American Institute of Mathematical Sciences

Advanced
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

[1]

Eric Foxall. Boundary dynamics of the replicator equations for neutral models of cyclic dominance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1061-1082. doi: 10.3934/dcdsb.2020153
[2]

Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529
[3]

Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251
[4]

Kun-Peng Jin, Jin Liang, Ti-Jun Xiao. Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077
[5]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[6]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021051
[7]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[8]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
[9]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
[10]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
[11]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[12]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
[13]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053
[14]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160
[15]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053
[16]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[17]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[18]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
[19]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[20]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy