Applications in Biology

  • Peter Schuster
Part of the Springer Series in Synergetics book series (SSSYN)


Stochastic phenomena are central to biological modeling. Small numbers of molecules regulate and control genetics, epigenetics, and cellular metabolism, and small numbers of well-adapted individuals drive evolution. As far as processes are concerned, the relation between chemistry and biology is a tight one. Reproduction, the basis of all processes in evolution, is understood as autocatalysis with an exceedingly complex mechanism in the language of chemists, and replication of the genetic molecules, DNA and RNA, builds the bridge between chemistry and biology. The earliest stochastic models in biology applied branching processes in order to give answers to genealogical questions like, for example, the fate of family names in pedigrees. Branching processes, birth-and-death processes, and related stochastic models are frequently used in biology and they are defined, analyzed, and applied to typical problems. Although the master equation is not so dominant in biology as it is in chemistry, it is a very useful tool for deriving analytical solutions, and most birth-and-death processes can be analyzed successfully by means of master equations. Kimura’s neutral theory of evolution makes use of a Fokker–Planck equation and describes population dynamics in the absence of fitness differences. A section on coalescent theory demonstrates the applicability of backwards modeling to the problem of reconstruction of phylogenies. Unlike in the previous chapter we shall present and discuss numerical simulations here together with the analytical approaches. Simulations of stochastic reaction networks in systems biology are a rapidly growing field and several new monographs have come out during the last few years. Therefore only a brief account and a collection of references are given.


  1. 1.
    Aase, K.: A note on a singular diffusion equation in population genetics. J. Appl. Probab. 13, 1–8 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 9.
    Alonso, D., McKane, A.J., Pascual, M.: Stochastic amplifications in epidemics. J. Roy. Soc. Interface 4, 575–582 (2007)CrossRefGoogle Scholar
  3. 15.
    Anderson, R.M., May, R.M.: Population biology of infectious diseases: Part I. Nature 280, 361–367 (1979)ADSCrossRefGoogle Scholar
  4. 16.
    Anderson, R.M., May, R.M.: Population biology of infectious diseases: Part II. Nature 280, 455–461 (1979)CrossRefGoogle Scholar
  5. 17.
    Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York (1991)Google Scholar
  6. 25.
    Arscott, F.M.: Heun’s equation. In: Ronveau, A. (ed.) Heun’s Differential Equations, pp. 3–86. Oxford University Press, New York (1955)Google Scholar
  7. 29.
    Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Heidelberg, DE (1972)zbMATHCrossRefGoogle Scholar
  8. 32.
    Bailey, N.T.J.: A simple stochastic epidemic. Biometrika 37, 193–202 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 33.
    Bailey, N.T.J.: The Elements of Stochastic Processes with Application in the Natural Sciences. Wiley, New York (1964)zbMATHGoogle Scholar
  10. 44.
    Bernoulli, D.: Essai d’une nouvelle analyse de la mortaltié causée par la petite vérole et des avantages de l’inoculation pour la prévenir. Mém. Math. Phys. Acad. Roy. Sci.,Paris T5, 1–45 (1766). English translation: ‘An Attempt at a New Analysis of the Mortality Caused by Smallpox and of the Advantages of Inoculation to Prevent It.’ In: L. Bradley, Smallpox Inoculation: An Eighteenth Century Mathematical Controversy. Adult Education Department: Nottingham 1971, p. 21Google Scholar
  11. 45.
    Bernoulli, D., Blower, S.: An attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it. Rev. Med. Virol. 14, 275–288 (2004)CrossRefGoogle Scholar
  12. 48.
    Biebricher, C.K., Eigen, M., William C. Gardiner, J.: Kinetics of RNA replication. Biochemistry 22, 2544–2559 (1983)Google Scholar
  13. 49.
    Bienaymé, I.J.: Da la loi de Multiplication et de la durée des familles. Soc. Philomath. Paris Extraits Ser. 5, 37–39 (1845)Google Scholar
  14. 56.
    Blythe, R.A., McKane, A.J.: Stochastic models of evolution in genetics, ecology and linguistics. J. Stat. Mech. Theor. Exp. (2007). P07018Google Scholar
  15. 71.
    Cann, R.L.: Y weigh in again on modern humans. Science 341, 465–467 (2013)ADSCrossRefGoogle Scholar
  16. 72.
    Cann, R.L., Stoneking, M., Wilson, A.C.: Mitochondrial DNA and human evolution. Nature 325, 31–36 (1987)ADSCrossRefGoogle Scholar
  17. 91.
    Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. Methuen, London (1965)zbMATHGoogle Scholar
  18. 96.
    Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Sinauer Associates, Sunderland (1970). Reprinted at The Blackburn Press, Caldwell (2009)Google Scholar
  19. 97.
    Cull, P., Flahive, M., Robson, R.: Difference Equations. From Rabbits to Chaos. Undergraduate Texts in Mathematics. Springer, New York (2005)zbMATHGoogle Scholar
  20. 102.
    De Candolle, A.: Zur Geschichte der Wissenschaften und Gelehrten seit zwei Jahrhunderten nebst anderen Studien über wissenschaftliche Gegenstände insbesondere über Vererbung und Selektion beim Menschen. Akademische Verlagsgesellschaft, Leipzig, DE (1921). Deutsche Übersetzung der Originalausgabe “Histoire des sciences et des savants depuis deux siècle”, Geneve 1873, durch Wilhelm Ostwald.Google Scholar
  21. 105.
    Demetrius, L., Schuster, P., Sigmund, K.: Polynucleotide evolution and branching processes. Bull. Math. Biol. 47, 239–262 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 107.
    Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley Series in Mathematical and Computational Biology. Princeton University Press, Hoboken (2000)zbMATHGoogle Scholar
  23. 108.
    Diekmann, O., Heesterbeek, J.A.P., Britton, T.: Mathematical Tools for Understanding Infectious Disease Dynamics. Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton (2012)zbMATHCrossRefGoogle Scholar
  24. 109.
    Dietz, K.: Epidemics and rumors: A survey. J. R. Stat. Soc. A 130, 505–528 (1967)MathSciNetCrossRefGoogle Scholar
  25. 110.
    Dietz, K., Heesterbeeck, J.A.P.: Daniel Bernoulli’s epidemiological model revisited. Math. Biosci. 180, 1–21 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 113.
    Domingo, E., Schuster, P. (eds.): Quasispecies: From Theory to Experimental Systems, Current Topics in Microbiology and Immunology, vol. 392. Springer, Berlin (2016)Google Scholar
  27. 114.
    Donnelly, P.J., Tavaré, S.: Coalescents and genealogical structure under neutrality. Annu. Rev. Genet. 29, 401–421 (1995)CrossRefGoogle Scholar
  28. 121.
    Edelson, D., Field, R.J., Noyes, R.M.: Mechanistic details of the Belousov-Zhabotinskii oscillations. Int. J. Chem. Kinet. 7, 417–423 (1975)CrossRefGoogle Scholar
  29. 130.
    Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58, 465–523 (1971)ADSCrossRefGoogle Scholar
  30. 131.
    Eigen, M., McCaskill, J., Schuster, P.: The molecular quasispecies. Adv. Chem. Phys. 75, 149–263 (1989)Google Scholar
  31. 132.
    Eigen, M., Schuster, P.: The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle. Naturwissenschaften 64, 541–565 (1977)ADSGoogle Scholar
  32. 141.
    Erlich, H.A. (ed.): PCR Technology. Principles and Applications for DNA Amplification. Stockton Press, New York (1989)Google Scholar
  33. 143.
    Everett, C.J., Ulam, S.: Multiplicative systems I. Proc. Natl. Acad. Sci. USA 34, 403–405 (1948)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 146.
    Everett, C.J., Ulam, S.M.: Multiplicative systems in several variables III. Tech. Rep. LA-707, Los Alamos Scientific Laboratory (1948)Google Scholar
  35. 147.
    Ewens, W.J.: Mathematical Population Genetics. I. Theoretical Introduction, 2nd edn. Interdisciplinary Applied Mathematics. Springer, Berlin (2004)Google Scholar
  36. 159.
    Feller, W.: Diffusion processes in genetics. In: Neyman, J. (ed.) Proc. 2nd Berkeley Symp. on Mathematical Statistics and Probability. University of Caifornia Press, Berkeley (1951)Google Scholar
  37. 162.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Sunderland (2004)Google Scholar
  38. 174.
    Fisher, R.A.: The Genetical Theory of Natural Selection. Oxford University Press, Oxford (1930)zbMATHCrossRefGoogle Scholar
  39. 181.
    Fletcher, R.I.: The quadratic law of damped exponential growth. Biometrics 20, 111–124 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 185.
    Francalacci, P., Morelli, L., Angius, A., Berutti, R., Reinier, F., Atzeni, R., Pilu, R., Busonero, F., Maschino, A., Zara, I., Sanna, D., Useli, A., Urru, M.F., Marcelli, M., Cusano, R., Oppo, M., Zoledziewska, M., Pitzalis, M., Deidda, F., Porcu, E., Poddie, F., Kang, H.M., Lyons, R., Tarrier, B., Gresham, J.B., Li, B., Tofanelli, S., Alonso, S., Dei, M., Lai, S., Mulas, A., Whalen, M.B., Uzzau, S., Jones, C., Schlessinger, D., Abecasis, G.R., Sanna, S., Sidore, C., Cucca, F.: Low-pass DNA sequencing of 1200 Sardinians reconstructs European Y-chrmosome phylogeny. Science 341, 565–569 (2013)Google Scholar
  41. 192.
    Galton, F.: Natural Inheritance, second american edn. Macmillan, London (1889). App. F, pp. 241–248Google Scholar
  42. 195.
    Gause, G.F.: Experimental studies on the struggle for existence. J. Exp. Biol. 9, 389–402 (1932)Google Scholar
  43. 196.
    Gause, G.F.: The Struggle for Existence. Willans & Wilkins, Baltimore (1934). Also published by Hafner, New York (1964) and Dover, Mineola (1971 and 2003)Google Scholar
  44. 216.
    Goel, N.S., Richter-Dyn, N.: Stochastic Models in Biology. Academic Press, New York (1974)Google Scholar
  45. 219.
    Gradstein, I.S., Ryshik, I.M.: Tables of Series, Products, and Integrals, vol. 1. Verlag Harri Deutsch, Thun, DE (1981). In German and English. Translated from Russian by Ludwig Boll, BerlinGoogle Scholar
  46. 234.
    Hammer, M.F.: A recent common ancestry for human Y chromosomes. Nature 378, 376–378 (1995)ADSCrossRefGoogle Scholar
  47. 235.
    Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29, 147–160 (1950)MathSciNetCrossRefGoogle Scholar
  48. 236.
    Hamming, R.W.: Coding and Information Theory, 2nd edn. Prentice-Hall, Englewood Cliffs (1986)zbMATHGoogle Scholar
  49. 239.
    Harris, T.E.: Branching Processes. Springer, Berlin (1963)zbMATHCrossRefGoogle Scholar
  50. 240.
    Harris, T.E.: The Theory of Branching Processes. Dover Publications, New York (1989)Google Scholar
  51. 241.
    Hartl, D.L., Clark, A.G.: Principles of Population Genetics, 3rd edn. Sinauer Associates, Sunderland (1997)Google Scholar
  52. 246.
    Hawkins, D., Ulam, S.: Theory of multiplicative processes I. Tech. Rep. LADC-265, Los Alamos Scientific Laboratory (1944)Google Scholar
  53. 248.
    Heathcote, C.R., Moyal, J.E.: The random walk (in continuous time) and its application to the theory of queues. Biometrika 46, 400–411 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 249.
    Heinrich, R., Sonntag, I.: Analysis of the selection equation for a multivariable population model. Deterministic anad stochastic solutios and discussion of the approach for populations of self-reproducing biochemical networks. J. Theor. Biol. 93, 325–361 (1981)MathSciNetGoogle Scholar
  55. 250.
    Heyde, C.C., Seneta, E.: Studies in the history of probability and statistics. xxxi. the simple branching porcess, a turning point test and a fundmanetal inequality: A historical note on I. J. Bienaymé. Biometrika 59, 680–683 (1972)MathSciNetzbMATHGoogle Scholar
  56. 256.
    Hofbauer, J., Schuster, P., Sigmund, K., Wolff, R.: Dynamical systems und constant organization II: Homogenoeous growth functions of degree p = 2. SIAM J. Appl. Math. 38, 282–304 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 259.
    Holder, M., Lewis, P.O.: Phylogeny estimation: Traditional and Bayesian approaches. Nat. Rev. Genet. 4, 275–284 (2003)CrossRefGoogle Scholar
  58. 261.
    Holsinger, K.E.: Lecture Notes in Population Genetics. University of Connecticut, Dept. of Ecology and Evolutionary Biology, Storrs, CT (2012). Licensed under the Creative Commons Attribution-ShareAlike License:
  59. 264.
    Houchmandzadeh, B., Vallade, M.: An alternative to the diffusion equation in population genetics. Phys. Rev. E 83, e051,913 (2010)Google Scholar
  60. 270.
    Inagaki, H.: Selection under random mutations in stochastic Eigen model. Bull. Math. Biol. 44, 17–28 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 275.
    Jackson, E.A.: Perspectives of Nonlinear Dynamics, vol. 1. Cambridge University Press, Cambridge (1989)zbMATHCrossRefGoogle Scholar
  62. 276.
    Jackson, E.A.: Perspectives of Nonlinear Dynamics, vol. 2. Cambridge University Press, Cambridge (1989)zbMATHCrossRefGoogle Scholar
  63. 282.
    Jensen, A.L.: Comparison of logistic equations for population growth. Biometrics 31, 853–862 (1975)zbMATHCrossRefGoogle Scholar
  64. 283.
    Jensen, L.: Solving a singular diffusion equation occurring in population genetics. J. Appl. Probab. 11, 1–15 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 286.
    Jones, B.L., Enns, R.H., Rangnekar, S.S.: On the theory of selection of coupled macromolecular systems. Bull. Math. Biol. 38, 15–28 (1976)zbMATHCrossRefGoogle Scholar
  66. 287.
    Jones, B.L., Leung, H.K.: Stochastic analysis of a non-linear model for selection of biological macromolecules. Bull. Math. Biol. 43, 665–680 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 288.
    Joyce, G.F.: Forty years of in vitro evolution. Angew. Chem. Internat. Ed. 46, 6420–6436 (2007)CrossRefGoogle Scholar
  68. 289.
    Karlin, S., McGregor, J.: On a genetics model of moran. Math. Proc. Camb. Philos. Soc. 58, 299–311 (1962)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  69. 293.
    Kendall, D.G.: Branching processes since 1873. J. Lond. Math. Soc. 41, 386–406 (1966)MathSciNetzbMATHGoogle Scholar
  70. 294.
    Kendall, D.G.: The genalogy of genealogy: Branching processes before (an after) 1873. Bull. Lond. Math. Soc. 7, 225–253 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 297.
    Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721 (1927)ADSzbMATHCrossRefGoogle Scholar
  72. 298.
    Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat. 37, 1211–1223 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 302.
    Kimura, M.: Solution of a process of random genetic drift with a continuous model. Proc. Natl. Acad. Sci. USA 41, 144–150 (1955)ADSzbMATHCrossRefGoogle Scholar
  74. 303.
    Kimura, M.: Diffusion models in population genetics. J. Appl. Probab. 1, 177–232 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 304.
    Kimura, M.: The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge (1983)CrossRefGoogle Scholar
  76. 305.
    Kingman, J.F.C.: Mathematics of Genetic Diversity. Society for Industrial and Applied Mathematics, Washigton, DC (1980)zbMATHCrossRefGoogle Scholar
  77. 306.
    Kingman, J.F.C.: The genealogy of large populations. J. Appl. Probab. 19 (Essays in Statistical Science), 27–43 (1982)Google Scholar
  78. 307.
    Kingman, J.F.C.: Origins of the coalescent: 1974 – 1982. Genetics 156, 1461–1463 (2000)Google Scholar
  79. 312.
    Kolmogorov, A.N., Dmitriev, N.A.: “Zur Lösung einer biologischen Aufgabe”. Isvestiya Nauchno-Issledovatel’skogo Instituta Matematiki i Mekhaniki pri Tomskom Gosudarstvennom Universitete 2, 1–12 (1938)zbMATHGoogle Scholar
  80. 313.
    Kolmogorov, A.N., Dmitriev, N.A.: Branching stochastic processes. Doklady Akad. Nauk U.S.S.R. 56, 5–8 (1947)Google Scholar
  81. 338.
    Leung, K.: Expansion of the master equation for a biomolecular selection model. Bull. Math. Biol. 47, 231–238 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 372.
    Maruyama, T.: Stochastic Problems in Population Genetics. Springer, Berlin (1977)zbMATHCrossRefGoogle Scholar
  83. 379.
    McCaskill, J.S.: A stochastic theory of macromolecular evolution. Biol. Cybern. 50, 63–73 (1984)zbMATHCrossRefGoogle Scholar
  84. 385.
    McVinish, R., Pollett, P.K.: A central limit theorem for a discrete time SIS model with individual variation. J. Appl. Probab. 49, 521–530 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 386.
    McVinish, R., Pollett, P.K.: The deterministic limit of a stochastic logistic model with individual variation. J. Appl. Probab. 241, 109–114 (2013)MathSciNetzbMATHGoogle Scholar
  86. 393.
    Meredith, M.: Born in Africa: The Quest for the Origins of Human Life. Public Affairs, New York (2011)Google Scholar
  87. 400.
    Mode, C.J., Sleeman, C.K.: Stochastic Processes in Genetics and Evolution. Computer Experiments in the Quantification of Mutation and Selection. World Scientific Publishing, Singapore (2012)CrossRefGoogle Scholar
  88. 404.
    Montroll, E.W.: Stochastic processes and chemical kinetics. In: Muller, W.M. (ed.) Energetics in Metallurgical Phenomenon, vol. 3, pp. 123–187. Gordon & Breach, New York (1967)Google Scholar
  89. 406.
    Montroll, E.W., Shuler, K.E.: The application of the theory of stochastic processes to chemical kinetics. Adv. Chem. Phys. 1, 361–399 (1958)MathSciNetGoogle Scholar
  90. 410.
    Moran, P.A.P.: Random processes in genetics. Proc. Camb. Philos. Soc. 54, 60–71 (1958)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  91. 411.
    Moran, P.A.P.: The Statistical Processes of Evolutionary Theroy. Clarendon Press, Oxford (1962)Google Scholar
  92. 412.
    Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. I. McGraw-Hill, Boston (1953)zbMATHGoogle Scholar
  93. 414.
    Mount, D.W.: Bioinformatics. Sequence and Genome Analysis, 2nd edn. Cold Spring Harbor Laboratory Press, Cold Spring Harbor (2004)Google Scholar
  94. 417.
