zbMATH — the first resource for mathematics

Discrete time spatial models arising in genetics, evolutionary game theory, and branching processes. (English) Zbl 0882.92022
Summary: A saddle point method is used to obtain the speed of first spread of new genotypes in genetic models and of new strategies in game theoretic models. It is also used to obtain the speed of the forward tail of the distribution of farthest spread for branching process models. The technique is applicable to a wide range of models. They include multiple allele and sex-linked models in genetics, multistrategy and bimatrix evolutionary games, and multitype and demographic branching processes. The speed of propagation has been obtained for genetic models (in simple cases only) by H. F. Weinberger [Lect. Notes Math. 648, 47-98 (1978; Zbl 0383.35034); SIAM J. Math. Anal. 13, 353-396 (1982; Zbl 0529.92010)] and R. Lui [SIAM J. Math. Anal. 13, No. 6, 913-953 (1982; Zbl 0508.45006 and 007); J. Math. Biol. 16, 199-220 (1983; Zbl 0514.45006); Math. Biosci. 93, No.2, 269-312 (1989; Zbl 0706.92014 and 015)], using exact analytical methods. The exact results were obtained only for two-allele, single-locus genetic models. The saddle point method agrees in these very simple cases with the results obtained by using the exact analytic methods. Of course, it can also be used in much more general situations far less tractable to exact analysis. The connection between genetic and game theoretic models is also briefly considered, as is the extent to which the exact analytic methods yield results for simple models in game theory.
Reviewer: Reviewer (Berlin)

92D10 Genetics and epigenetics
91A40 Other game-theoretic models
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
Full Text: DOI
[1] Weinberger, H.F., Asymptotic behavior of a model in population genetics, (), 47-98 · Zbl 0383.35034
[2] Weinberger, H.F., Long-time behavior of a class of biological models, SIAM J. math. anal., 13, 353-396, (1982) · Zbl 0529.92010
[3] Lui, R., A nonlinear integral operator arising from a model in population genetics, I: monotone initial data, SIAM J. math. anal., 13, 6, 913-937, (1982) · Zbl 0508.45006
[4] Lui, R., A nonlinear integral operator arising from a model in population genetics, II: initial data with compact support, SIAM J. math. anal., 13, 6, 938-953, (1982) · Zbl 0508.45007
[5] Lui, R., Existence and stability of travelling wave solutions of a nonlinear integral operator, J. math. biol., 16, 199-220, (1983) · Zbl 0514.45006
[6] Lui, R., Biological growth and spread modeled by systems of recursions, I: mathematical theory, Math. biosci., 93, 269-295, (1989) · Zbl 0706.92014
[7] Lui, R., Biological growth and spread modeled by systems of recursions, II: biological theory, Math. biosci., 93, 297-312, (1989) · Zbl 0706.92015
[8] Daniels, H.E., The deterministic spread of a simple epidemic, (), 689-701, distributed for Applied Probability Trust by Academic Press, London · Zbl 0388.60086
[9] Daniels, H.E., The advancing wave in a spatial birth process, J. appl. probability, 14, 689-701, (1977) · Zbl 0388.60086
[10] Radcliffe, J.; Rass, L., Saddle-point approximations in n-type epidemics and contact birth processes, Rocky mt. J. math., 14, 3, 599-617, (1984) · Zbl 0587.92021
[11] Radcliffe, J.; Rass, L., Reducible epidemics: choosing your saddle, Rocky mt. J. math., 23, 2, 725-752, (1993) · Zbl 0801.92022
[12] Aronson, D.G., The asymptotic speed of propagation of a simple epidemic, (), 1-23 · Zbl 0361.35011
[13] Diekmann, O., Run for your life: a note on the asymptotic speed of propagation of an epidemic, J. differential equations, 33, 1, 58-73, (1979) · Zbl 0377.45007
[14] Radcliffe, J.; Rass, L., The asymptotic speed of propagation of the deterministic nonreducible n-type epidemic, J. math. biol., 23, 341-359, (1986) · Zbl 0606.92019
[15] Radcliffe, J.; Rass, L., The asymptotic behavior of a reducible system of nonlinear integral equations, Rocky mt. J. math., 26, 2, 731-752, (1996) · Zbl 0885.92028
[16] Radcliffe, J.; Rass, L., Spatial branching and epidemic processes, (), 147-170 · Zbl 0882.92022
[17] Radcliffe, J.; Rass, L., Multitype contact branching processes, (), 169-179 · Zbl 0823.60076
[18] Ewens, W.J., Mathematical population genetics, (1979), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0422.92011
[19] Creegan, P.; Lui, R., Some remarks about the wave speed and traveling wave solutions of a nonlinear integral operator, J. math. biol., 20, 59-68, (1984) · Zbl 0556.45002
[20] Diekmann, O.; Kaper, H.G., On the bounded solutions of a nonlinear convolution equation, Nonlinear anal. theory appl., 2, 6, 721-737, (1978) · Zbl 0433.92028
[21] Maynard Smith, J., Evolution and the theory of games, (1982), Cambridge University Press · Zbl 0526.90102
[22] Zeeman, E.C., Dynamics of the evolution of animal conflicts, J. theor. biol., 89, 249-270, (1981)
[23] Hofbaeur, J.; Sigmund, K., The theory of evolution and dynamical systems, ()
[24] Hutson, V.C.L.; Vickers, G.T., Traveling waves and dominance of ESS’s, J. math. biol., 30, 457-471, (1992) · Zbl 0763.92007
[25] Vickers, G.T., Spatial patterns and ESS’s, J. theor. biol., 140, 129-135, (1989)
[26] Mode, C.J., Multitype branching processes theory and applications, (1971), American Elsevier New York · Zbl 0219.60061
[27] Dieudonné, J., Foundations of modern analysis, (1969), Chelsea New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.