August  2014, 7(4): 807-821. doi: 10.3934/dcdss.2014.7.807

On an initial value problem modeling evolution and selection in living systems

1. 

Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy, Italy

Received  September 2013 Revised  November 2013 Published  February 2014

This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is globally proliferative, see Theorem 3.3.
Citation: Patrizia Pucci, Maria Cesarina Salvatori. On an initial value problem modeling evolution and selection in living systems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807
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D. L. Abel and J. T. Trevors, Self-organization vs. self-ordering events in life-origin models, Physics of Life Reviews, 3 (2006), 221-228. doi: 10.1016/j.plrev.2006.07.003.

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G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems reprsentation, Kinet. Relat. Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249.

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N. Bellomo and B. Carbonaro, Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives, Physics of Life Reviews, 8 (2011), 1-18. doi: 10.1016/j.plrev.2010.12.001.

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N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity,'' and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X.

[5]

N. Bellomo, L. Preziosi and G. Forni, On a kinetic (cellular) theory for competition between tumors and the host immune system, J. Biological Systems, 4 (1996), 479-502. doi: 10.1142/S0218339096000326.

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), suppl. 1, 1140006, 29 pp. doi: 10.1142/S0218202511400069.

[7]

A. Belloquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Birkhäuser, Boston, 2006. doi: 10.1007/978-0-8176-4503-8.

[8]

A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), suppl. 1, 1140003, 35 pp. doi: 10.1142/S0218202511400033.

[9]

A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology, Math. Models Methods Appl. Sci., 23 (2013), 949-978. doi: 10.1142/S0218202512500650.

[10]

E. L. Cooper, Evolution of immune system from self/not self to danger to artificial immune system (AIS), Physics of Life Reviews, 7 (2010), 55-78. doi: 10.1016/j.plrev.2009.12.001.

[11]

M. Delitala, P. Pucci and M.C. Salvatori, From methods of the mathematical kinetic theory for active particles to modelling virus mutations, Math. Models Methods Appl. Sci., 21 (2011), suppl. 1, 843-870. doi: 10.1142/S0218202511005398.

[12]

S. De Lillo, M. Delitala and M. C. Salvatori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), suppl. 1, 1405-1425. doi: 10.1142/S0218202509003838.

[13]

K. Drucis, M. Kolev, W. Majda and B. Zubik-Kowal, Nonlinear modeling with mammographic evidence of carcinoma, Nonlinear Anal. Real World Appl., 11 (2010), 4326-4334. doi: 10.1016/j.nonrwa.2010.05.017.

[14]

R. A. Gatenby and T. L. Vincent, An evolutionary model of carcinogenesis, Cancer Research, 63 (2003), 6212-6220.

[15]

R. A. Gatenby, T. L. Vincent and R. J. Gillies, Evolutionary dynamics in carcinogenesis, Math. Models Methods Appl. Sci., 15 (2005), 1619-1638. doi: 10.1142/S0218202505000911.

[16]

D. H. Erwin, Extintion: How Life on Earth Nearly Ended 250 Million Years Ago, Princeton Univ. Press, 2006.

[17]

S. A. Frank, Dynamics of Cancer: Incidence, Inheritance and Evolution, Princeton University Press, 2007.

[18]

E. Gabetta and E. Regazzini, About the gene families size distribution in a recent model of genome evolution, Math. Models Methods Appl. Sci., 20 (2010), 1005-1020. doi: 10.1142/S0218202510004519.

[19]

D. Hanahan and R.A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 57-70. doi: 10.1016/S0092-8674(00)81683-9.

[20]

E. Mayr, What Evolution Is, Basic Books, New York, 2001.

[21]

M. A Nowak and K. Sigmund, Evolutionary dynamics of biological games, Science, 303 (2004), 793-799. doi: 10.1126/science.1093411.

[22]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, Harvard Univ. Press, 2006.

[23]

F. C. Santos, J. M. Pacheco and T. Lenaerts, Evolutionary dynamics of social dilemmas in structured heterogeneous populations, Proceeding of the National Academy of Sciences, 103 (2006), 3490-3494. doi: 10.1073/pnas.0508201103.

[24]

F. C. Santos, V. V. Vasconcelos, M. D. Santos, P. N. B. Neves and J. M. Pacheco, Evolutionary dynamics of climate change under collective-risk dilemmas, Math. Models Methods Appl. Sci., 22 (2012), 1140004, 17 pp. doi: 10.1142/S0218202511400045.

[25]

F. J. Weissing, Genetic versus phenotypic models of selection: Can genetics be neglected in a long-term perspective? J. Math. Biol., 34 (1996), 533-555. doi: 10.1007/BF02409749.

show all references

References:
[1]

D. L. Abel and J. T. Trevors, Self-organization vs. self-ordering events in life-origin models, Physics of Life Reviews, 3 (2006), 221-228. doi: 10.1016/j.plrev.2006.07.003.

[2]

G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems reprsentation, Kinet. Relat. Models, 1 (2008), 249-278. doi: 10.3934/krm.2008.1.249.

[3]

N. Bellomo and B. Carbonaro, Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives, Physics of Life Reviews, 8 (2011), 1-18. doi: 10.1016/j.plrev.2010.12.001.

[4]

N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity,'' and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913. doi: 10.1142/S021820251350053X.

[5]

N. Bellomo, L. Preziosi and G. Forni, On a kinetic (cellular) theory for competition between tumors and the host immune system, J. Biological Systems, 4 (1996), 479-502. doi: 10.1142/S0218339096000326.

[6]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), suppl. 1, 1140006, 29 pp. doi: 10.1142/S0218202511400069.

[7]

A. Belloquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Birkhäuser, Boston, 2006. doi: 10.1007/978-0-8176-4503-8.

[8]

A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci., 22 (2012), suppl. 1, 1140003, 35 pp. doi: 10.1142/S0218202511400033.

[9]

A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology, Math. Models Methods Appl. Sci., 23 (2013), 949-978. doi: 10.1142/S0218202512500650.

[10]

E. L. Cooper, Evolution of immune system from self/not self to danger to artificial immune system (AIS), Physics of Life Reviews, 7 (2010), 55-78. doi: 10.1016/j.plrev.2009.12.001.

[11]

M. Delitala, P. Pucci and M.C. Salvatori, From methods of the mathematical kinetic theory for active particles to modelling virus mutations, Math. Models Methods Appl. Sci., 21 (2011), suppl. 1, 843-870. doi: 10.1142/S0218202511005398.

[12]

S. De Lillo, M. Delitala and M. C. Salvatori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), suppl. 1, 1405-1425. doi: 10.1142/S0218202509003838.

[13]

K. Drucis, M. Kolev, W. Majda and B. Zubik-Kowal, Nonlinear modeling with mammographic evidence of carcinoma, Nonlinear Anal. Real World Appl., 11 (2010), 4326-4334. doi: 10.1016/j.nonrwa.2010.05.017.

[14]

R. A. Gatenby and T. L. Vincent, An evolutionary model of carcinogenesis, Cancer Research, 63 (2003), 6212-6220.

[15]

R. A. Gatenby, T. L. Vincent and R. J. Gillies, Evolutionary dynamics in carcinogenesis, Math. Models Methods Appl. Sci., 15 (2005), 1619-1638. doi: 10.1142/S0218202505000911.

[16]

D. H. Erwin, Extintion: How Life on Earth Nearly Ended 250 Million Years Ago, Princeton Univ. Press, 2006.

[17]

S. A. Frank, Dynamics of Cancer: Incidence, Inheritance and Evolution, Princeton University Press, 2007.

[18]

E. Gabetta and E. Regazzini, About the gene families size distribution in a recent model of genome evolution, Math. Models Methods Appl. Sci., 20 (2010), 1005-1020. doi: 10.1142/S0218202510004519.

[19]

D. Hanahan and R.A. Weinberg, The hallmarks of cancer, Cell, 100 (2000), 57-70. doi: 10.1016/S0092-8674(00)81683-9.

[20]

E. Mayr, What Evolution Is, Basic Books, New York, 2001.

[21]

M. A Nowak and K. Sigmund, Evolutionary dynamics of biological games, Science, 303 (2004), 793-799. doi: 10.1126/science.1093411.

[22]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, Harvard Univ. Press, 2006.

[23]

F. C. Santos, J. M. Pacheco and T. Lenaerts, Evolutionary dynamics of social dilemmas in structured heterogeneous populations, Proceeding of the National Academy of Sciences, 103 (2006), 3490-3494. doi: 10.1073/pnas.0508201103.

[24]

F. C. Santos, V. V. Vasconcelos, M. D. Santos, P. N. B. Neves and J. M. Pacheco, Evolutionary dynamics of climate change under collective-risk dilemmas, Math. Models Methods Appl. Sci., 22 (2012), 1140004, 17 pp. doi: 10.1142/S0218202511400045.

[25]

F. J. Weissing, Genetic versus phenotypic models of selection: Can genetics be neglected in a long-term perspective? J. Math. Biol., 34 (1996), 533-555. doi: 10.1007/BF02409749.

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