2015, 12(2): 291-310. doi: 10.3934/mbe.2015.12.291

Basic stage structure measure valued evolutionary game model

1. 

University of Wisconsin-Richland, 1200 Hwy 14 West, Richland Center, WI 53581-1399, United States

Received  April 2014 Revised  September 2014 Published  December 2014

The ideas and techniques developed in [12,3] are extended to a basic stage structured model. Each strategy consists of two stages: a Juvenile (L for larvae), and Adult (A). A general model of this basic stage structure is formulated as a dynamical system on the state space of finite signed measures. Nonnegativity, well-posedness and uniform eventual boundedness are established under biologically natural conditions on the rates. Similar to [12] we also have the unifying of discrete and continuous systems and the containment of the classic nonlinearities.
Citation: John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291
References:
[1]

A. S. Ackleh, B. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 917.  doi: 10.3934/dcdsb.2005.5.917.  Google Scholar

[2]

A. S. Ackleh, D. F. Marshall, H. E. Heatherly and B. G. Fitzpatrick, Survival of the fittest in a generalized logistic model,, Math. Models Methods Appl. Sci., 9 (1999), 1379.  doi: 10.1142/S0218202599000610.  Google Scholar

[3]

A. S. Ackleh, J. Cleveland and H. Thieme, Selection mutation equations on measure spaces,, submitted JDE., ().   Google Scholar

[4]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis,, Springer-Verlag, (1994).  doi: 10.1007/978-3-662-03004-2.  Google Scholar

[5]

M. Barfield, R. Holt and R. Gomulkiewicz, Evolution in stage-structured populations,, The American Naturalist, 177 (2011), 397.  doi: 10.1086/658903.  Google Scholar

[6]

H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence,, J. Math. Biol., 27 (1989), 179.  doi: 10.1007/BF00276102.  Google Scholar

[7]

J. S. Brown and B. J. McGill, Evolutionary game theory and adaptive dynamics of continuous traits,, Ann. Rev. Ecol. Evol. Syst., 38 (2007), 403.   Google Scholar

[8]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics,, J. Math. Biol., 48 (2004), 135.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[9]

A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection mutation equations,, J. Math. Biol., 54 (2007), 489.  doi: 10.1007/s00285-006-0056-4.  Google Scholar

[10]

N. Champagnat, R. Ferriere and S. Meleard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models,, Theoretical Population Biology, 69 (2006), 297.  doi: 10.1016/j.tpb.2005.10.004.  Google Scholar

[11]

B. Charlesworth, Evolution in Age-Structured Populations,, Cambridge University Press, (1994).  doi: 10.1017/CBO9780511525711.  Google Scholar

[12]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness,, Nonlinear Anal. Real World Appl., 14 (2013), 785.  doi: 10.1016/j.nonrwa.2012.08.002.  Google Scholar

[13]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources,, Math. Model. Nat. Phenom., 1 (2006), 65.  doi: 10.1051/mmnp:2006004.  Google Scholar

[14]

S. C. Hille and D. H. T. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures,, Integr. Equ. Oper. Theory, 63 (2009), 351.  doi: 10.1007/s00020-008-1652-z.  Google Scholar

[15]

P. Jabin and G. Raoul, On selection dynamics for competitive interactions,, Journal of mathematical biology, 63 (2011), 493.  doi: 10.1007/s00285-010-0370-8.  Google Scholar

[16]

G. P. Karev, A. S. Novozhilov and E. V. Koonin, Mathematical modeling of tumor therapy with oncolytic viruses: Effects of parametric heterogeneity on cell dynamics,, Biology Direct, 1 (2006), 1.   Google Scholar

[17]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters,, PNAS, 54 (1965), 731.  doi: 10.1073/pnas.54.3.731.  Google Scholar

[18]

R. Lande, A quantitative genetic theory of life history evolution,, Ecology, 63 (1982), 607.   Google Scholar

[19]

S. Lang, Undergraduate Analysis,, Secaucus, (1983).  doi: 10.1007/978-1-4757-1801-0.  Google Scholar

[20]

M. A. Nowak, Evolutionary Dynamics,, Belknap Press, (2006).   Google Scholar

[21]

B. Perthame, Transport Equation in Biology,, Frontiers in Mathematics series, (2007).   Google Scholar

[22]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations,, Monatsh. Math., 165 (2012), 117.  doi: 10.1007/s00605-011-0354-9.  Google Scholar

[23]

G. Raoul, Long time evolution of populations under selection and vanishing mutations,, Acta Appl. Math., 114 (2011), 1.  doi: 10.1007/s10440-011-9603-0.  Google Scholar

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.  doi: 10.1038/246015a0.  Google Scholar

