Stochasticity, invasions, and branching random walks

Theoretical Population Biology

Volumn 66, Issue 3, 2004, Pages 175-184

  Kot, Mark ^a   Medlock, Jan ^a   Reluga, Timothy ^a   Brian Walton, D ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Biological invasions; Branching random walks; Dispersal; Fat tailed kernels; Integrodifference equations

Indexed keywords

BIOLOGICAL INVASION; STOCHASTICITY; THEORETICAL STUDY;

ARTICLE; PROBABILITY; SPECIES DIFFERENCE; STATISTICS;

PROBABILITY; SPECIES SPECIFICITY; STOCHASTIC PROCESSES;

EID: 4644298001     PISSN: 00405809     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tpb.2004.05.005     Document Type: Article

References (70)

1. E.J. Allen, L.J.S. Allen, and X. Gilliam Dispersal and competition models for plants J. Math. Biol. 34 1996 455 481
2. J.C. Allen, C.C. Brewster, and D.H. Slone Spatially explicit ecological models a spatial convolution approach Chaos, Solitons and Fractals 12 2001 333 347
3. L.J.S. Allen An Introduction to Stochastic Processes with Applications to Biology 2003 Prentice-Hall Inc. Upper Saddle River, NJ, USA
4. L.J.S. Allen, and R.K. Ernest The impact of long-range dispersal on the rate of spread in population and epidemic models C. Castillo-Chavez S. Blower P. van den Driessche D. Kirschner A.-A. Yakubu Mathematical Approaches for Emerging and Reemerging Infectious Diseases An Introduction 2002 Springer NY, USA 183 197
5. L.J.S. Allen, E.J. Allen, and S. Ponweera A mathematical model for weed dispersal and control Bull. Math. Biol. 58 1996 815 834
6. M. Andersen Properties of some density-dependent integrodifference equation population models Math. Biosci. 104 1991 135 157
7. K.B. Athreya, and P.E. Ney Branching Processes 1972 Springer Berlin, Germany
8. A.J. Bateman Is gene dispersion normal? Heredity 4 1950 353 363
9. L. Berec Techniques of spatially explicit individual-based models construction, simulation, and mean-field analysis Ecol. Model. 150 2002 55 81
10. R.J.H. Beverton, and S.J. Holt On the dynamics of exploited fish populations Fish. Invest. Ser. II 19 1957 1 533
11. J.D. Biggins Chernoff's theorem in the branching random walk J. Appl. Probab. 14 1977 630 636
12. J.D. Biggins How fast does a general branching random walk spread? K.B. Athreya P. Jagers Classical and Modern Branching Processes 1996 Springer NY, USA 19 40
13. C.C. Brewster, and J.C. Allen Spatiotemporal model for studying insect dynamics in large-scale cropping systems Environ. Entomol. 26 1997 473 482
14. H. Caswell Matrix Population Models Construction, Analysis, and Interpretation 2001 Sinauer Associates, Inc. Sunderland, Massachusetts, USA
15. H. Caswell, M. Fujiwara, and S. Brault Declining survival probability threatens the North Atlantic right whale Proceedings of the National Academy of Sciences of the United States of America vol. 96 1999 3308 3313
16. H. Caswell, R. Lensink, and M.G. Neubert Demography and dispersal life table response experiments for invasion speed Ecology 84 2003 1968 1978
17. C.L. Chiang An Introduction to Stochastic Processes and Their Applications 1980 Robert E. Krieger Publishing Company Huntington, NY, USA
18. J.S. Clark Why trees migrate so fast confronting theory with dispersal biology and the paleorecord Am. Natur. 152 1998 204 224
19. J.S. Clark, M. Lewis, and L. Horvath Invasion by extremes population spread with variation in dispersal and reproduction Am. Natur. 157 2001 537 554
20. D.J. Daley, and D. Vere-Jones An Introduction to the Theory of Point Processes. Volume 1 Elementary Theory and Methods 2003 Springer NY, USA
21. F.M. Dekking Deterministic growth-dispersal models and branching random walk J. Math. Anal. Appl. 192 1995 529 538
22. U. Dieckmann, R. Law, and J.A.J. Metz The Geometry of Ecological Interactions Simplifying Spatial Complexity 2000 Cambridge University Press Cambridge, UK
23. V. Grimm Ten years of individual-based modelling in ecology what have we learned and what could we learn in the future? Ecol. Model. 115 1999 129 148
24. G.R. Grimmett, and D.R. Stirzaker Probability and Random Processes 1992 Oxford University Press Oxford, UK
25. T.E. Harris The Theory of Branching Processes 1963 Springer Berlin, Germany
26. D.R. Hart, and R.H. Gardner A spatial model for the spread of invading organisms subject to competition J. Math. Biol. 35 1997 935 948
27. A. Hastings, and K. Higgins Persistence of transients in spatially structured ecological models Science 263 1994 1133 1136
28. P. Kareiva Population dynamics in spatially complex environments theory and data Philos. Trans. R. Soc. London B 330 1990 175 190
29. M. Keeling Spatial models of interacting populations J. McGlade Advanced Ecological Theory Principles and Applications 1999 Blackwell Science Oxford, UK 64 99
30. H. Kierstead, and L.B. Slobodkin The size of water masses containing plankton blooms J. Mar. Res. 12 1953 141 147
31. M. Kimmel, and D.E. Axelrod Branching Processes in Biology 2002 Springer NY, USA
32. M. Kot Diffusion-driven period-doubling bifurcations BioSystems 22 1989 279 287
