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March  2014, 7(1): 109-119. doi: 10.3934/krm.2014.7.109

Stability of solutions of kinetic equations corresponding to the replicator dynamics

Mirosław Lachowicz 1, , Andrea Quartarone 2, and  Tatiana V. Ryabukha 1,

1. 

Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland, Poland
2. 

Scienze Matematiche e Informatiche, Universitá di Messina, Dipartimento di Matematica, Viale F. Stagno D’Alcontres, Messina 98166, Italy

Received  January 2013 Revised  April 2013 Published  December 2013

In the present paper we propose a class of kinetic type equations that describes the replicator dynamics at the mesoscopic level. The equations are highly nonlinear due to the dependence of the transition rates of distribution function. Under suitable assumptions we show the asymptotic (exponential) stability of the solutions to such kinetic equations.
Keywords: stability., integro--differential equations, Replicator equation, mesoscopic models.
Mathematics Subject Classification: 60J75, 92D25, 35Q92, 35R09, 37N25, 45K0.
Citation: Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109
References:
[1]

L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Letters, 25 (2012), 490-495. doi: 10.1016/j.aml.2011.09.043.  Google Scholar

[2]

J. Banasiak, V. Capasso, M. A. J. Chaplain, M. Lachowicz and J. Miękisz, Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic, Lecture Notes in Mathematics, 1940, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78362-6.  Google Scholar

[3]

N. Bellomo and B. Carbonaro, Toward a mathematical theory of living system focusing on developmental biology and evolution: A review and prospectives, Phys. Life Rev., 8 (2011), 1-18. Google Scholar

[4]

N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479. doi: 10.3934/krm.2013.6.459.  Google Scholar

[5]

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

[6]

C. Cattani and A. Ciancio, Hybrid two scales mathematical tools for active particles modelling complex systems with learning hiding dynamics, Math. Models Methods Appl. Sci., 17 (2007), 171-187. doi: 10.1142/S0218202507001875.  Google Scholar

[7]

A. Ciancio and A. Quartarone, A hibrid model for tumor-immune competition, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75 (2013), 125-136. Google Scholar

[8]

E. Carlen, P. Degond and B. Wennberg, Kinetic limits for pair-interaction driven master equation and biological swarm models, Math. Models Methods Appl. Sci., 23 (2012), 1339-1376. doi: 10.1142/S0218202513500115.  Google Scholar

[9]

R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press Series on Economic Learning and Social Evolution, 5, MIT Press, Cambridge, MA, 2003.  Google Scholar

[10]

R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Popul. Biol., 46 (1994), 363-394. doi: 10.1006/tpbi.1994.1032.  Google Scholar

[11]

Evolutionary Game Theory, Stanford Encyclopedia of Philosophy, 2009., Available from: , ().   Google Scholar

[12]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1934. Google Scholar

[13]

C. Hilbe, Local replicator dynamics: A simple link between deterministic and stochastic models of evolutionary game theory, Bull. Math. Biol., 73 (2011), 2068-2087. doi: 10.1007/s11538-010-9608-2.  Google Scholar

[14]

J. Hofbauer, P. Schuster and K. Sigmund, A note on evolutionary strategy and game dynamics, J. Theory Biol., 81 (1979), 609-612. doi: 10.1016/0022-5193(79)90058-4.  Google Scholar

[15]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  Google Scholar

[16]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N. S.), 40 (2003), 479-519. doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[17]

A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032.  Google Scholar

[18]

A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Math. Comput. Model., 51 (2010), 572-591. doi: 10.1016/j.mcm.2009.11.005.  Google Scholar

[19]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 54-60. doi: 10.1016/j.probengmech.2010.06.007.  Google Scholar

[20]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Anal. Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014.  Google Scholar

[21]

M. Lachowicz and D. Wrzosek, Nonlocal bilinear equations. Equilibrium solutions and diffusive limit, Math. Models Methods Appl. Sci., 11 (2001), 1393-1409. doi: 10.1142/S0218202501001380.  Google Scholar

[22]

M. Lachowicz and A. Quartarone, A general framework for modeling tumor-immune system competition at the mesoscopic level, Appl. Math. Letters, 25 (2012), 2118-2122. doi: 10.1016/j.aml.2012.04.021.  Google Scholar

[23]

