A structured-population model of Proteus mirabilis swarm-colony development

Journal of Mathematical Biology

Volumn 52, Issue 1, 2006, Pages 93-114

  Ayati, Bruce P ^a  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

Age structure; Partial differential equations; Proteus mirabilis; Space structure; Swarm colony

Indexed keywords

AGAR;

ARTICLE; BIOLOGICAL MODEL; CYTOLOGY; FLAGELLUM; GROWTH, DEVELOPMENT AND AGING; MATHEMATICS; MOVEMENT (PHYSIOLOGY); PHYSIOLOGY; PROTEUS MIRABILIS;

AGAR; FLAGELLA; MATHEMATICS; MODELS, BIOLOGICAL; MOVEMENT; PROTEUS MIRABILIS;

PROTEUS MIRABILIS;

EID: 29244435800     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-005-0345-3     Document Type: Article

References (49)

1. Aronson, D.G.: The role of diffusion in mathematical population biology: Skellam revisited. In: Mathematics in Biology and Medicine, V. Capasso, E. Grosso, S.L. Paveri-Fontana, (eds.), vol. 57 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1985, pp. 2-6
2. Ayati, B.P.: A variable time step method for an age-dependent population model with nonlinear diffusion. SIAM J. Numer. Anal. 37, 1571-1589 (2000)
3. Ayati, B.P., Dupont, T.F.: Galerkin methods in age and space for a population model with nonlinear diffusion. SIAM J. Numer. Anal. 40, 1064-1076 (2002)
4. Ayati, B.P., Dupont, T.F.: Convergence of a step-doubling Galerkin method for parabolic problems. Math. Comp. 74, 1053-1065 (2005) Article electronically published on September 10, 2004
5. Bellomo, N., Preziosi, L.: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Model. 32, 413-452 (2000)
6. Bertuzzi, A., Gandolfi, A.: Cell kinetics in a tumor cord. J. Theor. Biol. 204, 587-599 (2000)
7. Bolker, B.M., Pacala, S.W., Neuhauser, C.: Spatial dynamics in model plant communities: What do we really know? American Naturalist 162, 135-148 (2003)
8. Busenberg, S., Iannelli, M.: A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlin. Anal. Th. Meth. Appl. 7, 501-529 (1983)
9. Chiu, C.: A numerical method for nonlinear age dependent population models. Diff. Int. Eqns. 3, 767-782 (1990)
10. de Roos, A.M.: Numerical methods for structured population models: The escalator boxcar train. Num. Meth. Part. Diff. Eqns. 4, 173-195 (1989)
11. de Roos, A.M.: A gentle introduction to physiologically structured population models. In: Structured-population Models in Marine, Terrestrial, and Freshwater Systems, S. Tuljapurkar, H. Caswell, (eds.), vol. 18 of Population and Community Biology Series, Chapman & Hall, New York, 1997, ch. 5, pp. 119-204
12. de Roos, A.M., Metz, J.A.J.: Towards a numerical analysis of the escalator boxcar train. In: Differential Equations with Applications in Biology, Physics and Engineering, J.A. Goldstein, F. Kappel, W. Schappacher, (eds.), vol. 133 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1991, pp. 91-113
13. di Blasio, G.: Non-linear age-dependent population diffusion. J. Math. Bio. 8, 265-284 (1979)
14. di Blasio, G., Lamberti, L.: An initial-boundary problem for age-dependent population diffusion. SIAM J. Appl. Math. 35, 593-615 (1978)
15. Dyson, J., Villella-Bressan, R., Webb, G.F.: Asynchronous exponential growth in an age structured population of proliferating and quiescent cells. Math. Biosci. 177-178, 73-83 (2002)
16. Esipov, S.E., Shapiro, J.A.: Kinetic model of Proteus mirabilis swarm colony development. J. Math. Biol. 36, 249-268 (1998)
17. Fairweather, G., López-Marcos, J.C.: A box method for a nonlinear equation of population dynamics. IMA J. Numer. Anal. 11, 525-538 (1991)
18. Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, New Jersey, 1971
19. Gurtin, M.E.: A system of equations for age-dependent population diffusion. J. Theor. Biol. 40, 389-392 (1973)
20. Gurtin, M.E., MacCamy, R.C.: Non-linear age-dependent population dynamics. Arch. Rat. Mech. Anal. 54, 281-300 (1974)
21. Gurtin, M.E., MacCamy, R.C.: Diffusion models for age-structured populations. Math. Biosci. 54, 49-59 (1981)
22. Höfer, T., Sherratt, J.A., Maini, P.K.: Cellular pattern formation during Dictostelium aggregation. Physica D 85, 425-444 (1995)
23. n. Quart. Appl. Math. 52, 377-398 (1994)
24. Ionides, E.L., Fang, K.S., Isseroff, R.R., Oster, G.F.: Stochastic models for cell motion and taxis. J. Math. Biol. 48, 23-37 (2004)
25. Kim, M.-Y.: Galerkin methods for a model of population dynamics with nonlinear diffusion. Num. Meth. Part. Diff. Eqns. 12, 59-73 (1996)
26. Kim, M.-Y., Park, E.-J.: Mixed approximation of a population diffusion equation. Computers Math. Applic. 30, 23-33 (1995)
27. Kohyama, T.: Size-structured multi-species model of rain forest trees. Functional Ecology 6, 206-212 (1992)
28. Kohyama, T.: Size-structured tree populations in gap-dynamic forest - the forest architecture hypothesis for the stable coexistence of species. J. Ecology 81, 131-143 (1993)
29. Kooi, B.W. Kooijman, S.A.L.M.: Discrete event versus continuous approach to reproduction in structured population dynamics. Theor. Popul. Biol. 56, 91-105 (1999)
30. Kraemer, M.A., Kalachev, L.V., Coble, D.W.: A class of models describing age structure dynamics in a natural forest. Natural Resource Modeling 15, 149-200 (2002)
31. Kubo, K., Langlais, M.: Periodic solutions for a population dynamics problem with age-dependence and spatial structure. J. Math. Biol. 29, 363-378 (1991)
32. Kwon, Y., Cho, C.-K.: Second-order accurate difference methods for a one-sex model of population dynamics. SIAM J. Numer. Anal. 30, 1385-1399 (1993)
33. Langlais, M.: A nonlinear problem in age-dependent population diffusion. SIAM J. Math. Anal. 16, 510-529 (1985)
34. Langlais, M.: Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion. J. Math. Biol. 26, 319-346 (1988)
35. Lopez, L., Trigiante, D.: A finite difference scheme for a stiff problem arising in the numerical solution of a population dynamic model with spatial diffusion. Nonlin. Anal. Th. Meth. Appl. 9, 1-12 (1985)
36. López-Marcos, J.C.: An upwind scheme for a nonlinear hyperbolic integro-differential equation with integral boundary condition. Computers Math. Applic. 22, 15-28 (1991)
37. MacCamy, R.C.: A population model with nonlinear diffusion. J. Diff. Eqns. 39, 52-72 (1981)
38. Marcati, P.: Asymptotic behavior in age-dependent population dynamics with hereditary renewal law. SIAM J. Math. Anal. 12, 904-916 (1981)
39. Medvedev, G.S., Kaper, T.J., Kopell, N.: A reaction-diffusion equation with periodic front dynamics. SIAM J. Appl. Math. 60, 1601-1638 (2000)
40. Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, vol. 18 of Interdisciplinary Applied Mathematics, Springer-Verlag, Berlin, third ed., 2003
41. Nagylaki, T.: The diffusion model for migration and selection. In: Some Mathematical Questions in Biology: Models in Population Biology, A. Hastings, (ed.), vol. 20 of Lectures on Mathematics in the Life Sciences, American Mathematical Society, Providence, R.I., 1989, pp. 55-75
42. Pascual, M., Levin, S.A.: Spatial scaling in a benthic population model with density-dependent disturbance. Theor. Popul. Biol. 56, 106-122 (1999)
43. Preziosi, L. ed.: Cancer Modelling and Simulation. Chapman & Hall/CRC, 2003
44. Rauprich, O., Matsushita, M., Weijer, K., Siegert, F., Esipov, S.E., Shapiro, J.A.: Periodic phenomena in Proteus mirabilis swarm colony development. J. Bacteriol. 178, 6525-6538 (1996)
45. Rotenberg, M.: Theory of population transport. J. Theor. Biol. 37, 291-305 (1972)
46. Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196-218 (1951)
47. Sulsky, D.: Numerical solution of structured population models, I. age structure. J. Math. Biol. 31, 817-839 (1993)
48. Webb, G.F.: Theory of Nonlinear Age-dependent Population Dynamics. vol. 89 of Pure and Applied Mathematics, Marcel Dekker, New York, 1985
49. Williams, F.D., Schwarzhoff, R.H.: Nature of the swarming phenomenon in Proteus. Ann. Rev. Microbiol. 32, 101-122 (1978)