    Munz, P., Hudea, I., Imad, J., Smith, R.J.: When zombies attack: Mathematical modelling of an outbreak of zombie infection. In: Tchuenche, J.M., Chiyaka, C. (eds.) Infectious Disease Modelling Research Progress, chap. 4, pp. 133–156. Nova Science Publishers, Hauppauge (2009)Google Scholar
  95. 418.
    Nåsell, I.: On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40 (1999)MathSciNetCrossRefGoogle Scholar
  96. 419.
    Nåsell, I.: Extiction and quasi-stationarity in the Verhulst logistic model. J. Theor. Biol. 211, 11–27 (2001)CrossRefGoogle Scholar
  97. 421.
    Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems. Wiley, New York (1977)zbMATHGoogle Scholar
  98. 422.
    Nishiyama, K.: Stochastic approach to nonlinear chemical reactions having multiple steatdy states. J. Phys. Soc. Jpn. 37, 44–49 (1974)ADSCrossRefGoogle Scholar
  99. 426.
    Norden, R.H.: On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Probab. 14, 687–708 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 442.
    Pearson, J.A.: Advanced Statistical Physics. University of Manchester, Manchester, UK (2009). URL:
  101. 451.
    Phillipson, P.E., Schuster, P.: Modeling by Nonlinear Differential Equations. Dissipative and Conservative Processes, World Scientific Series on Nonlinear Science A, vol. 69. World Scientific, Singapore (2009)Google Scholar
  102. 452.
    Picard, P.: Sur les Modèles stochastiques logistiques en Démographie. Ann. Inst. H. Poincaré B II, 151–172 (1965)Google Scholar
  103. 457.
    Poznik, G.D., Henn, B.M., Yee, M.C., Sliwerska, E., Lin, A.A., Snyder, M., Quintana-Murci, L., Kidd, J.M., Underhill, P.A., Bustamante, C.D.: Sequencing Y chromosomes resolves discrepancy in time to common ancestor of males versus females. Science 341, 562–565 (2013)ADSCrossRefGoogle Scholar
  104. 479.
    Schlögl, F.: Chemical reaction models for non-equilibrium phase transitions. Z. Physik 253, 147–161 (1972)ADSCrossRefGoogle Scholar
  105. 482.
    Schuster, P.: Mathematical modeling of evolution. Solved and open problems. Theory Biosci. 130, 71–89 (2011)CrossRefGoogle Scholar
  106. 484.
    Schuster, P.: Quasispecies on fitness landscapes. In: Domingo, E., Schuster, P. (eds.) Quasispecies: From Theory to Experimental Systems, Current Topics in Microbiology and Immunology, vol. 392, chap. 4, pp. ppp–ppp. Springer, Berlin (2016). DOI 10.10007/82_2015_469Google Scholar
  107. 485.
    Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol. 100, 533–538 (1983)MathSciNetCrossRefGoogle Scholar
  108. 486.
    Schuster, P., Sigmund, K.: Random selection - A simple model based on linear birth and death processes. Bull. Math. Biol. 46, 11–17 (1984)zbMATHGoogle Scholar
  109. 492.
    Seneta, E.: Non-negative Matrices and Markov Chains, 2nd edn. Springer, New York (1981)zbMATHCrossRefGoogle Scholar
  110. 496.
    Seydel, R.: Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos, Interdisciplinary Applied Mathematics, vol. 5, 2nd edn. Springer, New York (1994)Google Scholar
  111. 502.
    Shuler, K.E., Weiss, G.H., Anderson, K.: Studies in nonequilibrium rate processes. V. The relaxation of moments derived from a master equation. J. Math. Phys. 3, 550–556 (1962)zbMATHGoogle Scholar
  112. 505.
    Steffensen, J.F.: “deux problème du calcul des probabilités”. Ann. Inst. Henri Poincaré 3, 319–344 (1933)MathSciNetzbMATHGoogle Scholar
  113. 513.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos. With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press at Perseus Books, Cambridge (1994)Google Scholar
  114. 520.
