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A nonconservative kinetic framework under the action of an external force field: Theoretical results with application inspired to ecology

Part of: General theory in ordinary differential equations Mathematical biology in general Ergodic theory

Published online by Cambridge University Press:  31 August 2023

Bruno Carbonaro  and
Marco Menale Open the ORCID record for Marco Menale [Opens in a new window]
Bruno Carbonaro
Affiliation:
Department of Mathematics and Physics, University of Campania “L. Vanvitelli”, Caserta, Italy
Marco Menale*
Affiliation:
Department of Mathematics and Physics, University of Campania “L. Vanvitelli”, Caserta, Italy
*
Corresponding author: Marco Menale; Email: marco.menale@unicampania.it
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Abstract

The present paper deals with the kinetic-theoretic description of the evolution of systems consisting of many particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the ‘forcing term’. Firstly, the interactions between pairs of particles are both conservative and nonconservative; the latter represents, among others, birth/death rates. The ‘forcing term’ does not express a ‘classical’ force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles, for instance, climate changes, that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, that in turn can be also perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, the interest towards these problems moves the mathematical analysis of the effects of different kinds of forcing terms on solutions to equations governing the collective (that is statistical) behaviour of such nonconservative many-particle systems. In the present paper, we offer a study of the basic mathematical properties of such solutions, along with some numerical simulations to show the effects of forcing terms for a classical prey–predator model in ecology.

Keywords

Kinetic theoryinteracting systemsexternal force fielddifferential equationsecology

