Details

Giuseppe Toscani^[a] and Mattia Zanella^[b]

On a kinetic description of Lotka-Volterra dynamics

Pages: 61-77
Received: 5 March 2023
Accepted: 11 July 2023.
Mathematics Subject Classification: 91B60, 82C40, 35B40.
Keywords: Kinetic models, Boltzmann equation, Wealth distribution, Taxation and redistribution, Lotka-Volterra equations.
Authors address:
[a],[b]:University of Pavia, Department of Mathematics, Via Ferrata 5, 27100 Pavia, Italy

Abstract: Owing to the analogies between the problem of wealth redistribution with taxation in a multi-agent society, we introduce and discuss a kinetic model describing the statistical distributions in time of the sizes of groups of biological systems with prey-predator dynamic. While the evolution of the mean values is shown to be driven by a classical Lotka-Volterra system of differential equations, it is shown that the time evolution of the probability distributions of the size of groups of the two interacting species is heavily dependent both on a kinetic redistribution operator and the degree of randomness present in the system. Numerical experiments are given to clarify the time-behavior of the distributions of groups of the species.

References

[1]: M. Bisi, G. Spiga and G. Toscani, Kinetic models of conservative economies with wealth redistribution, Commun. Math. Sci. 7 (2009), no. 4, 901-916. MR
[2]: M. Bisi, Some kinetic models for a market economy, Boll. Unione Mat. Ital. 10 (2017), no. 1, 143-158. MR
[3]: A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 7, Harwood Academic Publishers, Chur, 1988,111-233. MR
[4]: A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C 13 (2002), 1315-1321. DOI
[5]: A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B 17 (2000), 167-170. DOI
[6]: A. Chatterjee, B. K. Chakrabarti and S. S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A 335 (2004), 155-163. DOI
[7]: A. Chatterjee, S. Yarlagadda and B. K. Chakrabarti, eds. Econophysics of wealth distributions, New Economic Window Series, Springer-Verlag, Milan, 2005. DOI
[8]: A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E 72 (2005), 026126. DOI
[9]: S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys. 120 (2005), 253-277. DOI
[10]: P. Degond, J.-G. Liu and C. Ringhofer, Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria, J. Stat. Phys. 154 (2014), 751-780. DOI
[11]: G. Dimarco, L.Pareschi, G. Toscani and M. Zanella, Wealth distribution under the spread of infectious diseases, Phys. Rev. E 102 (2020), 022303. DOI
[12]: G. Dimarco, B.Perthame, G. Toscani and M. Zanella, Kinetic models for epidemic dynamics with social heterogeneity, J. Math. Biol. 83 (2021), art. no. 4. DOI
[13]: A. Dragulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B 17 (2000), 723-729. DOI
[14]: B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E 78 (2008), 056103. DOI
[15]: B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma (8) 1 (2009), 199-261. MR
[16]: G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Fokker-Planck equations in the modelling of socio-economic phenomena, Math. Models Methods Appl. Sci. 27 (2017), no.1, 115-158. MR
[17]: G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani, Non-Maxwellian kinetic equations modeling the evolution of wealth distribution, Math. Mod. Meth. Appl. Sci. 30 (2020), no. 4, 685-725. DOI
[18]: G. Gandolfo, Giuseppe Palomba and the Lotka-Volterra equations, Rend. Fis. Acc. Lincei 19 (2008), 347-357. DOI
[19]: R. M. Goodwin, A Growth Cycle, in C. H. Feinstein, ed.,"Socialism, Capitalism and Economic Growth'', Cambridge University Press, Cambridge 1967. DOI
[20]: B. Hayes, Follow the money, American Scientist 90 (2002), 400-405. DOI
[21]: S. Ispolatov, P. L. Krapivsky and S. Redner, Wealth distributions in asset exchange models, Eur. Phys. Jour. B 2 (1998), 267-276. DOI
[22]: O. Malcai, O. Biham, P. Richmond and S. Solomon, Theoretical analysis and simulations of the generalized Lotka-Volterra model, Phys. Rev. E 66 (2002), 031102. DOI
[23]: D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys. 130 (2008), 1087-1117. DOI
[24]: A. Medaglia, A. Tosin and M. Zanella, Monte Carlo stochastic Galerkin methods for non-Maxwellian kinetic models of multiagent systems with uncertainties, Part. Diff. Equat. Appl. 3 (2022), art. no. 51. DOI
[25]: J. D. Murray, Mathematical Biology I: An Introduction., Interdisciplinary Applied Mathematics, Springer, New York, 2002. DOI
[26]: G. Palomba, Introduzione allo studio della dinamica economica, Napoli: Jovene, Napoli, 1939.
[27]: L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2013. Zbmath
[28]: L. Pareschi and M. Zanella, Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput. 74 (2018), 1575-1600. DOI
[29]: G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci. 4 (2006), no. 3, 481-496. DOI
[30]: G. Toscani, A. Tosin and M. Zanella, Kinetic modelling of multiple interactions in socio-economic systems, Netw. & Heter. Media 15 (2020), no. 3, 519-542. DOI
[31]: F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E 69 (2004), 046102. DOI
[32]: S. Solomon and P. Richmond, Stable power laws in variable economies; Lotka-Volterra implies Pareto-Zipf, Eur. Phys. J. B 27 (2002), 257-262. DOI