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Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

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Abstract

We study the numerical approximation of the solutions of a class of nonlinear reaction–diffusion systems modelling predator–prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction–diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621–646, 2005).

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References

  1. Adams R.A. and Fournier J. (1977). Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61: 713–734

    Article  MATH  MathSciNet  Google Scholar 

  2. Aide M., Osaki K., Tsujikawa T., Yagi A. and Mimura M. (2005). Chemotaxis and growth system with singular sensitivity function. Nonlinear Anal. Real World Appl. 6(2): 323–336

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  4. Barrett J.W., Blowey J.F. and Garcke H. (1998). Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80(4): 525–556

    Article  MATH  MathSciNet  Google Scholar 

  5. Barrett J.W., Blowey J.F. and Garcke H. (2001). On fully practical finite element approximations of degenerate Cahn–Hilliard systems. M2AN Math. Model. Numer. Anal. 35(4): 713–748

    Article  MATH  MathSciNet  Google Scholar 

  6. Begon M., Harper J.L. and Townsend C.R. (1990). Ecology: Individuals, Populations and Communities. Blackwell, Boston

    Google Scholar 

  7. Bonami A., Hihorst D., Logak E. and Mimura M. (2001). Singular limit of a chemotaxis-growth model. Adv. Differ. Equ. 6(10): 1173–1218

    MATH  Google Scholar 

  8. Ciarlet, P.G.: The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1979)

  9. Ciarlet P.G. and Raviart P.G. (1972). General Lagrange and Hermite interpolation in R n with applications to finite element methods. Arch. Rational. Mech. Anal. 46: 177–199

    Article  MATH  MathSciNet  Google Scholar 

  10. Crooks E.C.M., Dancer E.N., Hilhorst D., Mimura M. and Ninomiya H. (2004). Spatial segregation limit of a competition–diffusion system with dirichlet boundary conditions. Nonlinear Anal. Real World Appl. 5(4): 645–665

    Article  MATH  MathSciNet  Google Scholar 

  11. Dancer E.N., Hilhorst D., Mimura M. and Peletier L.A. (1999). Spatial segregation limit of a competition– diffusion system. Eur. J. Appl. Math. 10(2): 97–115

    Article  MATH  MathSciNet  Google Scholar 

  12. Ei S.-I., Ikota R. and Mimura M. (1999). Segregating partition problem in competition–diffusion systems. Interfaces Free Bound. 1(1): 57–80

    MATH  MathSciNet  Google Scholar 

  13. Ei S.-I., Mimura M. and Nagayama M. (2002). Pulse–pulse interaction in reaction–diffusion systems. Phys. D 165(3-4): 176–198

    Article  MATH  MathSciNet  Google Scholar 

  14. Elliott C.M. (1987). Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7: 61–71

    Article  MATH  MathSciNet  Google Scholar 

  15. Feireisl E., Hilhorst D., Mimura M. and Weidenfeld R. (2003). On a nonlinear diffusion system with resource–consumer interaction. Hiroshima Math. J. 33(2): 253–295

    MATH  MathSciNet  Google Scholar 

  16. Freedman, H.: Deterministic mathematical models in population ecology. Monographs and Textbooks in Pure and Applied Mathematics, vol. 57. Marcel Dekker, New York (1980)

  17. Funaki M., Mimura M. and Tsujikawa T. (2006). Travelling front solutions arising in the chemotaxis-growth model. Interfaces Free Bound. 8(2): 223–245

    Article  MATH  MathSciNet  Google Scholar 

  18. Garvie, M.R.: Analysis of a reaction–diffusion system of λ − ω type, Ph.D. thesis. University of Durham (2003)

  19. Garvie M.R. and Blowey J.F. (2005). A reaction–diffusion system of λ − ω type. Part II: Numerical analysis. Eur. J. Appl. Math. 16(5): 621–646

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  22. Gutierrez A.P. (1996). Applied Population Ecology: A Supply–Demand Approach. Wiley, New York

    Google Scholar 

  23. Hilhorst D., Iida M., Mimura M. and Ninomiya H. (2001). A competition–diffusion system approximation to the classical two-phase stefan problem. Japan J. Ind. Appl. Math. 18(2): 161–180

    Article  MATH  MathSciNet  Google Scholar 

  24. Hilhorst D., Iida M., Mimura M. and Ninomiya H. (2001). A reaction–diffusion system approximation to the two-phase stefan problem. Nonlinear Anal. 47(2): 801–812

    Article  MATH  MathSciNet  Google Scholar 

  25. Hilhorst D., Mimura M. and Schätzle R. (2003). Vanishing latent heat limit in a Stefan-like problem arising in biology. Anal. Real World Appl. 4(2): 261–285

    Article  MATH  MathSciNet  Google Scholar 

  26. Holling C.S. (1959). Some characteristics of simple types of predation and parasitism. Can. Entomol. 91: 385–398

    Google Scholar 

  27. Holling C.S. (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 

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

    Article  Google Scholar 

  29. Iida M., Mimura M. and Ninomiya H. (2006). Diffusion, cross-diffusion and competitive interaction. J. Math. Biol. 53(4): 617–641

