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A note on modelling with measures: Two-features balance equations
Basic stage structure measure valued evolutionary game model
1. | University of Wisconsin-Richland, 1200 Hwy 14 West, Richland Center, WI 53581-1399, United States |
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] | |
[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] | |
[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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