American Institute of Mathematical Sciences

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

Basic stage structure measure valued evolutionary game model

John Cleveland 1,

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.
Keywords: continuous dependence, well-posedness, Evolutionary game models, stage structure., selection-mutation, juvenile adult, space of finite signed measure.
Mathematics Subject Classification: Primary: 91A22, 34G20, 37C25, 92D2.
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), 917-928. doi: 10.3934/dcdsb.2005.5.917.

[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), 1379-1391. doi: 10.1142/S0218202599000610.

[3]

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

[4]

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

[5]

M. Barfield, R. Holt and R. Gomulkiewicz, Evolution in stage-structured populations, The American Naturalist, 177 (2011), 397-409. doi: 10.1086/658903.

[6]

H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol., 27 (1989), 179-190. doi: 10.1007/BF00276102.

[7]

J. S. Brown and B. J. McGill, Evolutionary game theory and adaptive dynamics of continuous traits, Ann. Rev. Ecol. Evol. Syst., 38 (2007), 403-435.

[8]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol., 48 (2004), 135-159. doi: 10.1007/s00285-003-0226-6.

[9]

A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection mutation equations, J. Math. Biol., 54 (2007), 489-511. doi: 10.1007/s00285-006-0056-4.

[10]

N. Champagnat, R. Ferriere and S. Meleard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theoretical Population Biology, 69 (2006), 297-321. doi: 10.1016/j.tpb.2005.10.004.

[11]

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

[12]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, Nonlinear Anal. Real World Appl., 14 (2013), 785-797. doi: 10.1016/j.nonrwa.2012.08.002.

[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), 65-82. doi: 10.1051/mmnp:2006004.

[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), 351-371. doi: 10.1007/s00020-008-1652-z.

[15]

P. Jabin and G. Raoul, On selection dynamics for competitive interactions, Journal of mathematical biology, 63 (2011), 493-517. doi: 10.1007/s00285-010-0370-8.

[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), 1-19.

[17]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, PNAS, 54 (1965), 731-736. doi: 10.1073/pnas.54.3.731.

[18]

R. Lande, A quantitative genetic theory of life history evolution, Ecology, 63 (1982), 607-615.

[19]

S. Lang, Undergraduate Analysis, Secaucus, New Jersey, Springer Verlag, 1983. doi: 10.1007/978-1-4757-1801-0.

[20]

M. A. Nowak, Evolutionary Dynamics, Belknap Press, 2006.

[21]

B. Perthame, Transport Equation in Biology, Frontiers in Mathematics series, Birkhauser, 2007.

[22]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations, Monatsh. Math., 165 (2012), 117-144. doi: 10.1007/s00605-011-0354-9.

[23]

G. Raoul, Long time evolution of populations under selection and vanishing mutations, Acta Appl. Math., 114 (2011), 1-14. doi: 10.1007/s10440-011-9603-0.

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0.

[25]

H. Thieme, Mathematics in Population Biology, Princeton University Press, 2003.

[26]

H. R. Thieme and J. Yang, An endemic model with variable re-infection rate and application to influenza, Math. Biosci., 180 (2002), 207-235. doi: 10.1016/S0025-5564(02)00102-5.

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), 917-928. doi: 10.3934/dcdsb.2005.5.917.

[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), 1379-1391. doi: 10.1142/S0218202599000610.

[3]

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

[4]

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

[5]

M. Barfield, R. Holt and R. Gomulkiewicz, Evolution in stage-structured populations, The American Naturalist, 177 (2011), 397-409. doi: 10.1086/658903.

[6]

H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol., 27 (1989), 179-190. doi: 10.1007/BF00276102.

[7]

J. S. Brown and B. J. McGill, Evolutionary game theory and adaptive dynamics of continuous traits, Ann. Rev. Ecol. Evol. Syst., 38 (2007), 403-435.

[8]

A. Calsina and S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol., 48 (2004), 135-159. doi: 10.1007/s00285-003-0226-6.

[9]

A. Calsina and S. Cuadrado, Asymptotic stability of equilibria of selection mutation equations, J. Math. Biol., 54 (2007), 489-511. doi: 10.1007/s00285-006-0056-4.

[10]

N. Champagnat, R. Ferriere and S. Meleard, Unifying evolutionary dynamics: From individual stochastic processes to macroscopic models, Theoretical Population Biology, 69 (2006), 297-321. doi: 10.1016/j.tpb.2005.10.004.

[11]

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

[12]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, Nonlinear Anal. Real World Appl., 14 (2013), 785-797. doi: 10.1016/j.nonrwa.2012.08.002.

[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), 65-82. doi: 10.1051/mmnp:2006004.

[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), 351-371. doi: 10.1007/s00020-008-1652-z.

[15]

P. Jabin and G. Raoul, On selection dynamics for competitive interactions, Journal of mathematical biology, 63 (2011), 493-517. doi: 10.1007/s00285-010-0370-8.

[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), 1-19.

[17]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, PNAS, 54 (1965), 731-736. doi: 10.1073/pnas.54.3.731.

[18]

R. Lande, A quantitative genetic theory of life history evolution, Ecology, 63 (1982), 607-615.

[19]

S. Lang, Undergraduate Analysis, Secaucus, New Jersey, Springer Verlag, 1983. doi: 10.1007/978-1-4757-1801-0.

[20]

M. A. Nowak, Evolutionary Dynamics, Belknap Press, 2006.

[21]

B. Perthame, Transport Equation in Biology, Frontiers in Mathematics series, Birkhauser, 2007.

[22]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations, Monatsh. Math., 165 (2012), 117-144. doi: 10.1007/s00605-011-0354-9.

[23]

G. Raoul, Long time evolution of populations under selection and vanishing mutations, Acta Appl. Math., 114 (2011), 1-14. doi: 10.1007/s10440-011-9603-0.

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0.

[25]

H. Thieme, Mathematics in Population Biology, Princeton University Press, 2003.

[26]

H. R. Thieme and J. Yang, An endemic model with variable re-infection rate and application to influenza, Math. Biosci., 180 (2002), 207-235. doi: 10.1016/S0025-5564(02)00102-5.

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