, 34, 148-176. https://doi.org/10.1007/BF00178771
Ascher, U.M., Ruuth, S.J. and Wetton, B.T. (1995) Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. SIAM Journal on Numerical Analysis, 32, 797-823.
Moler, C. and Van Loan, C. (2003) Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, 45, 3-49. https://doi.org/10.1137/S00361445024180