메뉴 건너뛰기




Volumn 69, Issue 3, 2007, Pages 931-956

Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in Matlab

Author keywords

Finite difference method; Matlab; Predator prey interaction; Reaction diffusion system

Indexed keywords

ALGORITHM; ANIMAL; ARTICLE; BIOLOGICAL MODEL; ECOSYSTEM; KINETICS; MATHEMATICAL COMPUTING; PREDATION;

EID: 33947514402     PISSN: 00928240     EISSN: 15229602     Source Type: Journal    
DOI: 10.1007/s11538-006-9062-3     Document Type: Article
Times cited : (260)

References (55)
  • 1
    • 0036141801 scopus 로고    scopus 로고
    • Mutual interference between predators can give rise to Turing spatial patterns
    • Alonso, D., Bartumeus, F., Catalan, J., 2002. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83(1), 28-34.
    • (2002) Ecology , vol.83 , Issue.1 , pp. 28-34
    • Alonso, D.1    Bartumeus, F.2    Catalan, J.3
  • 2
    • 0001737039 scopus 로고
    • Implicit-explicit methods for time-dependent partial differential equations
    • Ascher, U., Ruuth, S., Wetton, B., 1995. Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797-823.
    • (1995) SIAM J. Numer. Anal , vol.32 , Issue.3 , pp. 797-823
    • Ascher, U.1    Ruuth, S.2    Wetton, B.3
  • 3
    • 0035361441 scopus 로고    scopus 로고
    • On a uniformly accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution
    • 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.
    • (2001) J. Comput. Appl. Math , vol.131 , pp. 381-405
    • Beckett, G.1    Mackenzie, J.2
  • 4
    • 0003210742 scopus 로고
    • The Mathematical Theory of Finite Element Methods
    • Springer, New York
    • Brenner, S., Scott, L., 1994. The Mathematical Theory of Finite Element Methods. Vol. 15: Texts in Applied Mathematics. Springer, New York.
    • (1994) Texts in Applied Mathematics , vol.15
    • Brenner, S.1    Scott, L.2
  • 5
    • 0040504616 scopus 로고
    • The Finite Element Method for Elliptic Problems
    • North-Holland, Amsterdam
    • Ciarlet, P., 1979. The Finite Element Method for Elliptic Problems. Vol. 4: Studies in Mathematics and its Applications. North-Holland, Amsterdam.
    • (1979) Studies in Mathematics and its Applications , vol.4
    • Ciarlet, P.1
  • 6
    • 0027811215 scopus 로고
    • The global dynamics of discrete semilinear parabolic equations
    • Elliott, C., Stuart, A., 1993. The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30(6), 1622-1663.
    • (1993) SIAM J. Numer. Anal , vol.30 , Issue.6 , pp. 1622-1663
    • Elliott, C.1    Stuart, A.2
  • 7
  • 8
    • 58149255804 scopus 로고    scopus 로고
    • Analysis of two generic spatially extended predator-prey models
    • submitted for publication
    • Garvie, M., Trenchea, C., 2005a. Analysis of two generic spatially extended predator-prey models. Nonlinear Anal. Real World Appl., submitted for publication.
    • (2005) Nonlinear Anal. Real World Appl
    • Garvie, M.1    Trenchea, C.2
  • 9
    • 33947538669 scopus 로고    scopus 로고
    • Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response
    • 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.
    • (2005) Numer. Math
    • Garvie, M.1    Trenchea, C.2
  • 10
    • 0348077446 scopus 로고    scopus 로고
    • Functional responses for zooplankton feeding on multiple resources: A review of assumptions and biological dynamics
    • 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.
    • (2003) Deep Sea Res. II , vol.50 , pp. 2847-2875
    • Gentleman, W.1    Leising, A.2    Frost, B.3    Strom, S.4    Murray, J.5
  • 11
    • 0031662774 scopus 로고    scopus 로고
    • Circles and spirals: Population persistence in a spatially explicit predator-prey model
    • 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.
    • (1998) Ecology , vol.79 , Issue.7 , pp. 2516-2530
    • Gurney, W.1    Veitch, A.2    Cruickshank, I.3    McGeachin, G.4
  • 13
    • 0000185875 scopus 로고
    • Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations
    • 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.
    • (1978) SIAM J. Numer. Anal , vol.15 , Issue.6 , pp. 1161-1177
    • Hoff, D.1
  • 14
    • 84880026161 scopus 로고
    • Some characteristics of simple types of predation and parasitism
    • Holling, C., 1959. Some characteristics of simple types of predation and parasitism. Can. Entomol. 91, 385-398.
    • (1959) Can. Entomol , vol.91 , pp. 385-398
    • Holling, C.1
  • 15
    • 0002366911 scopus 로고
    • The functional response of predators to prey density and its role in mimicry and population regulation
    • 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.
    • (1965) Mem. Entomol. Soc. Can , vol.45 , pp. 1-60
    • Holling, C.1
  • 16
    • 0028176208 scopus 로고
    • Partial differential equations in ecology: Spatial interactions and population dynamics
    • Holmes, E., Lewis, M., Banks, J., Veit, R., 1994. Partial differential equations in ecology: Spatial interactions and population dynamics. Ecology 75(1), 17-29.
    • (1994) Ecology , vol.75 , Issue.1 , pp. 17-29
    • Holmes, E.1    Lewis, M.2    Banks, J.3    Veit, R.4
  • 19
    • 0021553164 scopus 로고
    • Fully discrete stability and invariant rectangular regions for reaction-diffusion systems
    • Jerome, J., 1984. Fully discrete stability and invariant rectangular regions for reaction-diffusion systems. SIAM J. Numer. Anal. 21(6), 1054-1065.
    • (1984) SIAM J. Numer. Anal , vol.21 , Issue.6 , pp. 1054-1065
    • Jerome, J.1
  • 20
    • 0036162215 scopus 로고    scopus 로고
    • Predator functional responses: Discriminating between handling and digesting prey
    • Jeschke, J., Kopp, M., Tollrian, R., 2002. Predator functional responses: Discriminating between handling and digesting prey. Ecol. Monogr. 72(1), 95-112.
    • (2002) Ecol. Monogr , vol.72 , Issue.1 , pp. 95-112
    • Jeschke, J.1    Kopp, M.2    Tollrian, R.3
  • 21
    • 0039291305 scopus 로고
    • Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction-diffusion equations
    • 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.
    • (1994) J. Aust. Math. Soc. Ser. B , vol.36 , Issue.2 , pp. 234-241
    • Li, N.1    Steiner, J.2    Tang, S.-M.3
  • 22
    • 0037119527 scopus 로고    scopus 로고
    • Dynamical stabilization of an unstable equilibrium in chemical and biological systems
    • Malchow, H., Petrovskii, S., 2002. Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Math. Comput. Model. 36, 307-319.
    • (2002) Math. Comput. Model , vol.36 , pp. 307-319
    • Malchow, H.1    Petrovskii, S.2
  • 24
  • 25
    • 0037284570 scopus 로고    scopus 로고
    • A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion
    • Mickens, R., 2003. A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion. Comput. Math. Appl. 45, 429-436.
    • (2003) Comput. Math. Appl , vol.45 , pp. 429-436
    • Mickens, R.1
  • 27
    • 0003363592 scopus 로고
    • Mathematical Biology
    • Springer, Berlin
    • Murray, J., 1993. Mathematical Biology. Vol. 19: Biomathematics Texts. Springer, Berlin.
    • (1993) Biomathematics Texts , vol.19
    • Murray, J.1
  • 28
    • 0036161027 scopus 로고    scopus 로고
    • Transient dynamics and pattern formation: Reactivity is necessary for Turing instabilities
    • Neubert, M., Caswell, H., Murray, J., 2002. Transient dynamics and pattern formation: Reactivity is necessary for Turing instabilities. Math. Biosci. 175, 1-11.
    • (2002) Math. Biosci , vol.175 , pp. 1-11
    • Neubert, M.1    Caswell, H.2    Murray, J.3
  • 29
    • 0032391453 scopus 로고    scopus 로고
    • Accelerated monotone iterative methods for finite difference equations of reaction-diffusion
    • Pao, C., 1998. Accelerated monotone iterative methods for finite difference equations of reaction-diffusion. Numer. Math. 79, 261-281.
    • (1998) Numer. Math , vol.79 , pp. 261-281
    • Pao, C.1
  • 30
    • 0001326901 scopus 로고    scopus 로고
    • Numerical analysis of coupled systems of nonlinear parabolic equations
    • Pao, C., 1999. Numerical analysis of coupled systems of nonlinear parabolic equations. SIAM J. Numer. Anal. 36(2), 393-416.
    • (1999) SIAM J. Numer. Anal , vol.36 , Issue.2 , pp. 393-416
    • Pao, C.1
  • 31
    • 0037104634 scopus 로고    scopus 로고
    • Finite difference reaction-diffusion systems with coupled boundary conditions and time delays
    • Pao, C., 2002. Finite difference reaction-diffusion systems with coupled boundary conditions and time delays. J. Math. Anal. 272, 407-434.
    • (2002) J. Math. Anal , vol.272 , pp. 407-434
    • Pao, C.1
  • 32
    • 0027806283 scopus 로고
    • Diffusion-induced chaos in a spatial predator-prey system
    • Pascual, M., 1993. Diffusion-induced chaos in a spatial predator-prey system. Proc. R. Soc. Lond. Ser. B 251, 1-7.
    • (1993) Proc. R. Soc. Lond. Ser. B , vol.251 , pp. 1-7
    • Pascual, M.1
  • 33
    • 0033120656 scopus 로고    scopus 로고
    • A minimal model of pattern formation in a prey-predator system
    • Petrovskii, S., Malchow, H., 1999. A minimal model of pattern formation in a prey-predator system. Math. Comput. Model. 29, 49-63.
    • (1999) Math. Comput. Model , vol.29 , pp. 49-63
    • Petrovskii, S.1    Malchow, H.2
  • 34
    • 0035718414 scopus 로고    scopus 로고
    • Wave of chaos: New mechanism of pattern formation in spatiotemporal population dynamics
    • Petrovskii, S., Malchow, H., 2001. Wave of chaos: New mechanism of pattern formation in spatiotemporal population dynamics. Theor. Populat. Biol. 59, 157-174.
    • (2001) Theor. Populat. Biol , vol.59 , pp. 157-174
    • Petrovskii, S.1    Malchow, H.2
  • 36
    • 0037608679 scopus 로고    scopus 로고
    • 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.
    • 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.
  • 37
    • 1642382839 scopus 로고    scopus 로고
    • Is diffusion-induced chaos robust?
    • Rai, V., Jayaraman, G., 2003. Is diffusion-induced chaos robust? Curr. Sci. India 84(7), 925-929.
    • (2003) Curr. Sci. India , vol.84 , Issue.7 , pp. 925-929
    • Rai, V.1    Jayaraman, G.2
  • 39
    • 0001524523 scopus 로고
    • Graphical representation and stability conditions for predator-prey interaction
    • Rosenzweig, M., MacArthur, R., 1963. Graphical representation and stability conditions for predator-prey interaction. Am. Nat. 97, 209-223.
    • (1963) Am. Nat , vol.97 , pp. 209-223
    • Rosenzweig, M.1    MacArthur, R.2
  • 40
    • 0001248878 scopus 로고
    • Implicit-explicit methods for reaction-diffusion problems in pattern formation
    • Ruuth, J., 1995. Implicit-explicit methods for reaction-diffusion problems in pattern formation. J. Math. Biol. 34, 148-176.
    • (1995) J. Math. Biol , vol.34 , pp. 148-176
    • Ruuth, J.1
  • 42
    • 0000048673 scopus 로고
    • GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems
    • Saad, Y., Schultz, M., 1986. GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856-869.