[25]

H. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003).   Google Scholar

[26]

H. R. Thieme and J. Yang, An endemic model with variable re-infection rate and application to influenza,, Math. Biosci., 180 (2002), 207.  doi: 10.1016/S0025-5564(02)00102-5.  Google Scholar

show all references

References:
[1]

A. S. Ackleh, B. G. Fitzpatrick and H. R. Thieme, Rate distributions and survival of the fittest: A formulation on the space of measures,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 917.  doi: 10.3934/dcdsb.2005.5.917.  Google Scholar

[2]

A. S. Ackleh, D. F. Marshall, H. E. Heatherly and B. G. Fitzpatrick, Survival of the fittest in a generalized logistic model,, Math. Models Methods Appl. Sci., 9 (1999), 1379.  doi: 10.1142/S0218202599000610.  Google Scholar

[3]

A. S. Ackleh, J. Cleveland and H. Thieme, Selection mutation equations on measure spaces,, submitted JDE., ().   Google Scholar

[4]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis,, Springer-Verlag, (1994).  doi: 10.1007/978-3-662-03004-2.  Google Scholar

[5]

M. Barfield, R. Holt and R. Gomulkiewicz, Evolution in stage-structured populations,, The American Naturalist, 177 (2011), 397.  doi: 10.1086/658903.  Google Scholar

[6]

H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence,, J. Math. Biol., 27 (1989), 179.  doi: 10.1007/BF00276102.  Google Scholar

[7]

J. S. Brown and B. J. McGill, Evolutionary game theory and adaptive dynamics of continuous traits,, Ann. Rev. Ecol. Evol. Syst., 38 (2007), 403.   Google Scholar

[8]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics,, J. Math. Biol., 48 (2004), 135.  doi: 10.1007/s00285-003-0226-6.  Google Scholar

[9]

A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection mutation equations,, J. Math. Biol., 54 (2007), 489.  doi: 10.1007/s00285-006-0056-4.  Google Scholar

[10]

N. Champagnat, R. Ferriere and S. Meleard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models,, Theoretical Population Biology, 69 (2006), 297.  doi: 10.1016/j.tpb.2005.10.004.  Google Scholar

[11]

B. Charlesworth, Evolution in Age-Structured Populations,, Cambridge University Press, (1994).  doi: 10.1017/CBO9780511525711.  Google Scholar

[12]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness,, Nonlinear Anal. Real World Appl., 14 (2013), 785.  doi: 10.1016/j.nonrwa.2012.08.002.  Google Scholar

[13]

S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources,, Math. Model. Nat. Phenom., 1 (2006), 65.  doi: 10.1051/mmnp:2006004.  Google Scholar

[14]

S. C. Hille and D. H. T. Worm, Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures,, Integr. Equ. Oper. Theory, 63 (2009), 351.  doi: 10.1007/s00020-008-1652-z.  Google Scholar

[15]

P. Jabin and G. Raoul, On selection dynamics for competitive interactions,, Journal of mathematical biology, 63 (2011), 493.  doi: 10.1007/s00285-010-0370-8.  Google Scholar

[16]

G. P. Karev, A. S. Novozhilov and E. V. Koonin, Mathematical modeling of tumor therapy with oncolytic viruses: Effects of parametric heterogeneity on cell dynamics,, Biology Direct, 1 (2006), 1.   Google Scholar

[17]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters,, PNAS, 54 (1965), 731.  doi: 10.1073/pnas.54.3.731.  Google Scholar

[18]

R. Lande, A quantitative genetic theory of life history evolution,, Ecology, 63 (1982), 607.   Google Scholar

[19]

S. Lang, Undergraduate Analysis,, Secaucus, (1983).  doi: 10.1007/978-1-4757-1801-0.  Google Scholar

[20]

M. A. Nowak, Evolutionary Dynamics,, Belknap Press, (2006).   Google Scholar

[21]

B. Perthame, Transport Equation in Biology,, Frontiers in Mathematics series, (2007).   Google Scholar

[22]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations,, Monatsh. Math., 165 (2012), 117.  doi: 10.1007/s00605-011-0354-9.  Google Scholar

[23]

G. Raoul, Long time evolution of populations under selection and vanishing mutations,, Acta Appl. Math., 114 (2011), 1.  doi: 10.1007/s10440-011-9603-0.  Google Scholar

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15.  doi: 10.1038/246015a0.  Google Scholar

[25]

H. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003).   Google Scholar

[26]

H. R. Thieme and J. Yang, An endemic model with variable re-infection rate and application to influenza,, Math. Biosci., 180 (2002), 207.  doi: 10.1016/S0025-5564(02)00102-5.  Google Scholar

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