Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T14:56:30.211Z Has data issue: false hasContentIssue false
  • English
  • Français

A survey of stepping-stone models in population dynamics

Published online by Cambridge University Press:  01 July 2016

Eric Renshaw
Eric Renshaw*
Affiliation:
University of Edinburgh
*
∗Postal address: Department of Statistics, University of Edinburgh, King&s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

A survey is presented of stochastic and deterministic developments in the study of the effects of nearest-neighbour ‘migration’ between spatially separated ‘colonies’. Such processes are of general applicability, and are relevant to any vector process X(t) = (X1(t), · ··, XN(t)) in which the arrival, departure and transfer rates for the states {X(t) = n} may be written in the form α i(ni), βi(ni) and γ ij(ni, nj), respectively, where n = (n1, · ··, nN). Whilst the main body of results are described in terms of birth-death, genetic and epidemic situations, the final section examines within colony interaction in the context of spatial predator-prey processes.

Keywords

BIRTH, DEATH, MIGRATIONDIFFUSIONSPATIAL PROCESSESEPIDEMICSGENETICSPREDATOR-PREYCONTACT DISTRIBUTIONSSADDLEPOINT APPROXIMATIONSVELOCITIES OF PROPAGATION
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 581 - 627
Copyright
Copyright © Applied Probability Trust 1986 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions (Dover Edition). AMS 55, U.S. Department of Commerce.Google Scholar
2. Adke, S. R. (1964) The generalized birth and death process and Gaussian diffusion. J. Math. Anal. Appl. 9, 336–340.Google Scholar
3. Adke, S. R. (1964) A stochastic population diffusing on a finite interval. J. Ind. Statist. Assoc. 2, 32–40.Google Scholar
4. Adke, S. R. (1969) A birth, death and migration process. J. Appl. Prob. 6, 687–691.CrossRefGoogle Scholar
5. Adke, S. R. and Moyal, J. E. (1963) A birth, death and diffusion process. J. Math. Anal. Appl. 7, 209–224.Google Scholar
6. Aksland, M. (1975) A birth, death and migration process with immigration. Adv. Appl. Prob. 7, 44–60.Google Scholar
7. Anderson, R. M., (ed.) (1982) The Population Dynamics of Infectious Diseases: Theory and Applications. Chapman and Hall, London.Google Scholar
8. Arley, N and Borchsenius, V. (1945) On the theory of infinite systems of differential equations and their application to the theory of stochastic processes and the perturbation theory of quantum mechanics. Acta Math. 76, 261–322.Google Scholar
9. Armitage, P. (1952) The statistical theory of bacterial populations subject to mutation. J. R. Statist. Soc. B 14, 1–33.Google Scholar
10. Arnason, A. N. (1971) Migration Models for Animal Populations. Ph.D. thesis (unpublished), University of Edinburgh.Google Scholar
11. Arnold, L. (1974) Stochastic Differential Equations: Theory and Applications. Wiley, New York.Google Scholar
12. Bailey, N. T. J. (1967) The simulation of stochastic epidemics in two dimensions. Proc. 5th Berkeley Symp. Math. Statist. Prob. 4, 237–257.Google Scholar
13. Bailey, N. T. J. (1968) Stochastic birth, death and migration processes for spatially distributed populations. Biometrika 55, 189–198.CrossRefGoogle Scholar
14. Bailey, N. T. J. (1975) The Mathematical Theory of Infectious Diseases and its Applications , 2nd edn. Griffin, London.Google Scholar
15. Bartholomew, D. J. (1967) Stochastic Models for Social Processes , 2nd edn. Wiley, London.Google Scholar
16. Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
17. Bartlett, M. S. (1956) Deterministic and stochastic models for recurrent epidemics. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 81–109.Google Scholar
18. Bartlett, M. S. (1957) On theoretical models for competitive and predatory biological systems. Biometrika 44, 27–43.Google Scholar
19. Bartlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
20. Beneš, V. E. (1965) Mathematical Theory of Connecting Networks and Telephone Traffic. Academic Press, London.Google Scholar
21. Bramson, M. (1983) Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44 (285), 1–190.Google Scholar
22. Broadbent, S. R. and Kendall, D. G. (1953) The random walk of Trichostrongylus retortaeformis. Biometrics 9, 460–466.CrossRefGoogle Scholar
23. Chewning, W. C. (1975) Migratory effects in predator-prey models. Math. Biosci. 23, 253–262.Google Scholar
24. Chiang, C. L. (1968) Introduction to Stochastic Processes in Biostatistics. Wiley, New York.Google Scholar
25. Chiang, C. L. (1980) An Introduction to Stochastic Processes and their Applications. Krieger, New York.Google Scholar
26. Cliff, A. D., Haggett, P., Ord, J. K. and Versey, G. R. (1981) Spatial Diffusion: An Historical Geography of Epidemics in an Island Community. Cambridge University Press, Cambridge.Google Scholar
27. Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Methuen, London.Google Scholar
28. Cox, D. R. and Smith, W. L. (1957) On the distribution of Tribolium confusum in a container. Biometrika 44, 328–335.Google Scholar
29. Crump, K. S. (1970) Migratory populations in branching processes. J. Appl. Prob. 7, 565–572.Google Scholar
30. Daniels, H. E. (1975) The deterministic spread of a simple epidemic. In Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett , ed. Gani, J., Applied Probability Trust, Sheffield 373–386.Google Scholar
31. Daniels, H. E. (1977) The advancing wave in a spatial birth process. J. Appl. Prob. 14, 689–701.Google Scholar
32. Davis, A. W. (1965) On the theory of birth, death and diffusion processes. J. Appl. Prob. 2, 293–322.Google Scholar
33. Davis, A. W. (1967) Branching-diffusion processes with no absorbing boundaries. I. J. Math. Anal. Appl. 18, 276–296.Google Scholar
34. Davis, A. W. (1967) Branching–diffusion processes with no absorbing boundaries. II. J. Math. Anal. Appl. 19, 1–25.Google Scholar
35. Davis, A. W. (1970) Some generalizations of Bailey&s birth, death and migration model. Adv. Appl. Prob. 2, 83–109.Google Scholar
36. Dubois, D. M. (1975) A model of patchiness for prey-predator plankton populations. Ecol. Mod. 1, 67–80.CrossRefGoogle Scholar
37. Dubois, D. M. and Monfort, G. (1978) Stochastic simulation of a space-time dependent predator-prey model. Compstat 78, 384–390.Google Scholar
38. Durrett, R. (1979) Maxima of branching random walks versus independent random walks. Stoch. Proc. Appl. 9, 117–136.Google Scholar
39. Durrett, R. (1981) An introduction to infinite particle systems. Stoch. Proc. Appl. 11, 109–150.CrossRefGoogle Scholar
40. Faddy, M. J. (1977) Stochastic compartmental models as approximations to more general stochastic systems with the general stochastic epidemic as an example. Adv. Appl. Prob. 9, 448–461.Google Scholar
41. Faddy, M. J. (1983) The outcome of a general spatial epidemic on the line. J. Appl. Prob. 20, 715–727.Google Scholar
42. Faddy, M. J. and Slorach, I. H. (1980) Bounds on the velocity of spread of infection for a spatially connected epidemic process. J. Appl. Prob. 17, 839–845.Google Scholar
43. Feller, W. (1939) Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in Wahrscheinlichkeitstheoretischer Behandlung. Acta Bioth. 5, 11–40.Google Scholar
44. Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369.Google Scholar
45. Fleming, W. H. and Su, C. (1974) Some one-dimensional migration models in population genetics theory. Theoret. Popn Biol. 5, 431–449.CrossRefGoogle ScholarPubMed
46. Gani, J. (1962) An approximate model for phage reproduction in a bacterium. J. Austral. Math. Soc. 2, 478–483.Google Scholar
47. Gani, J. (1962) The extinction of a bacterial colony by phages: a branching process with deterministic removals. Biometrika 49, 272–276.Google Scholar
48. Gani, J. (1965) Stochastic models for bacteriophage. J. Appl. Prob. 2, 225–268.Google Scholar
49. Gani, J. and Yeo, G. F. (1965) Some birth-death and mutation models for phage reproduction. J. Appl. Prob. 2, 150–161.Google Scholar
50. Gans, P. J. (1960) Open first order stochastic processes. J. Chem. Phys. 33, 691–694.Google Scholar
51. Griffeath, D. (1981) The basic contact process. Stoch. Proc. Appl. 11, 151–185.Google Scholar
52. Griffiths, D. A. (1972) A bivariate birth-death process which approximates to the spread of a disease involving a vector. J. Appl. Prob. 9, 65–75.CrossRefGoogle Scholar
53. Griffiths, D. A. (1973) Multivariate birth-and-death processes as approximations to epidemic processes. J. Appl. Prob. 10, 15–26.Google Scholar
54. Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
55. Helland, I. S. (1975) The condition for extinction with probability one in a birth, death and migration process. Adv. Appl. Prob. 7, 61–65.Google Scholar
56. Holgate, P. (1968) Interaction between migration and breeding studied by means of genetic algebras. J. Appl. Prob. 5, 1–8.Google Scholar
57. Horn, H. S. and Macarthur, R. H. (1972) Competition among fugitive species in a harlequin environment. Ecology 53, 749–752.Google Scholar
58. Huffaker, C. B. (1958) Experimental studies on predation: dispersion factors and predator-prey oscillations. Hilgardia 27, 343–383.Google Scholar
59. Itatsu, S. (1981) Ergodic properties of the equilibrium measure of the stepping-stone model in population genetics. Nagoya Math. J. 83, 37–51.Google Scholar
60. Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
61. Jernigan, R. W. and Tsokos, C. P. (1980) A linear stochastic model for phytoplankton production in a marine ecosystem. Ecol. Mod. 10, 1–12.Google Scholar
62. Karlin, S. and Mcgregor, J. L. (1957) The differential equations of birth and death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
63. Karlin, S. and Mcgregor, J. L. (1957) The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
64. Kelker, D. (1973) A random walk epidemic simulation. J. Amer. Statist. Assoc. 68, 821–823.Google Scholar
65. Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
66. Kendall, D. G. (1948) On the generalized birth and death process. Ann. Math. Statist. 19, 1–15.Google Scholar
67. Kendall, D. G. (1948) On the role of variable generation time in the development of a stochastic birth process. Biometrika 35, 316–330.Google Scholar
68. Kendall, D. G. (1965) Mathematical models of the spread of infection. In Mathematics and Computer Science in Biology and Medicine , HMSO, London, 213–225.Google Scholar
69. Kenyon, E. A. (1982) Predator-prey Processes. M. Phil. Thesis (unpublished), University of Edinburgh.Google Scholar
70. Kimura, M. (1953) ‘Stepping stone’ model of population. Ann. Rept. Nat. Inst. Genetics, Japan 3, 62–63.Google Scholar
71. Kimura, M. and Crow, J. F. (1963) The measurement of effective population number. Evolution 17, 279–288.Google Scholar
72. Kimura, M. and Weiss, G. (1964) The stepping-stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 561–576.CrossRefGoogle ScholarPubMed
73. Kingman, J. F. C. (1969) Markov population processes. J. Appl. Prob. 6, 1–18.Google Scholar
74. Kolmogorov, A. N., Petrovsky, I. G. and Piscounov, N. S. (1937) Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. Bull. Univ. d’État Moscou (sér. Intern.) A 1 (6), 1–25.Google Scholar
75. Krieger, I. M. and Gans, P. J. (1960) First order stochastic processes. J. Chem. Phys. 32, 247–250.Google Scholar
76. Kuulasmaa, K. (1982) The spatial general epidemic and locally dependent random graphs. J. Appl. Prob. 19, 745–758.Google Scholar
77. Leslie, P. H. and Gower, J. C. (1958) The properties of a stochastic model for two competing species. Biometrika 45, 316–330.Google Scholar
78. Levin, S. A. (1974) Dispersion and population interactions. Amer. Natur. 108, 207–228.Google Scholar
79. Levin, S. A. (1978) Population models and community structure in heterogeneous environments. In Mathematical Association of America Study in Mathematical Biology, Vol. II: Populations and Communities , ed. Levin, S. A., 439–476. Math. Assoc. Amer., Washington.Google Scholar
80. Levins, R. and Culver, D. (1971) Regional coexistence of species and competition between rare species. Proc. Nat. Acad. Sci. U.S.A. 68, 1246–1248.Google Scholar
81. Liggett, T. M. (1973) A characterization of the invariant measures for an infinite particle system with interactions, I. Trans. Amer. Math. Soc. 179, 433–453.Google Scholar
82. Liggett, T. M. (1974) A characterization of the invariant measures for an infinite particle system with interactions, II. Trans. Amer. Math. Soc. 198, 201–213.Google Scholar
83. Liggett, T. M. (1985) Interacting Particle Systems. Springer-Verlag, New York.Google Scholar
84. Lotka, A. J. (1925) Elements of Physical Biology. Williams and Wilkins, Baltimore.Google Scholar
85. Macarthur, R. and Wilson, E. O. (1967) Theory of Island Biogeography. Princeton University Press, Princeton NJ. Google Scholar
86. Maruyama, T. (1969) Genetic correlation in the stepping-stone model with nonsymmetrical migration rates. J. Appl. Prob. 6, 463–477.Google Scholar
87. Maruyama, T. (1970) Stepping-stone models of finite length. Adv. Appl. Prob. 2, 229–258.CrossRefGoogle Scholar
88. Matis, J. H. and Hartley, H. O. (1971) Stochastic compartmental analysis: model and least squares estimation from time series data. Biometrics 27, 77–102.Google Scholar
89. Maynard Smith, J. (1974) Models in Ecology. Cambridge University Press, Cambridge.Google Scholar
90. Milch, P. R. (1968) A multi-dimensional linear growth birth and death process. Ann. Math. Statist. 39, 727–754.Google Scholar
91. Mimura, M. and Murray, J. D. (1978) On a diffusive prey-predator model which exhibits patchiness. J. Theoret. Biol. 75, 249–262.Google Scholar
92. Mode, C. J. (1962) Some multi-dimensional birth and death processes and their application in population genetics. Biometrics 18, 543–567.Google Scholar
93. Mode, C. J. (1971) Multitype Branching Processes. American Elsevier, New York.Google Scholar
94. Mode, C. J. (1971) Multitype age-dependent branching processes and cell cycle analysis. Math. Biosci. 10, 177–190.Google Scholar
95. Mollison, D. (1972) The rate of spatial propagation of simple epidemics. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 579–614.Google Scholar
96. Mollison, D. (1972) Possible velocities for a simple epidemic. Adv. Appl. Prob. 4, 233–257.Google Scholar
97. Mollison, D. (1977) Spatial contact models for ecological and epidemic spread (with discussion). J. R. Statist. Soc. B 39, 283–326.Google Scholar
98. Morgan, B. J. T. and Hinde, J. P. (1976) On an approximation made when analysing stochastic processes. J. Appl. Prob. 13, 672–683.Google Scholar
99. Morgan, R. W. and Welsh, D. J. A. (1965) A two-dimensional Poisson growth process. J. R. Statist. Soc. B 27, 497–504.Google Scholar
100. Neyman, J., Park, T. and Scott, E. L. (1956) Struggle for existence. The Tribolium model: biological and statistical aspects. Proc. 3rd Berkeley Symp. Math. Statist. Prob. 4, 41–79.Google Scholar
101. Neyman, J. and Scott, E. L. (1957) On a mathematical theory of populations conceived as conglomerations of clusters. Cold Spring Harbour Symp. Quant. Biol. 22, 109–120.Google Scholar
102. Nisbet, R. M. and Gurney, W. S. C. (1982) Modelling Fluctuating Populations. Wiley, New York.Google Scholar
103. Notohara, M. and Shiga, T. (1980) Convergence to genetically uniform state in stepping stone models of population genetics. J. Math. Biol. 10, 281–294.Google Scholar
104. Okubo, A. (1980) Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin.Google Scholar
105. Patil, V. T. (1957) The consistency and adequacy of the Poisson-Markoff model for density fluctuations. Biometrika 44, 43–56.Google Scholar
106. Puri, P. S. (1968) Interconnected birth and death processes. J. Appl. Prob. 5, 334–349.Google Scholar
107. Radcliffe, J. and Staff, P. J. (1969) First-order conservative processes with multiple latent roots. J. Appl. Prob. 6, 186–194.Google Scholar
108. Raman, S. and Chiang, C. L. (1973) On a solution of the migration process and the application to a problem in epidemiology. J. Appl. Prob. 10, 718–727.Google Scholar
109. Reid, W. T. (1959) Solutions of the Riccati matrix equation as functions of initial values. J. Math. Mech. 8, 221–230.Google Scholar
110. Renshaw, E. (1972) Birth, death and migration processes. Biometrika 59, 49–60.Google Scholar
111. Renshaw, E. (1973) Interconnected population processes. J. Appl. Prob. 10, 1–14.Google Scholar
112. Renshaw, E. (1973) The effect of migration between two developing populations. Proc. 39th Session Internat. Statist. Inst. 2, 294–298.Google Scholar
113. Renshaw, E. (1974) Stepping-stone models for population growth. J. Appl. Prob. 11, 16–31.Google Scholar
114. Renshaw, E. (1976) Spatial Population Processes. Ph.D. Thesis (unpublished), University of Edinburgh.Google Scholar
115. Renshaw, E. (1977) Velocities of propagation for stepping-stone models of population growth. J. Appl. Prob. 14, 591–597.Google Scholar
116. Renshaw, E. (1979) Waveforms and velocities for non-nearest-neighbour contact distributions. J. Appl. Prob. 16, 1–11.Google Scholar
117. Renshaw, E. (1980) The spatial distribution of Tribolium confusum. J. Appl. Prob. 17, 895–911.Google Scholar
118. Renshaw, E. (1981) Waveforms and velocities for models of spatial infection. J. Appl. Prob. 18, 715–720.Google Scholar
119. Renshaw, E. (1982) The development of a spatial predator-prey process on interconnected sites. J. Theoret. Biol. 94, 355–365.Google Scholar
120. Ruben, H. (1962) Some aspects of the emigration-immigration process. Ann. Math. Statist. 33, 119–129.Google Scholar
121. Rushton, S. and Mautner, A. J. (1955) The deterministic model of a simple epidemic for more than one community. Biometrika 42, 126–132.Google Scholar
122. Sawyer, S. (1976) Results for the stepping stone model for migration in population genetics. Ann. Prob. 4, 699–728.Google Scholar
123. Sawyer, S. (1977) Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv. Appl. Prob. 9, 268–282.Google Scholar
124. Sawyer, S. and Felsenstein, J. (1983) Isolation by distance in a hierarchically clustered population. J. Appl. Prob. 20, 1–10.Google Scholar
125. Severo, N. C. (1967) Two theorems on solutions of differential-difference equations and applications to epidemic theory. J. Appl. Prob. 4, 271–280.CrossRefGoogle Scholar
126. Severo, N. C. (1969) A recursion theorem on solving differential–difference equations and applications to some stochastic processes. J. Appl. Prob. 6, 673–681.Google Scholar
127. Sherman, B. (1956) The limiting distribution of Brownian motion on a finite interval with instantaneous return. Westinghouse Research Laboratory Scientific Paper 60-94698-3-P3.Google Scholar
128. Sherman, B. (1958) The limiting distribution of Brownian motion in a bounded region with instantaneous return. Ann. Math. Statist. 29, 267–273.Google Scholar
129. Shiga, T. (1980) An interacting system in population genetics I. J. Math. Kyoto Univ. 20, 212–242.Google Scholar
130. Shiga, T. (1980) An interacting system in population genetics II. J. Math. Kyoto Univ. 20, 723–733.Google Scholar
131. Shiga, T. (1982) Continuous time multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ. 22, 1–40.Google Scholar
132. Shiga, T. (1985) Mathematical results on the stepping stone model in population genetics. In Population Genetics and Molecular Evolution , ed. Ohta, T. and Aoki, K.. Japan Scientific Societies Press, Tokyo.Google Scholar
133. Siegart, A. J. F. (1949) On the approach to statistical equilibrium. Phys. Rev. 76, 1708–1714.Google Scholar
134. Skellam, J. G. (1951) Random dispersal in theoretical populations. Biometrika 38, 196–218.Google Scholar
135. Smith, R. H. and Mead, R. (1979) On predicting extinction in simple population models. I. Stochastic linearisation. J. Theoret. Biol. 80, 189–203.Google Scholar
136. Spitzer, F. (1970) Interaction of Markov processes. Adv. Math. 5, 246–290.Google Scholar
137. Tallis, G. M. (1966) A migration model. Biometrics 22, 409–412.Google Scholar
138. Turing, A. M. (1952) The chemical basis of morphogenesis. Phil. Trans. R. Soc. London B 237, 37–72.Google Scholar
139. Uchiyama, K. (1978) The behaviour of solutions of some non-linear diffusion equations for large time. J. Math. Kyoto Univ. 18, 453–508.Google Scholar
140. Uchiyama, K. (1981) Spatial growth of solutions of a non-linear equation. Proc. Japan Acad. A 57, 90–94.Google Scholar
141. Uchiyama, K. (1982) Spatial growth of a branching process of particles living in Rd. Ann. Prob. 10, 896–918.Google Scholar
142. Usher, M. B. and Williamson, M. H. (1970) A deterministic matrix model for handling the birth, death and migration processes of spatially distributed populations. Biometrics 23, 1–12.Google Scholar
143. Vandermeer, J. H. (1973) On the regional stabilization of locally unstable predator-prey relationships. J. Theoret. Biol. 41, 161–170.Google Scholar
144. Volterra, V. (1931) Leçons sur la Théorie mathématique de la Lutte pour la Vie. Gauthier-Villars, Paris.Google Scholar
145. Weiss, G. and Kimura, M. (1965) A mathematical analysis of the stepping-stone model of genetic correlation. J. Appl. Prob. 2, 129–149.Google Scholar
146. Whittle, P. (1954) On stationary processes in the plane. Biometrika 41, 434–447.Google Scholar
147. Whittle, P. (1967) Non-linear migration processes. Proc. 36th Session Internat. Statist. Inst. 2, 642–646.Google Scholar
148. Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.Google Scholar
149. Wiggins, A. D. (1960) On a multicompartment migration model with chronic feeding. Biometrics 16, 642–658.Google Scholar
150. Wright, S. (1943) Isolation by distance. Genetics 28, 114–138.Google Scholar