# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807
##### 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. 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. 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. 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. 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. 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). doi: 10.1142/S0218202511400069. [7] A. Belloquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,, Birkhäuser, (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). 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. 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. 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), 843. 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), 1405. 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. 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. [15] R. A. Gatenby, T. L. Vincent and R. J. Gillies, Evolutionary dynamics in carcinogenesis,, Math. Models Methods Appl. Sci., 15 (2005), 1619. 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. doi: 10.1142/S0218202510004519. [19] D. Hanahan and R.A. Weinberg, The hallmarks of cancer,, Cell, 100 (2000), 57. doi: 10.1016/S0092-8674(00)81683-9. [20] E. Mayr, What Evolution Is,, Basic Books, (2001). [21] M. A Nowak and K. Sigmund, Evolutionary dynamics of biological games,, Science, 303 (2004), 793. 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. 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). 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. 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. 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. 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. 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. 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. 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). doi: 10.1142/S0218202511400069. [7] A. Belloquid and M. Delitala, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,, Birkhäuser, (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). 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. 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. 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), 843. 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), 1405. 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. 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. [15] R. A. Gatenby, T. L. Vincent and R. J. Gillies, Evolutionary dynamics in carcinogenesis,, Math. Models Methods Appl. Sci., 15 (2005), 1619. 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. doi: 10.1142/S0218202510004519. [19] D. Hanahan and R.A. Weinberg, The hallmarks of cancer,, Cell, 100 (2000), 57. doi: 10.1016/S0092-8674(00)81683-9. [20] E. Mayr, What Evolution Is,, Basic Books, (2001). [21] M. A Nowak and K. Sigmund, Evolutionary dynamics of biological games,, Science, 303 (2004), 793. 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. 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). 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. doi: 10.1007/BF02409749.
 [1] Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 847-863. doi: 10.3934/dcdsb.2013.18.847 [2] Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895 [3] Martin Parisot, Mirosław Lachowicz. A kinetic model for the formation of swarms with nonlinear interactions. Kinetic & Related Models, 2016, 9 (1) : 131-164. doi: 10.3934/krm.2016.9.131 [4] N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59 [5] Nicolas Besse, Florent Berthelin, Yann Brenier, Pierre Bertrand. The multi-water-bag equations for collisionless kinetic modeling. Kinetic & Related Models, 2009, 2 (1) : 39-80. doi: 10.3934/krm.2009.2.39 [6] Michele Gianfelice, Enza Orlandi. Dynamics and kinetic limit for a system of noiseless $d$-dimensional Vicsek-type particles. Networks & Heterogeneous Media, 2014, 9 (2) : 269-297. doi: 10.3934/nhm.2014.9.269 [7] Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Networks & Heterogeneous Media, 2013, 8 (4) : 879-912. doi: 10.3934/nhm.2013.8.879 [8] Roberto Alicandro, Giuliano Lazzaroni, Mariapia Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Networks & Heterogeneous Media, 2018, 13 (1) : 1-26. doi: 10.3934/nhm.2018001 [9] Mats Gyllenberg, Yi Wang. Periodic tridiagonal systems modeling competitive-cooperative ecological interactions. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 289-298. doi: 10.3934/dcdsb.2005.5.289 [10] Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541 [11] Sarbaz H. A. Khoshnaw. Reduction of a kinetic model of active export of importins. Conference Publications, 2015, 2015 (special) : 705-722. doi: 10.3934/proc.2015.0705 [12] Jean-Philippe Bernard, Emmanuel Frénod, Antoine Rousseau. Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects. Conference Publications, 2013, 2013 (special) : 69-76. doi: 10.3934/proc.2013.2013.69 [13] Minbo Yang, Jianjun Zhang, Yimin Zhang. Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (2) : 493-512. doi: 10.3934/cpaa.2017025 [14] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [15] Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785 [16] MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 [17] Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233 [18] Robert A. Gatenby, B. Roy Frieden. The Role of Non-Genomic Information in Maintaining Thermodynamic Stability in Living Systems. Mathematical Biosciences & Engineering, 2005, 2 (1) : 43-51. doi: 10.3934/mbe.2005.2.43 [19] Alexander Mielke. Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 479-499. doi: 10.3934/dcdss.2013.6.479 [20] José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357

2018 Impact Factor: 0.545