    Article  MATH  MathSciNet  Google Scholar 

  30. Ikeda H. and Mimura M. (1991). Stability analysis of stationary solutions of bistable reaction–variable diffusion systems. SIAM J. Math. Anal. 22(6): 1651–1678

    Article  MATH  MathSciNet  Google Scholar 

  31. Ikeda T., Ikeda H. and Mimura M. (2000). Hopf bifurcation of travelling pulses in some bistable reaction– diffusion systems. Methods Appl. Anal. 7(1): 165–193

    MATH  MathSciNet  Google Scholar 

  32. Ivlev V.S. (1961). Experimental Ecology of the Feeding Fishes. Yale University Press, New Haven

    Google Scholar 

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

    Google Scholar 

  34. Juliano S.A. (1993). Nonlinear curve fitting: Predation and functional response curves. In: Scheiner, S.M. and Gurevitch, J. (eds) Design and Analysis of Ecological Experiments, pp. Chapman & Hall, New York

    Google Scholar 

  35. Malchow H. and Petrovskii S.V. (2002). Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Math. Comput. Model. 36: 307–319

    Article  MATH  MathSciNet  Google Scholar 

  36. May R.M. (1974). Stability and Complexity in Model Ecosystems. Princeton University Press, NJ

    Google Scholar 

  37. May R.M. (1976). Models for two interacting populations. In: May, R.M. (eds) Theoretical Ecology: Principles and Applications., pp 1–2. W.B. Saunders, Philadelphia

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Mimura M. (2001). Dynamics of patterns and interfaces in some reaction–diffusion systems from chemical and biological viewpoints. Methods Appl. Anal. 8(3): 497–514

    MATH  MathSciNet  Google Scholar 

  40. Mimura M. and Kawasaki K. (1980). Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9(1): 49–64

    Article  MATH  MathSciNet  Google Scholar 

  41. Mimura M., Nagayama M. and Ohta T. (2002). Non-annihilation of travelling pulses in a reaction–diffusion system. Methods Appl. Anal. 9(4): 493–515

    MathSciNet  MATH  Google Scholar 

  42. Murray J.D. (1993). Mathematical biology Biomathematics Texts, vol. 19.. Springer, Berlin

    Google Scholar 

  43. Nishiura Y., Mimura M., Ikeda H. and Fujii H. (1990). Singular limit analysis of stability of traveling wave solutions in bistable reaction–diffusion systems. SIAM J. Math. Anal. 21(1): 85–122

    Article  MATH  MathSciNet  Google Scholar 

  44. Nochetto, R.H.: Finite element methods for parabolic free boundary problems. In: Light, W. (ed.) Advances in Numerical Analysis: Nonlinear Partial Differential Equations and Dynamical Systems, vol. 1, pp. 34–95. Oxford University Press, New York (1991)

  45. Nochetto R.H. and Verdi C. (1996). Combined effect of explicit time-stepping and quadrature for curvature driven flows. Numer. Math. 74: 105–136

    Article  MATH  MathSciNet  Google Scholar 

  46. Petrovskii S.V. and 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 

  47. Petrovskii S.V. and Malchow H. (2001). Wave of chaos: New mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol. 59: 157–174

    Article  MATH  Google Scholar 

  48. Petrovskii S.V. and Malchow H. (2002). Critical phenomena in plankton communities: KISS model revisited. Nonlinear Anal. Real. 1: 37–51

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  50. Raviart P.A. (1973). The use of numerical integration in finite element methods for solving parabolic equations. In: Miller, J. (eds) Topics in Numerical Analysis, pp 233–264. Academic, New York

    Google Scholar 

  51. Rodrigues J. (1987). Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam

    MATH  Google Scholar 

  52. Rosenzweig M.L. and MacArthur R.H. (1963). Graphical representation and stability conditions for predator–prey interaction. Am. Nat. 97: 209–223

    Article  Google Scholar 

  53. Savill N.J. and Hogeweg P. (1999). Competition and dispersal in predator–prey waves. Theor. Popul. Biol. 56: 243–263

    Article  MATH  Google Scholar 

  54. Sherratt J.A., Eagan B.T. and Lewis M.A. (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 

  55. Sherratt J.A., Lambin X., Thomas C.J. and Sherratt T.N. (2002). Generation of periodic waves by landscape features in cyclic predator–prey systems. Proc. R. Soc. Lond. B 269: 327–334

    Article  Google Scholar 

  56. Sherratt J.A., Lewis M.A. and Fowler A.C. (1995). Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. 92: 2524–2528

    Article  MATH  Google Scholar 

  57. Skalski G.T. and 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 

  58. Smoller J. (1983). Shock waves and reaction–diffusion equations. Grundlehren der mathematischen Wissenschaften, vol. 258. Springer, New York

    Google Scholar 

  59. Thomée V. (1997). Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, ON, N1G 2W1, Canada

    Marcus R. Garvie

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Catalin Trenchea

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  1. Marcus R. Garvie
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  2. Catalin Trenchea
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Correspondence to Marcus R. Garvie.

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Garvie, M.R., Trenchea, C. Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response. Numer. Math. 107, 641–667 (2007). https://doi.org/10.1007/s00211-007-0106-x

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