MSC classification

Primary: 37A50: Relations with probability theory and stochastic processes 34A34: Nonlinear equations and systems, general 92B10: Taxonomy, cladistics, statistics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 6 , December 2023 , pp. 1170 - 1186
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Bellomo, N. & Bellouquid, A. (2004) From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete Contin. Dyn. Syst. Ser. B 4(1), 59–80.CrossRefGoogle Scholar
Bianca, C. (2012) Thermostatted kinetic equations as models for complex systems in physics and life sciences. Phys. Life Rev. 9(4), 359–399.CrossRefGoogle ScholarPubMed
Bellomo, N. & Pulvirenti, M. (2013) Modeling in Applied Sciences: A Kinetic Theory Approach, Springer, New York.Google Scholar
Burini, D., Gibelli, L. & Outada, N. (2017) A kinetic theory approach to the modeling of complex living systems. In: Active Particles, Vol. 1, Springer, New York, pp. 229–258.CrossRefGoogle Scholar
Puppo, G., Semplice, M., Tosin, A. & Visconti, G. (2017) Analysis of a multi-population kinetic model for traffic flow. Commun. Math. Sci. 15(2), 379–412.CrossRefGoogle Scholar
Cercignani, C., Illner, R. & Pulvirenti, M. (2013) The Mathematical Theory of Dilute Gases, Vol. 106, Springer, New York.Google Scholar
Bellouquid, A. & Delitala, M. (2005) Mathematical methods and tools of kinetic theory towards modelling complex biological systems. Math. Models Methods Appl. Sci. 15(11), 1639–1666.CrossRefGoogle Scholar
Conte, M. & Loy, N. (2022) Multi-cue kinetic model with non-local sensing for cell migration on a fiber network with chemotaxis. Bull. Math. Biol. 84(3), 1–46.CrossRefGoogle ScholarPubMed
Knopoff, D. A., Nieto, J. & Urrutia, L. (2019) Numerical simulation of a multiscale cell motility model based on the kinetic theory of active particles. Symmetry 11(8), 1003.CrossRefGoogle Scholar
Albi, G., Bellomo, N., Fermo, L., Ha, S-Y., Kim, J., Pareschi, L., Poyato, D. & Soler, J. (2019) Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives. Math. Models Methods Appl. Sci. 29(10), 1901–2005.CrossRefGoogle Scholar
Carbonaro, B. & Serra, N. (2002) Towards mathematical models in psychology: a stochastic description of human feelings. Math. Models Methods Appl. Sci. 12(10), 1453–1490.CrossRefGoogle Scholar
Dolfin, M. & Lachowicz, M. (2015) Modeling opinion dynamics: how the network enhances consensus. Netw. Heterog. Media 10(4), 877–896.CrossRefGoogle Scholar
Delitala, M. & Tosin, A. (2007) Mathematical modeling of vehicular traffic: a discrete kinetic theory approach. Math. Models Methods Appl. Sci. 17(06), 901–932.CrossRefGoogle Scholar
Fermo, L. & Tosin, A. (2013) A fully-discrete-state kinetic theory approach to modeling vehicular traffic. SIAM J. Appl. Math. 73(4), 1533–1556.CrossRefGoogle Scholar
Marca, R. D., Loy, N. & Tosin, A. (2021) An SIR-like kinetic model tracking individuals’ viral load. arXiv preprint arXiv: 2106.14480 .Google Scholar
Bertotti, M. L. & Modanese, G. (2011) From microscopic taxation and redistribution models to macroscopic income distributions. Phys. A Stat. Mech. Appl. 390(21-22), 3782–3793.CrossRefGoogle Scholar
Bertotti, M. L. & Modanese, G. (2014) Micro to macro models for income distribution in the absence and in the presence of tax evasion. Appl. Math. Comput. 244, 836–846.CrossRefGoogle Scholar
Marca, R. D., Loy, N. & Menale, M. (2022) Intransigent vs. volatile opinions in a kinetic epidemic model with imitation game dynamics. Math. Med. Biol. J. IMA 40(2), 111–140.CrossRefGoogle Scholar
Pareschi, L., Vellucci, P. & Zanella, M. (2017) Kinetic models of collective decision-making in the presence of equality bias. Phys. A Stat. Mech. Appl. 467, 201–217.CrossRefGoogle Scholar
Albi, G., Bertaglia, G., Boscheri, W., Dimarco, G., Pareschi, L., Toscani, G. & Zanella, M. (2022) Kinetic modelling of epidemic dynamics: Social contacts, control with uncertain data, and multiscale spatial dynamics. In: Predicting Pandemics in a Globally Connected World, Volume 1: Toward a Multiscale, Multidisciplinary Framework through Modeling and Simulation, Springer, New York, pp. 43–108.CrossRefGoogle Scholar
Bernardi, E., Pareschi, L., Toscani, G. & Zanella, M. (2022) Effects of vaccination efficacy on wealth distribution in kinetic epidemic models. Entropy 24(2), 216.CrossRefGoogle ScholarPubMed
Bertaglia, G. & Pareschi, L. (2021) Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods. ESAIM Math. Model. Numer. Anal. 55(2), 381–407.CrossRefGoogle Scholar
Dimarco, G., Pareschi, L., Toscani, G. & Zanella, M. (2020) Wealth distribution under the spread of infectious diseases. Phys. Rev. E 102(2), 022303.CrossRefGoogle ScholarPubMed
Loy, N. & Tosin, A. (2021) A viral load-based model for epidemic spread on spatial networks. arXiv preprint arXiv: 2104.12107 .CrossRefGoogle Scholar
d’Onofrio, A., Duarte, J., Januário, C. & Martins, N. (2022) A sir forced model with interplays with the external world and periodic internal contact interplays. Phys. Lett. A 454, 128498.CrossRefGoogle Scholar
Arlotti, L. & Lachowicz, M. (1996) Qualitative analysis of a nonlinear integrodifferential equation modeling tumor-host dynamics. Math. Comput. Model. 23(6), 11–29.CrossRefGoogle Scholar
Luisa Arlotti, N. B. & Latrach, K. (1999) From the jager and segel model to kinetic population dynamics nonlinear evolution problems and applications. Math. Comput. Model. 30(1-2), 15–40.CrossRefGoogle Scholar
Bianca, C. (2012) An existence and uniqueness theorem to the cauchy problem for thermostatted-KTAP models. Int. J. Math. Anal. 6(17-20), 813–824.Google Scholar
Bianca, C. (2012) Kinetic theory for active particles modelling coupled to gaussian thermostats. Appl. Math. Sci. 6(13-16), 651–660.Google Scholar
Bianca, C. (2012) Modeling complex systems by functional subsystems representation and thermostatted-ktap methods. Appl. Math. Inf. Sci. 6(3), 495–499.Google Scholar
Bianca, C. & Menale, M. (2019) Existence and uniqueness of nonequilibrium stationary solutions in discrete thermostatted models. Commun. Nonlinear Sci. Numer. Simul. 73, 25–34.CrossRefGoogle Scholar
Bellomo, N., Bianca, C. & Delitala, M. (2009) