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Applications in Biology

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Stochasticity in Processes

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

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.

Nothing in biology makes sense except in the light of evolution.

Theodosius Dobzhansky 1972.

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Notes

  1. 1.

    Throughout this monograph we use subspecies in the sense of molecular species or variant for the components of a population \(\boldsymbol{\Pi } =\{ \mathsf{X}_{1},\mathsf{X}_{2},\mathop{\ldots },\mathsf{X}_{n}\}\). We express its numbers by the random vector \(\mathcal{N} = \vert \boldsymbol{\Pi }\vert = (\mathcal{N}_{1},\mathcal{N}_{2},\mathop{\ldots },\mathcal{N}_{n})\) and indicate by using the notion biological species when we mean species in biology.

  2. 2.

    Verhulst himself gave no biological interpretation of the logistic equation and its parameters in terms of a carrying capacity. For a brief historical account on the origin of this equation, see [282]. The Malthus parameter is commonly denoted by r. Since r is used as the flow rate in the CSTR, we choose f here in order to avoid confusion and to stress the close relationship between the Malthus parameter and fitness.

  3. 3.

    In this section we shall use n for the number of molecules involved in the autocatalytic reaction, as well as for the numbers of stochastic variables.

  4. 4.

    Termolecular and higher reaction steps are neglected in mass action kinetics, but they are nevertheless frequently used in models and simplified kinetic mechanisms. Examples are the Schlögl model aut]Schlögl, F. [479] and the Brusselator model [421] (Sect. 4.6.4). Thomas Wilhelm aut]Wilhelm, T. and Reinhart Heinrich aut]Heinrich, R. provided a rigorous proof that the smallest oscillating system with only mono- and bimolecular reaction steps has to be at least three-dimensional and must contain one bimolecular term [571]. A similar proof for the smallest system showing bistability can be found in [570].

  5. 5.

    This relation is a result of mass conservation in the closed system.

  6. 6.

    As for the time dependence in the closed system expressed by (5.5c), we make use of the uncommon implicit function \(r = f(\bar{x})\) rather than the direct relation \(\bar{x} = f(r)\).

  7. 7.

    Bifurcation analysis is a standard topic in the theory of nonlinear systems. Monographs oriented towards practical applications are, for example, [275, 276, 496, 513].

  8. 8.

    Although confusion is highly improbable, we remark that the use of n as the exponent of the growth function in x n and as the number of particles in P n is ambiguous here.

  9. 9.

    Reproduction is understood here as asexual reproduction, which under pseudo-first order conditions gives rise to linear reaction kinetics. Sexual reproduction requires two partners and gives rise to a special process of reaction order two (Table 4.2).

  10. 10.

    Here we use the symbols commonly applied in biology: λ (n) for birth, μ (n) for death, ν for immigration, and ρ for emigration. Other notions and symbols are used in chemistry: f ≡ λ for birth corresponding to the production of a molecule and d ≡ μ for death understood as decomposition or degradation through a chemical reaction. Inflow and outflow are the equivalents of immigration and emigration. Pure immigration and emigration give rise to Poisson processes and continuous time random walks, which have been discussed extensively in Chap. 3 There we denoted the parameters by γ and ϑ, instead of ν and ρ.

  11. 11.

    Here the conjugated Laplace variable is denoted by \(\hat{s}\) in order to avoid confusion with the dummy variable s in the generating function.

  12. 12.

    The notion of allele was invented in genetics as a short form of allelomorph, which means other form, for the variants of a gene.

  13. 13.

    Here we use 2N for the number of alleles in a population of size N, which refers to diploid organisms. For haploid organisms, 2N has to be replaced by N. In real populations, the population size is corrected for various other factors and taken to be 2N e or N e , respectively.

  14. 14.

    The selection coefficient is denoted here by ϱ instead of s in order to avoid confusion with the auxiliary variable of the probability generation function. The definition here is the same as used by Kimura [96, 304]: ϱ > 0 implies greater fitness than the reference and an advantageous allele, ϱ < 0 reduced fitness and a deleterious allele. We remark that the conventional definition in population genetics uses the opposite sign: s = 1 means fitness zero, no progeny, and a lethal variant. In either case, selective neutrality occurs for s = 0 or ϱ = 0 (see also Sect. 5.3.3).

  15. 15.

    The definition of Gegenbauer polynomials here is slightly different from the one given in Sect. 4.3.3: T n (β)(z) = (2β − 1)! ! C n (β+1∕2)(z).

  16. 16.

    The definitions of the product terms of the ratios π k and ϱ k differ from those used in Sect. 3.2.3.

  17. 17.

    The law of the mean expresses the difference in the values of a function f(x) in terms of the derivative at one particular point x = x 1 and the difference in the arguments

    $$\displaystyle{ f(b) - f(a) = (b - a)\left.\frac{\partial f} {\partial x}\right \vert _{x=x_{1}}\;,\quad a < x < b. }$$

    The law of the mean is satisfied for at least one point x 1 on the arc between a and b.

  18. 18.

    In real systems, we are always dealing with finite populations in finite time, and then expectation values do not diverge (but see, for example, the unrestricted birth-and-death process in Sect. 5.2.2).

  19. 19.

    Situations may exist where it is for all practical purposes impossible to reach one population from another one through a chain of mutations in any reasonable time span. Then M is not irreducible in reality, and we are dealing with two independently mutating populations. In particular, when more involved mutation mechanisms comprising point mutations, deletions, and insertions are considered, it may be advantageous to deal with disjoint sets of subspecies.

  20. 20.

    In the case of the mathematically equivalent Crow–Kimura mutation–selection equation [96, p. 265, Eq. 6.4.1], additivity of the fitness and mutation matrix is assumed rather than factorizability (see, e.g., Sect. 5.3.3 and [484]).

  21. 21.

    The selection–mutation equation (5.46a) in the original formulation [130, 132] aut]Eigen, M. also contains a degradation term d j x j , and the corresponding definition of the value matrix reads W = (w ij  = Q ij f j − d j ). If all individuals follow the same death law, i.e., d j  = d, \(\forall \,j\), the parameter d can be absorbed into the population size conservation relation and need not be considered separately.

  22. 22.

    The notation applied here is the conventional way of writing transitions in physics: W nm is the probability of the transition n ← m, whereas many mathematicians would write W mn , indicating m → n.

  23. 23.

    When doing actual calculations, one has to use the convention 00 = 1 used in probability theory and combinatorics, but not usually in analysis, where 00 is an indefinite expression.

  24. 24.

    A matrix W with this property is called a stochastic matrix.

  25. 25.

    The second procedure can be visualized by a somewhat strange but nevertheless precise model assumption: after the replication event, the parent but not the offspring is put back into the pool from which the individual, which is doomed to die, is chosen in the second draw.

  26. 26.

    In the non-degenerate case, stationary states do not depend on initial conditions, but this is no longer true for linear combinations of degenerate eigenvectors: α and β, and ϑ are functions of the initial state.

  27. 27.

    In population genetics, the fitness parameter is conventionally denoted by s, but here we use ϱ in order to avoid confusion with the auxiliary variable s.

  28. 28.

    The function3 F 2 belongs to the class of extended hypergeometric functions, referred to in Mathematica as HypergeometricPFQ.

  29. 29.

    The result for ɛ is easily obtained by making use of the infinite series

    $$\displaystyle{\sqrt{1 + x} = 1 + \frac{1} {2}x -\frac{1} {8}x^{2} + \frac{1} {16}x^{3} + \cdots \;,\quad 1/\sqrt{1 + x} = 1 -\frac{1} {2}x + \frac{3} {8}x^{2} - \frac{5} {16}x^{3} + \cdots \;,}$$

    for small x.

  30. 30.

    Nucleic acids are linear polymers and have two different ends with the hydroxy group in the 5′ position or in the 3′ position, respectively.

  31. 31.

    The Hamming distance between two end-to-end aligned strings is the number of digits in which the two strings differ [235, 236].

  32. 32.

    Time is running backwards from the present τ = −t, with today as the origin (Fig. 5.29).

  33. 33.

    The Y-chromosome in males is haploid and non-recombining.

  34. 34.