M. Lachowicz and T. Ryabukha, Equilibrium solutions for microscopic stochastic systems in population dynamics, Math. Biosci. Eng., 10 (2013), 777-786. doi: 10.3934/mbe.2013.10.777.  Google Scholar

[24]

M. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006.  Google Scholar

[25]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[26]

J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.  Google Scholar

show all references

References:
[1]

L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Letters, 25 (2012), 490-495. doi: 10.1016/j.aml.2011.09.043.  Google Scholar

[2]

J. Banasiak, V. Capasso, M. A. J. Chaplain, M. Lachowicz and J. Miękisz, Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic, Lecture Notes in Mathematics, 1940, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78362-6.  Google Scholar

[3]

N. Bellomo and B. Carbonaro, Toward a mathematical theory of living system focusing on developmental biology and evolution: A review and prospectives, Phys. Life Rev., 8 (2011), 1-18. Google Scholar

[4]

N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts: Looking for the black swan, Kinet. Relat. Models, 6 (2013), 459-479. doi: 10.3934/krm.2013.6.459.  Google Scholar

[5]

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

[6]

C. Cattani and A. Ciancio, Hybrid two scales mathematical tools for active particles modelling complex systems with learning hiding dynamics, Math. Models Methods Appl. Sci., 17 (2007), 171-187. doi: 10.1142/S0218202507001875.  Google Scholar

[7]

A. Ciancio and A. Quartarone, A hibrid model for tumor-immune competition, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75 (2013), 125-136. Google Scholar

[8]

E. Carlen, P. Degond and B. Wennberg, Kinetic limits for pair-interaction driven master equation and biological swarm models, Math. Models Methods Appl. Sci., 23 (2012), 1339-1376. doi: 10.1142/S0218202513500115.  Google Scholar

[9]

R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press Series on Economic Learning and Social Evolution, 5, MIT Press, Cambridge, MA, 2003.  Google Scholar

[10]

R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Popul. Biol., 46 (1994), 363-394. doi: 10.1006/tpbi.1994.1032.  Google Scholar

[11]

Evolutionary Game Theory, Stanford Encyclopedia of Philosophy, 2009., Available from: , ().   Google Scholar

[12]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1934. Google Scholar

[13]

C. Hilbe, Local replicator dynamics: A simple link between deterministic and stochastic models of evolutionary game theory, Bull. Math. Biol., 73 (2011), 2068-2087. doi: 10.1007/s11538-010-9608-2.  Google Scholar

[14]

J. Hofbauer, P. Schuster and K. Sigmund, A note on evolutionary strategy and game dynamics, J. Theory Biol., 81 (1979), 609-612. doi: 10.1016/0022-5193(79)90058-4.  Google Scholar

[15]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  Google Scholar

[16]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bull. Amer. Math. Soc. (N. S.), 40 (2003), 479-519. doi: 10.1090/S0273-0979-03-00988-1.  Google Scholar

[17]

A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032.  Google Scholar

[18]

A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Math. Comput. Model., 51 (2010), 572-591. doi: 10.1016/j.mcm.2009.11.005.  Google Scholar

[19]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 54-60. doi: 10.1016/j.probengmech.2010.06.007.  Google Scholar

[20]

M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology, Nonlinear Anal. Real World Appl., 12 (2011), 2396-2407. doi: 10.1016/j.nonrwa.2011.02.014.  Google Scholar

[21]

M. Lachowicz and D. Wrzosek, Nonlocal bilinear equations. Equilibrium solutions and diffusive limit, Math. Models Methods Appl. Sci., 11 (2001), 1393-1409. doi: 10.1142/S0218202501001380.  Google Scholar

[22]

M. Lachowicz and A. Quartarone, A general framework for modeling tumor-immune system competition at the mesoscopic level, Appl. Math. Letters, 25 (2012), 2118-2122. doi: 10.1016/j.aml.2012.04.021.  Google Scholar

[23]

M. Lachowicz and T. Ryabukha, Equilibrium solutions for microscopic stochastic systems in population dynamics, Math. Biosci. Eng., 10 (2013), 777-786. doi: 10.3934/mbe.2013.10.777.  Google Scholar

[24]

M. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006.  Google Scholar

[25]

P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156. doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[26]

J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.  Google Scholar

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