    Swetina, J., Schuster, P.: Self-replication with errors - A model for polynucleotide replication. Biophys. Chem. 16, 329–345 (1982)CrossRefGoogle Scholar
  115. 521.
    Szathmáry, E., Gladkih, I.: Sub-exponential growth and coexistence of non-enzymatically replicating templates. J. Theor. Biol. 138, 55–58 (1989)CrossRefGoogle Scholar
  116. 522.
    Tang, H., Siegmund, D.O., Shen, P., Oefner, P.J., Feldman, M.W.: Frequentist estimation of coalescence times from nucleotide sequence data using a tree-based partition. Genetics 161, 448–459 (2002)Google Scholar
  117. 525.
    Tavaré, S.: Line-of-descent and genealogical processes, and their application in population genetics models. Theor. Popul. Biol. 26, 119–164 (1984)zbMATHCrossRefGoogle Scholar
  118. 530.
    Thompson, C.J., McBride, J.L.: On Eigen’s theory of the self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21, 127–142 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 536.
    Ullah, M., Wolkenhauer, O.: Stochastic Approaches for Systems Biology. Springer, New York (2011)zbMATHCrossRefGoogle Scholar
  120. 539.
    Van Doorn, E.A.: Quasi-stationary distribution and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab. 23, 683–700 (1991)zbMATHCrossRefGoogle Scholar
  121. 541.
    van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245–309 (1976)Google Scholar
  122. 550.
    Verhulst, P.: Notice sur la loi que la population pursuit dans son accroisement. Corresp. Math. Phys. 10, 113–121 (1838)Google Scholar
  123. 555.
    von Kiedrowski, G.: A self-replicating hexanucleotide. Angew. Chem. Internat. Ed. 25, 932–935 (1986)CrossRefGoogle Scholar
  124. 556.
    von Kiedrowski, G., Wlotzka, B., Helbig, J., Matzen, M., Jordan, S.: Parabolic growth of a self-replicating hexanucleotide bearing a 3’-5’-phosphoamidate linkage. Angew. Chem. Int. Ed. 30, 423–426 (1991)CrossRefGoogle Scholar
  125. 562.
    Watson, H.W., Galton, F.: On the probability of the extinction of families. J. Anthropol. Inst. G. Br. Irel. 4, 138–144 (1875)Google Scholar
  126. 566.
    Weiss, G.H., Dishon, M.: On the asympotitic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  127. 570.
    Wilheim, T.: The smallest chemical rwaction system with bistability. BMC Syst. Biol. 3, e90 (2009)CrossRefGoogle Scholar
  128. 571.
    Wilheim, T., Heinrich, R.: Smallest chemical rwaction system with Hopf bifurcation. J. Math. Chem. 17, 1–14 (1995)MathSciNetCrossRefGoogle Scholar
  129. 573.
    Wilkinson, D.J.: Stochastic Modelling for Systems Biology, 2nd edn. Chapman & Hall/CRC Press – Taylor and Francis Group, Boca Raton (2012)zbMATHGoogle Scholar
  130. 575.
    Wills, P.R., Kauffman, S.A., Stadler, B.M.R., Stadler, P.F.: Selection dynamics in autocatalytic systems: Templates replicating through binary ligation. Bull. Math. Biol. 60, 1073–1098 (1998)zbMATHCrossRefGoogle Scholar
  131. 579.
    Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)Google Scholar
  132. 580.
    Wright, S.: The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Jones, D.F. (ed.) Int. Proceedings of the Sixth International Congress on Genetics, vol. 1, pp. 356–366. Brooklyn Botanic Garden, Ithaca (1932)Google Scholar
  133. 583.
    Zhabotinsky, A.M.: A history of chemical oscillations and waves. Chaos 1, 379–386 (1991)ADSCrossRefGoogle Scholar
  134. 585.
    Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Peter Schuster
    • 1
  1. 1.Institut für Theoretische ChemieUniversität WienWienAustria

Personalised recommendations