Skip to main content
SpringerLink
Log in

Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We present two finite-difference algorithms for studying the dynamics of spatially extended predator–prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB. Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator–prey interactions.

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

  • Alonso, D., Bartumeus, F., Catalan, J., 2002. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83(1), 28–34.

    Google Scholar 

  • Ascher, U., Ruuth, S., Wetton, B., 1995. Implicit–explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823.

    Article  MATH  MathSciNet  Google Scholar 

  • Beckett, G., Mackenzie, J., 2001. On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution. J. Comput. Appl. Math. 131, 381–405.

    Article  MATH  MathSciNet  Google Scholar 

  • Brenner, S., Scott, L., 1994. The Mathematical Theory of Finite Element Methods. Vol. 15: Texts in Applied Mathematics. Springer, New York.

    Google Scholar 

  • Ciarlet, P., 1979. The Finite Element Method for Elliptic Problems. Vol. 4: Studies in Mathematics and its Applications. North-Holland, Amsterdam.

    Google Scholar 

  • Elliott, C., Stuart, A., 1993. The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30(6), 1622–1663.

    Article  MATH  MathSciNet  Google Scholar 

  • Freedman, H., 1980. Deterministic Mathematical Models in Population Ecology. Vol. 57: Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York.

    Google Scholar 

  • Garvie, M., Trenchea, C., 2005a. Analysis of two generic spatially extended predator–prey models. Nonlinear Anal. Real World Appl., submitted for publication.

  • Garvie, M., Trenchea, C., 2005b. Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response. Numer. Math., submitted for publication.

  • Gentleman, W., Leising, A., Frost, B., Strom, S., Murray, J., 2003. Functional responses for zooplankton feeding on multiple resources: A review of assumptions and biological dynamics. Deep Sea Res. II 50, 2847–2875.

    Article  Google Scholar 

  • Gurney, W., Veitch, A., Cruickshank, I., McGeachin, G., 1998. Circles and spirals: Population persistence in a spatially explicit predator–prey model. Ecology 79(7), 2516–2530.

    Google Scholar 

  • Hildebrand, F., 1968. Finite-Difference Equations and Simulations. Prentice-Hall, Englewood Cliffs, NJ.

    MATH  Google Scholar 

  • Hoff, D., 1978. Stability and convergence of finite difference methods for systems of nonlinear reaction–diffusion equations. SIAM J. Numer. Anal. 15(6), 1161–1177.

    Article  MATH  MathSciNet  Google Scholar 

  • Holling, C., 1959. Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385–398.

    Article  Google Scholar 

  • Holling, C., 1965. The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 45, 1–60.

    Google Scholar 

  • Holmes, E., Lewis, M., Banks, J., Veit, R., 1994. Partial differential equations in ecology: Spatial interactions and population dynamics. Ecology 75(1), 17–29.

    Article  Google Scholar 

  • Isaacson, E., Keller, H., 1966. Analysis of Numerical Methods. Wiley, New York.

    MATH  Google Scholar 

  • Ivlev, V., 1961. Experimental Ecology of the Feeding Fishes. Yale University Press, New Haven.

    Google Scholar 

  • Jerome, J., 1984. Fully discrete stability and invariant rectangular regions for reaction–diffusion systems. SIAM J. Numer. Anal. 21(6), 1054–1065.

    Article  MATH  MathSciNet  Google Scholar 

  • Jeschke, J., Kopp, M., Tollrian, R., 2002. Predator functional responses: Discriminating between handling and digesting prey. Ecol. Monogr. 72(1), 95–112.

    Google Scholar 

  • Li, N., Steiner, J., Tang, S.-M., 1994. Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction–diffusion equations. J. Aust. Math. Soc. Ser. B 36(2), 234–241.

    Article  MATH  MathSciNet  Google Scholar 

  • Malchow, H., Petrovskii, S., 2002. Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Math. Comput. Model. 36, 307–319.

    Article  MATH  MathSciNet  Google Scholar 

  • May, R., 1974. Stability and Complexity in Model Ecosystems. Princeton University Press, New Jersey.

    Google Scholar 

  • Medvinsky, A., Petrovskii, S., Tikhonova, I., Malchow, H., Li, B.-L., 2002. Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370.

    Article  MATH  MathSciNet  Google Scholar 

  • Mickens, R., 2003. A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion. Comput. Math. Appl. 45, 429–436.

    Article  MATH  MathSciNet  Google Scholar 

  • Morton, K., Mayers, D., 1996. Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge.

    Google Scholar 

  • Murray, J., 1993. Mathematical Biology. Vol. 19: Biomathematics Texts. Springer, Berlin.

    Google Scholar 

  • Neubert, M., Caswell, H., Murray, J., 2002. Transient dynamics and pattern formation: Reactivity is necessary for Turing instabilities. Math. Biosci. 175, 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  • Pao, C., 1998. Accelerated monotone iterative methods for finite difference equations of reaction–diffusion. Numer. Math. 79, 261–281.

    Article  MATH  MathSciNet  Google Scholar 

  • Pao, C., 1999. Numerical analysis of coupled systems of nonlinear parabolic equations. SIAM J. Numer. Anal. 36(2), 393–416.

