# American Institute of Mathematical Sciences

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), 917-928. 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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##### 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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