    This means that reproduction lies in the domain of neutral evolution, i.e., all fitness values are assumed to be the same, or in other words no effects of selection are observable and the numbers of descendants of the individual alleles \(\mathcal{N}_{1},\mathcal{N}_{2},\mathop{\ldots },\mathcal{N}_{N}\), are entirely determined by random events.

  35. 35.

    Assume that X i (n) has the ancestor X i (n + 1). The probability that X j (n) has the same ancestor is simply one out of N, i.e., 1∕N.

References

  1. Aase, K.: A note on a singular diffusion equation in population genetics. J. Appl. Probab. 13, 1–8 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alonso, D., McKane, A.J., Pascual, M.: Stochastic amplifications in epidemics. J. Roy. Soc. Interface 4, 575–582 (2007)

    Article  Google Scholar 

  3. Anderson, R.M., May, R.M.: Population biology of infectious diseases: Part I. Nature 280, 361–367 (1979)

    Article  ADS  Google Scholar 

  4. Anderson, R.M., May, R.M.: Population biology of infectious diseases: Part II. Nature 280, 455–461 (1979)

    Article  Google Scholar 

  5. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, New York (1991)

    Google Scholar 

  6. 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. Athreya, K.B., Ney, P.E.: Branching Processes. Springer, Heidelberg, DE (1972)

    Book  MATH  Google Scholar 

  8. Bailey, N.T.J.: A simple stochastic epidemic. Biometrika 37, 193–202 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bailey, N.T.J.: The Elements of Stochastic Processes with Application in the Natural Sciences. Wiley, New York (1964)

    MATH  Google Scholar 

  10. 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. 21

    Google Scholar 

  11. 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)

    Article  Google Scholar 

  12. Biebricher, C.K., Eigen, M., William C. Gardiner, J.: Kinetics of RNA replication. Biochemistry 22, 2544–2559 (1983)

    Google Scholar 

  13. 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. Blythe, R.A., McKane, A.J.: Stochastic models of evolution in genetics, ecology and linguistics. J. Stat. Mech. Theor. Exp. (2007). P07018

    Google Scholar 

  15. Cann, R.L.: Y weigh in again on modern humans. Science 341, 465–467 (2013)

    Article  ADS  Google Scholar 

  16. Cann, R.L., Stoneking, M., Wilson, A.C.: Mitochondrial DNA and human evolution. Nature 325, 31–36 (1987)

    Article  ADS  Google Scholar 

  17. Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. Methuen, London (1965)

    MATH  Google Scholar 

  18. 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. Cull, P., Flahive, M., Robson, R.: Difference Equations. From Rabbits to Chaos. Undergraduate Texts in Mathematics. Springer, New York (2005)

    MATH  Google Scholar 

  20. 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. Demetrius, L., Schuster, P., Sigmund, K.: Polynucleotide evolution and branching processes. Bull. Math. Biol. 47, 239–262 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    MATH  Google Scholar 

  23. 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)

    Book  MATH  Google Scholar 

  24. Dietz, K.: Epidemics and rumors: A survey. J. R. Stat. Soc. A 130, 505–528 (1967)

    Article  MathSciNet  Google Scholar 

  25. Dietz, K., Heesterbeeck, J.A.P.: Daniel Bernoulli’s epidemiological model revisited. Math. Biosci. 180, 1–21 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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. Donnelly, P.J., Tavaré, S.: Coalescents and genealogical structure under neutrality. Annu. Rev. Genet. 29, 401–421 (1995)

    Article  Google Scholar 

  28. Edelson, D., Field, R.J., Noyes, R.M.: Mechanistic details of the Belousov-Zhabotinskii oscillations. Int. J. Chem. Kinet. 7, 417–423 (1975)

    Article  Google Scholar 

  29. Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58, 465–523 (1971)

    Article  ADS  Google Scholar 

  30. Eigen, M., McCaskill, J., Schuster, P.: The molecular quasispecies. Adv. Chem. Phys. 75, 149–263 (1989)

    Google Scholar 

  31. Eigen, M., Schuster, P.: The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle. Naturwissenschaften 64, 541–565 (1977)

    ADS  Google Scholar 

  32. Erlich, H.A. (ed.): PCR Technology. Principles and Applications for DNA Amplification. Stockton Press, New York (1989)

    Google Scholar 

  33. Everett, C.J., Ulam, S.: Multiplicative systems I. Proc. Natl. Acad. Sci. USA 34, 403–405 (1948)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Everett, C.J., Ulam, S.M.: Multiplicative systems in several variables III. Tech. Rep. LA-707, Los Alamos Scientific Laboratory (1948)

    Google Scholar 

  35. Ewens, W.J.: Mathematical Population Genetics. I. Theoretical Introduction, 2nd edn. Interdisciplinary Applied Mathematics. Springer, Berlin (2004)

    Google Scholar 

  36. 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. Felsenstein, J.: Inferring Phylogenies. Sinauer Associates, Sunderland (2004)

    Google Scholar 

  38. Fisher, R.A.: The Genetical Theory of Natural Selection. Oxford University Press, Oxford (1930)

    Book  MATH  Google Scholar 

  39. Fletcher, R.I.: The quadratic law of damped exponential growth. Biometrics 20, 111–124 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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. Galton, F.: Natural Inheritance, second american edn. Macmillan, London (1889). App. F, pp. 241–248

    Google Scholar 

  42. Gause, G.F.: Experimental studies on the struggle for existence. J. Exp. Biol. 9, 389–402 (1932)

    Google Scholar 

  43. 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. Goel, N.S., Richter-Dyn, N.: Stochastic Models in Biology. Academic Press, New York (1974)

    Google Scholar 

  45. 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, Berlin

    Google Scholar 

  46. Hammer, M.F.: A recent common ancestry for human Y chromosomes. Nature 378, 376–378 (1995)

    Article  ADS  Google Scholar 

  47. Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29, 147–160 (1950)

    Article  MathSciNet  Google Scholar 

  48. Hamming, R.W.: Coding and Information Theory, 2nd edn. Prentice-Hall, Englewood Cliffs (1986)

    MATH  Google Scholar 

  49. Harris, T.E.: Branching Processes. Springer, Berlin (1963)

    Book  MATH  Google Scholar 

  50. Harris, T.E.: The Theory of Branching Processes. Dover Publications, New York (1989)

    Google Scholar 

  51. Hartl, D.L., Clark, A.G.: Principles of Population Genetics, 3rd edn. Sinauer Associates, Sunderland (1997)

    Google Scholar 

  52. Hawkins, D., Ulam, S.: Theory of multiplicative processes I. Tech. Rep. LADC-265, Los Alamos Scientific Laboratory (1944)

    Google Scholar 

  53. 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)

    Article  MathSciNet  MATH  Google Scholar 

  54. 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)

    MathSciNet  Google Scholar 

  55. 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)

    MathSciNet  MATH  Google Scholar 

  56. 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)

    Article  MathSciNet  MATH  Google Scholar 

  57. Holder, M., Lewis, P.O.: Phylogeny estimation: Traditional and Bayesian approaches. Nat. Rev. Genet. 4, 275–284 (2003)

    Article  Google Scholar 

  58. 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: http://creativecommons.org/licenses/by-sa/3.0/

  59. Houchmandzadeh, B., Vallade, M.: An alternative to the diffusion equation in population genetics. Phys. Rev. E 83, e051,913 (2010)

    Google Scholar 

  60. Inagaki, H.: Selection under random mutations in stochastic Eigen model. Bull. Math. Biol. 44, 17–28 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. Jackson, E.A.: Perspectives of Nonlinear Dynamics, vol. 1. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  62. Jackson, E.A.: Perspectives of Nonlinear Dynamics, vol. 2. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  63. Jensen, A.L.: Comparison of logistic equations for population growth. Biometrics 31, 853–862 (1975)

    Article  MATH  Google Scholar 

  64. Jensen, L.: Solving a singular diffusion equation occurring in population genetics. J. Appl. Probab. 11, 1–15 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  65. 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)

    Article  MATH  Google Scholar 

  66. 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)

    Article  MathSciNet  MATH  Google Scholar 

  67. Joyce, G.F.: Forty years of in vitro evolution. Angew. Chem. Internat. Ed. 46, 6420–6436 (2007)

    Article  Google Scholar 

  68. Karlin, S., McGregor, J.: On a genetics model of moran. Math. Proc. Camb. Philos. Soc. 58, 299–311 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. Kendall, D.G.: Branching processes since 1873. J. Lond. Math. Soc. 41, 386–406 (1966)

    MathSciNet  MATH  Google Scholar 

  70. Kendall, D.G.: The genalogy of genealogy: Branching processes before (an after) 1873. Bull. Lond. Math. Soc. 7, 225–253 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  71. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A 115, 700–721 (1927)

    Article  ADS  MATH  Google Scholar 

  72. Kesten, H., Stigum, B.P.: A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Stat. 37, 1211–1223 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  73. Kimura, M.: Solution of a process of random genetic drift with a continuous model. Proc. Natl. Acad. Sci. USA 41, 144–150 (1955)

    Article  ADS  MATH  Google Scholar 

  74. Kimura, M.: Diffusion models in population genetics. J. Appl. Probab. 1, 177–232 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  75. Kimura, M.: The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge (1983)

    Book  Google Scholar 

  76. Kingman, J.F.C.: Mathematics of Genetic Diversity. Society for Industrial and Applied Mathematics, Washigton, DC (1980)

    Book  MATH  Google Scholar 

  77. Kingman, J.F.C.: The genealogy of large populations. J. Appl. Probab. 19 (Essays in Statistical Science), 27–43 (1982)

    Google Scholar 

  78. Kingman, J.F.C.: Origins of the coalescent: 1974 – 1982. Genetics 156, 1461–1463 (2000)

    Google Scholar 

  79. 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)

    MATH  Google Scholar 

  80. Kolmogorov, A.N., Dmitriev, N.A.: Branching stochastic processes. Doklady Akad. Nauk U.S.S.R. 56, 5–8 (1947)

    Google Scholar 

  81. Leung, K.: Expansion of the master equation for a biomolecular selection model. Bull. Math. Biol. 47, 231–238 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  82. Maruyama, T.: Stochastic Problems in Population Genetics. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  83. McCaskill, J.S.: A stochastic theory of macromolecular evolution. Biol. Cybern. 50, 63–73 (1984)

    Article  MATH  Google Scholar 

  84. 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)

    Article  MathSciNet  MATH  Google Scholar 

  85. McVinish, R., Pollett, P.K.: The deterministic limit of a stochastic logistic model with individual variation. J. Appl. Probab. 241, 109–114 (2013)

    MathSciNet  MATH  Google Scholar 

  86. Meredith, M.: Born in Africa: The Quest for the Origins of Human Life. Public Affairs, New York (2011)

    Google Scholar 

  87. 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)

    Book  Google Scholar 

  88. 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. Montroll, E.W., Shuler, K.E.: The application of the theory of stochastic processes to chemical kinetics. Adv. Chem. Phys. 1, 361–399 (1958)

    MathSciNet  Google Scholar 

  90. Moran, P.A.P.: Random processes in genetics. Proc. Camb. Philos. Soc. 54, 60–71 (1958)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  91. Moran, P.A.P.: The Statistical Processes of Evolutionary Theroy. Clarendon Press, Oxford (1962)

    Google Scholar 

  92. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. I. McGraw-Hill, Boston (1953)

    MATH  Google Scholar 

  93. Mount, D.W.: Bioinformatics. Sequence and Genome Analysis, 2nd edn. Cold Spring Harbor Laboratory Press, Cold Spring Harbor (2004)

    Google Scholar 

  94. 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. Nåsell, I.: On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40 (1999)

    Article  MathSciNet  Google Scholar 

  96. Nåsell, I.: Extiction and quasi-stationarity in the Verhulst logistic model. J. Theor. Biol. 211, 11–27 (2001)

    Article  Google Scholar 

  97. Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems. Wiley, New York (1977)

    MATH  Google Scholar 

  98. Nishiyama, K.: Stochastic approach to nonlinear chemical reactions having multiple steatdy states. J. Phys. Soc. Jpn. 37, 44–49 (1974)

    Article  ADS  Google Scholar 

  99. Norden, R.H.: On the distribution of the time to extinction in the stochastic logistic population model. Adv. Appl. Probab. 14, 687–708 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  100. Pearson, J.A.: Advanced Statistical Physics. University of Manchester, Manchester, UK (2009). URL: http://www.joffline.com/

  101. 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. Picard, P.: Sur les Modèles stochastiques logistiques en Démographie. Ann. Inst. H. Poincaré B II, 151–172 (1965)

    Google Scholar 

  103. 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)

    Article  ADS  Google Scholar 

  104. Schlögl, F.: Chemical reaction models for non-equilibrium phase transitions. Z. Physik 253, 147–161 (1972)

    Article  ADS  Google Scholar 

  105. Schuster, P.: Mathematical modeling of evolution. Solved and open problems. Theory Biosci. 130, 71–89 (2011)

    Article  Google Scholar 

  106. 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_469

    Google Scholar 

  107. Schuster, P., Sigmund, K.: Replicator dynamics. J. Theor. Biol. 100, 533–538 (1983)

    Article  MathSciNet  Google Scholar 

  108. Schuster, P., Sigmund, K.: Random selection - A simple model based on linear birth and death processes. Bull. Math. Biol. 46, 11–17 (1984)

    MATH  Google Scholar 

  109. Seneta, E.: Non-negative Matrices and Markov Chains, 2nd edn. Springer, New York (1981)

    Book  MATH  Google Scholar 

  110. 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. 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)

    MATH  Google Scholar 

  112. Steffensen, J.F.: “deux problème du calcul des probabilités”. Ann. Inst. Henri Poincaré 3, 319–344 (1933)

    MathSciNet  MATH  Google Scholar 

  113. 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. Swetina, J., Schuster, P.: Self-replication with errors - A model for polynucleotide replication. Biophys. Chem. 16, 329–345 (1982)

    Article  Google Scholar 

  115. Szathmáry, E., Gladkih, I.: Sub-exponential growth and coexistence of non-enzymatically replicating templates. J. Theor. Biol. 138, 55–58 (1989)

    Article  Google Scholar 

  116. 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. Tavaré, S.: Line-of-descent and genealogical processes, and their application in population genetics models. Theor. Popul. Biol. 26, 119–164 (1984)

    Article  MATH  Google Scholar 

  118. 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)

    Article  MathSciNet  MATH  Google Scholar 

  119. Ullah, M., Wolkenhauer, O.: Stochastic Approaches for Systems Biology. Springer, New York (2011)

    Book  MATH  Google Scholar 

  120. Van Doorn, E.A.: Quasi-stationary distribution and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab. 23, 683–700 (1991)

    Article  MATH  Google Scholar 

  121. van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245–309 (1976)

    Google Scholar 

  122. Verhulst, P.: Notice sur la loi que la population pursuit dans son accroisement. Corresp. Math. Phys. 10, 113–121 (1838)

    Google Scholar 

  123. von Kiedrowski, G.: A self-replicating hexanucleotide. Angew. Chem. Internat. Ed. 25, 932–935 (1986)

    Article  Google Scholar 

  124. 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)

    Article  Google Scholar 

  125. 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. Weiss, G.H., Dishon, M.: On the asympotitic behavior of the stochastic and deterministic models of an epidemic. Math. Biosci. 11, 261–265 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  127. Wilheim, T.: The smallest chemical rwaction system with bistability. BMC Syst. Biol. 3, e90 (2009)

    Article  Google Scholar 

  128. Wilheim, T., Heinrich, R.: Smallest chemical rwaction system with Hopf bifurcation. J. Math. Chem. 17, 1–14 (1995)

    Article  MathSciNet  Google Scholar 

  129. Wilkinson, D.J.: Stochastic Modelling for Systems Biology, 2nd edn. Chapman & Hall/CRC Press – Taylor and Francis Group, Boca Raton (2012)

    MATH  Google Scholar 

  130. 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)

    Article  MATH  Google Scholar 

  131. Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)

    Google Scholar 

  132. 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. Zhabotinsky, A.M.: A history of chemical oscillations and waves. Chaos 1, 379–386 (1991)

    Article  ADS  Google Scholar 

  134. Zwillinger, D.: Handbook of Differential Equations, 3rd edn. Academic Press, San Diego (1998)

    MATH  Google Scholar 

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Schuster, P. (2016). Applications in Biology. In: Stochasticity in Processes. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-39502-9_5

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