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Stability of solutions of kinetic equations corresponding to the replicator dynamics
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press Series on Economic Learning and Social Evolution, 5, MIT Press, Cambridge, MA, 2003. |
[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. |
[11] |
Evolutionary Game Theory, Stanford Encyclopedia of Philosophy, 2009. Available from: http://plato.stanford.edu/archives/fall2009/entries/game-evolutionary/. |
[12] |
G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1934. |
[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. |
[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. |
[15] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
M. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. |
[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. |
[26] |
J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
R. Cressman, Evolutionary Dynamics and Extensive Form Games, MIT Press Series on Economic Learning and Social Evolution, 5, MIT Press, Cambridge, MA, 2003. |
[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. |
[11] |
Evolutionary Game Theory, Stanford Encyclopedia of Philosophy, 2009. Available from: http://plato.stanford.edu/archives/fall2009/entries/game-evolutionary/. |
[12] |
G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge University Press, 1934. |
[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. |
[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. |
[15] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
M. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, The Belknap Press of Harvard University Press, Cambridge, MA, 2006. |
[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. |
[26] |
J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995. |
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