33. M. Kot Discrete-time travelling waves ecological examples J. Math. Biol. 30 1992 413 436
34. M. Kot Do invading organsims do the wave? Canad. Appl. Math. Quarterly 10 2003 139 170
35. M. Kot, and W.M. Schaffer Discrete-time growth-dispersal models Math. Biosci. 80 1986 109 136
36. M. Kot, M.A. Lewis, and P. van den Driessche Dispersal data and the spread of invading organisms Ecology 77 1996 2027 2042
37. J. Latore, P. Gould, and A.M. Mortimer Spatial dynamics and critical patch size of annual plant populations J. Theoret. Biol. 190 1998 277 285
38. J. Latore, P. Gould, and A.M. Mortimer Effects of habitat heterogeneity and dispersal strategies on population persistence in annual plants Ecol. Model. 123 1999 127 139
39. B.C. Legaspi, J.C. Allen, C.C. Brewster, J.A. Morales-Ramos, and E.G. King Areawide management of the cotton boll weevil use of a spatio-temporal model in augmentative biological control Ecol. Model. 110 1998 151 164
40. M.A. Lewis Variability, patchiness, and jump dispersal in the spread of an invading population D. Tilman P. Kareiva Spatial Ecology The Role of Space in Population Dynamics and Interspecific Interactions 1997 Princeton University Press Princeton, NJ, USA 46 69
41. M.A. Lewis Spread rate for a nonlinear stochastic invasion J. Math. Biol. 41 2000 430 454
42. M.A. Lewis, and S. Pacala Modeling and analysis of stochastic invasion processes J. Math. Biol. 41 2000 387 429
43. D.R. Lockwood, A. Hastings, and L.W. Botsford The effects of dispersal patterns on marine reserves does the tail wag the dog? Theoret. Popul. Biol. 61 2002 297 309
44. R. Lui A nonlinear integral operator arising from a model in population genetics, II. Initial data with compact support SIAM J. Math. Anal. 13 1982 938 953
45. R. Lui Existence and stability of travelling wave solutions of a nonlinear integral operator J. Math. Biol. 16 1983 199 220
46. R.M. May On relationships among various types of population models Am. Natur. 107 1972 46 57
47. R.M. May, and G.F. Oster Bifurcations and dynamic complexity in simple ecological models Am. Natur. 110 1976 573 599
48. J. Maynard Smith Mathematical Ideas in Biology 1968 Cambridge University Press NY, USA
49. H.P. McKean Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piscunov Commun. Pure Appl. Math. 28 1975 323 331
50. D. Mollison The rate of spatial propagation of simple epidemics Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability vol. 3 1972 579 614
51. D. Mollison Spatial contact models for ecological and epidemic spread J. Roy. Statist. Soc. B 39 1977 283 326
52. D. Mollison Dependence of epidemic and population velocities on basic parameters Math. Biosci. 107 1991 255 287
53. D. Mollison, and H. Daniels The "deterministic simple epidemic" unmasked Math. Biosci. 117 1993 147 153
54. M.G. Neubert, and H. Caswell Dispersal and demography calculation and sensitivity analysis of invasion speed for structured populations Ecology 81 2000 1613 1628
55. M.G. Neubert, M. Kot, and M.A. Lewis Dispersal and pattern formation in a discrete-time predator-prey model Theoret. Popul. Biol. 48 1995 7 43
56. M.G. Neubert, M. Kot, and M.A. Lewis Invasion speeds in fluctuating environments Proc. R. Soc. London B 267 2000 1603 1610 (Errata: 267, 2568-2569)
57. A. Okubo, and S.A. Levin Diffusion and Ecological Problems Modern Perspectives 2001 Springer NY, USA
58. E.C. Pielou Mathematical Ecology 1977 Wiley NY, USA
59. J. Radcliffe, and L. Rass Discrete time spatial models arising in genetics, evolutionary game theory, and branching processes Math. Biosci. 140 1997 101 129
60. W.E. Ricker Stock and recruitment J. Fish. Res. Board Can. 11 1954 559 623
61. J.A. Sherratt, B.T. Eagan, and M.A. Lewis Oscillations and chaos behind predator-prey invasion mathematical artifact or ecological reality? Proc. R. Soc. London B 352 1997 21 38
62. N. Shigesada, and K. Kawasaki Invasion and range expansion of species effects of long-distance dispersal J.M. Bullock R.E. Kenward R.S. Hails Dispersal Ecology 2002 Blackwell Science, Malden MA, USA 350 373
63. J.G. Skellam Random dispersal in theoretical populations Biometrika 38 1951 196 218
64. R.E. Snyder How demographic stochasticity can slow biological invasions Ecology 84 2003 1333 1339
65. R.W. Van Kirk, and M.A. Lewis Integrodifference models for persistence in fragmented habitats Bull. Math. Biol. 59 1997 107 137
66. V.A. Vatutin, and A.M. Zubkov Branching Processes. II J. Soviet Math. 67 1993 3407 3485
67. R.R. Veit, and M.A. Lewis Dispersal, population growth, and the Allee effect dynamics of the House Finch invasion of Eastern North America Am. Natur. 148 1996 255 274
68. M.H. Wang, M. Kot, and M.G. Neubert Integrodifference equations, Allee effects, and invasions J. Math. Biol. 44 2002 150 168
69. H.F. Weinberger Asymptotic behavior of a model of population genetics Lecture Notes in Mathematics vol. 648 1978 47 96
70. H.F. Weinberger Long-time behavior of a class of biological models SIAM J. Math. Anal. 13 1982 353 396