    • (1986) SIAM J. Sci. Stat. Comput , vol.7 , Issue.3 , pp. 856-869
    • Saad, Y.1    Schultz, M.2
  • 43
    • 0033383624 scopus 로고    scopus 로고
    • Competition and dispersal in predator-prey waves
    • Savill, N., Hogeweg, P., 1999. Competition and dispersal in predator-prey waves. Theor. Populat. Biol. 56, 243-263.
    • (1999) Theor. Populat. Biol , vol.56 , pp. 243-263
    • Savill, N.1    Hogeweg, P.2
  • 44
    • 0015442349 scopus 로고
    • Dissipative structure: An explanation and an ecological example
    • Segel, L., Jackson, J., 1972. Dissipative structure: An explanation and an ecological example. J. Theor. Biol. 37, 545-559.
    • (1972) J. Theor. Biol , vol.37 , pp. 545-559
    • Segel, L.1    Jackson, J.2
  • 45
    • 0035127918 scopus 로고    scopus 로고
    • Periodic travelling waves in cyclic predator-prey systems
    • Sherratt, J., 2001. Periodic travelling waves in cyclic predator-prey systems. Ecol. Lett. 4, 30-37.
    • (2001) Ecol. Lett , vol.4 , pp. 30-37
    • Sherratt, J.1
  • 46
    • 0030814768 scopus 로고    scopus 로고
    • Oscillations and chaos behind predator-prey invasion: Mathematical artifact or ecological reality?
    • 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.
    • (1997) Phil. Trans. R. Soc. Lond. B , vol.352 , pp. 21-38
    • Sherratt, J.1    Eagan, B.2    Lewis, M.3
  • 47
    • 53149140283 scopus 로고    scopus 로고
    • Generation of periodic waves by landscape features in cyclic predator-prey systems
    • 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.
    • (2002) Proc. R. Soc. Lond. Ser. B , vol.269 , pp. 327-334
    • Sherratt, J.1    Lambin, X.2    Thomas, C.3    Sherratt, T.4
  • 49
    • 0035658752 scopus 로고    scopus 로고
    • Functional responses with predator interference: Viable alternatives to the Holling type II model
    • Skalski, G., Gilliam, J.F., 2001. Functional responses with predator interference: Viable alternatives to the Holling type II model. Ecology 82(11), 3083-3092.
    • (2001) Ecology , vol.82 , Issue.11 , pp. 3083-3092
    • Skalski, G.1    Gilliam, J.F.2
  • 50
    • 0003233281 scopus 로고
    • Shock Waves and Reaction-Diffusion Equations
    • Wissenschaften. Springer-Verlag, New York
    • Smoller, J., 1983. Shock Waves and Reaction-Diffusion Equations. Vol. 258: Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York.
    • (1983) Grundlehren der mathematischen , vol.258
    • Smoller, J.1
  • 51
    • 0024682304 scopus 로고
    • Nonlinear instability in dissipative finite difference schemes
    • Stuart, A., 1989. Nonlinear instability in dissipative finite difference schemes. SIAM Rev. 31(2), 191-220.
    • (1989) SIAM Rev , vol.31 , Issue.2 , pp. 191-220
    • Stuart, A.1
  • 53
    • 0002011401 scopus 로고
    • The chemical basis of morphogenesis
    • Turing, A., 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37-72.
    • (1952) Phil. Trans. R. Soc. Lond. B , vol.237 , pp. 37-72
    • Turing, A.1
  • 54
    • 0344375578 scopus 로고
    • Global asymptotic behavior of iterative implicit schemes
    • Yee, H., Sweby, P., 1994. Global asymptotic behavior of iterative implicit schemes. Int. J. Bifurcat. Chaos 4(6), 1579-1611.
    • (1994) Int. J. Bifurcat. Chaos , vol.4 , Issue.6 , pp. 1579-1611
    • Yee, H.1    Sweby, P.2
  • 55
    • 84954811121 scopus 로고
    • Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations II. Global asymptotic behaviour of time discretizations
    • 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.
    • (1995) Comp. Fluid Dyn , vol.4 , pp. 219-283
    • Yee, H.1    Sweby, P.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.