    Article  MathSciNet  Google Scholar 

  • Pao, C., 2002. Finite difference reaction–diffusion systems with coupled boundary conditions and time delays. J. Math. Anal. 272, 407–434.

    Article  MATH  MathSciNet  Google Scholar 

  • Pascual, M., 1993. Diffusion-induced chaos in a spatial predator–prey system. Proc. R. Soc. Lond. Ser. B 251, 1–7.

    Article  Google Scholar 

  • Petrovskii, S., Malchow, H., 1999. A minimal model of pattern formation in a prey–predator system. Math. Comput. Model. 29, 49–63.

    Article  MATH  MathSciNet  Google Scholar 

  • Petrovskii, S., Malchow, H., 2001. Wave of chaos: New mechanism of pattern formation in spatio-temporal population dynamics. Theor. Populat. Biol. 59, 157–174.

    Article  MATH  Google Scholar 

  • Petrovskii, S., Malchow, H., 2002. Critical phenomena in plankton communities: KISS model revisited. Nonlinear Anal. Real 1, 37–51.

    Article  MathSciNet  Google Scholar 

  • Pujol, M., Grimalt, P., 2002. A non-linear model for cerebral diffusion: Stability of finite differences method and resolution using the Adomian method. Int. J. Numer. Method H 13(4), 473–485.

    Article  MathSciNet  Google Scholar 

  • Rai, V., Jayaraman, G., 2003. Is diffusion-induced chaos robust? Curr. Sci. India 84(7), 925–929.

    Google Scholar 

  • Richtmyer, R., Morton, K., 1967. Difference Methods for Initial Value Problems. Vol. 4: Interscience Tracts in Pure and Applied Mathematics. Wiley-Interscience, New York.

    Google Scholar 

  • Rosenzweig, M., MacArthur, R., 1963. Graphical representation and stability conditions for predator–prey interaction. Am. Nat. 97, 209–223.

    Article  Google Scholar 

  • Ruuth, J., 1995. Implicit–explicit methods for reaction–diffusion problems in pattern formation. J. Math. Biol. 34, 148–176.

    Article  MATH  MathSciNet  Google Scholar 

  • Saad, Y., 2003. Iterative methods for sparse linear systems. SIAM.

  • Saad, Y., Schultz, M., 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869.

    Article  MATH  MathSciNet  Google Scholar 

  • Savill, N., Hogeweg, P., 1999. Competition and dispersal in predator–prey waves. Theor. Populat. Biol. 56, 243–263.

    Article  MATH  Google Scholar 

  • Segel, L., Jackson, J., 1972. Dissipative structure: An explanation and an ecological example. J. Theor. Biol. 37, 545–559.

    Article  Google Scholar 

  • Sherratt, J., 2001. Periodic travelling waves in cyclic predator–prey systems. Ecol. Lett. 4, 30–37.

    Article  Google Scholar 

  • Sherratt, J., Eagan, B., Lewis, M., 1997. Oscillations and chaos behind predator–prey invasion: Mathematical artifact or ecological reality? Phil. Trans. R. Soc. Lond. B 352, 21–38.

    Article  Google Scholar 

  • Sherratt, J., Lambin, X., Thomas, C., Sherratt, T., 2002. Generation of periodic waves by landscape features in cyclic predator–prey systems. Proc. R. Soc. Lond. Ser. B 269, 327–334.

    Article  Google Scholar 

  • Sherratt, J., Lewis, M., Fowler, A., 1995. Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. U.S.A. 92, 2524–2528.

    Article  MATH  Google Scholar 

  • Skalski, G., Gilliam, J.F., 2001. Functional responses with predator interference: Viable alternatives to the Holling type II model. Ecology 82(11), 3083–3092.

    Article  Google Scholar 

  • Smoller, J., 1983. Shock Waves and Reaction–Diffusion Equations. Vol. 258: Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York.

    Google Scholar 

  • Stuart, A., 1989. Nonlinear instability in dissipative finite difference schemes. SIAM Rev. 31(2), 191–220.

    Article  MATH  MathSciNet  Google Scholar 

  • Stuart, A., Humphries, A., 1998. Dynamical Systems and Numerical Analysis. Vol. 2: Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge.

    Google Scholar 

  • Turing, A., 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72.

    Article  Google Scholar 

  • Yee, H., Sweby, P., 1994. Global asymptotic behavior of iterative implicit schemes. Int. J. Bifurcat. Chaos 4(6), 1579–1611.

    Article  MATH  MathSciNet  Google Scholar 

  • Yee, H., Sweby, P., 1995. Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations II. Global asymptotic behaviour of time discretizations. Comp. Fluid Dyn. 4, 219–283.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computational Science, Florida State University, Tallahassee, FL, 32306-4120, USA

    Marcus R. Garvie

Authors
  1. Marcus R. Garvie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcus R. Garvie.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garvie, M.R. Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB . Bull. Math. Biol. 69, 931–956 (2007). https://doi.org/10.1007/s11538-006-9062-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-006-9062-3

Keywords

Search

Navigation