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.

[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.

[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. 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. 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.

[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.

[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.

[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.

[11]

B. Charlesworth, Evolution in Age-Structured Populations,, Cambridge University Press, (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. 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. 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. 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. 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.

[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.

[18]

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

[19]

S. Lang, Undergraduate Analysis,, Secaucus, (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, (2007).

[22]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations,, Monatsh. Math., 165 (2012), 117. 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. doi: 10.1007/s10440-011-9603-0.

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15. 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. 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. 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. 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. 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. 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.

[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.

[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.

[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.

[11]

B. Charlesworth, Evolution in Age-Structured Populations,, Cambridge University Press, (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. 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. 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. 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. 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.

[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.

[18]

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

[19]

S. Lang, Undergraduate Analysis,, Secaucus, (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, (2007).

[22]

G. Raoul, Local stability of evolutionary attractors for continuous structured populations,, Monatsh. Math., 165 (2012), 117. 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. doi: 10.1007/s10440-011-9603-0.

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict,, Nature, 246 (1973), 15. 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. doi: 10.1016/S0025-5564(02)00102-5.

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