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

Numerische Mathematik

Volumn 107, Issue 4, 2007, Pages 641-667

  Garvie, Marcus R ^a   Trenchea, Catalin ^b  

^a UNIVERSITY OF GUELPH   (Canada)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34848888912     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-007-0106-x     Document Type: Article

References (59)

1. Adams R.A. and Fournier J. (1977). Cone conditions and properties of Sobolev spaces. J. Math. Anal. Appl. 61: 713-734
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
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
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
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
6. Begon M., Harper J.L. and Townsend C.R. (1990). Ecology: Individuals, Populations and Communities. Blackwell, Boston
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
8. Ciarlet, P.G.: The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1979)
9. n with applications to finite element methods. Arch. Rational. Mech. Anal. 46: 177-199
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
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
12. Ei S.-I., Ikota R. and Mimura M. (1999). Segregating partition problem in competition-diffusion systems. Interfaces Free Bound. 1(1): 57-80
13. Ei S.-I., Mimura M. and Nagayama M. (2002). Pulse-pulse interaction in reaction-diffusion systems. Phys. D 165(3-4): 176-198
14. Elliott C.M. (1987). Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7: 61-71
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
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
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
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
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
22. Gutierrez A.P. (1996). Applied Population Ecology: A Supply-Demand Approach. Wiley, New York
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
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
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
26. Holling C.S. (1959). Some characteristics of simple types of predation and parasitism. Can. Entomol. 91: 385-398
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
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
29. Iida M., Mimura M. and Ninomiya H. (2006). Diffusion, cross-diffusion and competitive interaction. J. Math. Biol. 53(4): 617-641
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
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
32. Ivlev V.S. (1961). Experimental Ecology of the Feeding Fishes. Yale University Press, New Haven
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
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
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
36. May R.M. (1974). Stability and Complexity in Model Ecosystems. Princeton University Press, NJ
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
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
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
40. Mimura M. and Kawasaki K. (1980). Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9(1): 49-64
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
42. Murray J.D. (1993). Mathematical biology Biomathematics Texts, vol. 19. Springer, Berlin
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
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
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
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
48. Petrovskii S.V. and Malchow H. (2002). Critical phenomena in plankton communities: KISS model revisited. Nonlinear Anal. Real. 1: 37-51
49. Rai V. and Jayaraman G. (2003). Is diffusion-induced chaos robust?. Curr. Sci. India 84(7): 925-929
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
51. Rodrigues J. (1987). Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam
52. Rosenzweig M.L. and MacArthur R.H. (1963). Graphical representation and stability conditions for predator-prey interaction. Am. Nat. 97: 209-223
53. Savill N.J. and Hogeweg P. (1999). Competition and dispersal in predator-prey waves. Theor. Popul. Biol. 56: 243-263
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
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
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
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
58. Smoller J. (1983). Shock waves and reaction-diffusion equations. Grundlehren der mathematischen Wissenschaften, vol. 258. Springer, New York
59. Thomée V. (1997). Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, vol. 25. Springer, Berlin