American Institute of Mathematical Sciences

Advanced
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.  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), 1379-1391. 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), 397-409. 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), 179-190. 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), 403-435. 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), 135-159. 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), 489-511. 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), 297-321. doi: 10.1016/j.tpb.2005.10.004.  Google Scholar

[11]

B. Charlesworth, Evolution in Age-Structured Populations, Cambridge University Press, Cambridge, 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), 785-797. 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), 65-82. 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), 351-371. 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), 493-517. 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), 1-19. Google Scholar

[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.  Google Scholar

[18]

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

[19]

S. Lang, Undergraduate Analysis, Secaucus, New Jersey, Springer Verlag, 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, Birkhauser, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. 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), 207-235. 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), 917-928. 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), 1379-1391. 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), 397-409. 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), 179-190. 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), 403-435. 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), 135-159. 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), 489-511. 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), 297-321. doi: 10.1016/j.tpb.2005.10.004.  Google Scholar

[11]

B. Charlesworth, Evolution in Age-Structured Populations, Cambridge University Press, Cambridge, 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), 785-797. 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), 65-82. 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), 351-371. 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), 493-517. 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), 1-19. Google Scholar

[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.  Google Scholar

[18]

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

[19]

S. Lang, Undergraduate Analysis, Secaucus, New Jersey, Springer Verlag, 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, Birkhauser, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Maynard Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. 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), 207-235. doi: 10.1016/S0025-5564(02)00102-5.  Google Scholar

[1]

Pierre-Emmanuel Jabin. Small populations corrections for selection-mutation models. Networks & Heterogeneous Media, 2012, 7 (4) : 805-836. doi: 10.3934/nhm.2012.7.805
[2]

Azmy S. Ackleh, Shuhua Hu. Comparison between stochastic and deterministic selection-mutation models. Mathematical Biosciences & Engineering, 2007, 4 (2) : 133-157. doi: 10.3934/mbe.2007.4.133
[3]

Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169
[4]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[5]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187
[6]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361
[7]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[8]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[9]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2039-2064. doi: 10.3934/cpaa.2021057
[10]

J. M. Cushing, Simon Maccracken Stump. Darwinian dynamics of a juvenile-adult model. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1017-1044. doi: 10.3934/mbe.2013.10.1017
[11]

Andrew R. Becklin, Mohammad A. Rammaha. Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms. Evolution Equations & Control Theory, 2021, 10 (4) : 797-836. doi: 10.3934/eect.2020093
[12]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[13]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095
[14]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
[15]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737
[16]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
[17]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
[18]

Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531
[19]

Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015
[20]

Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy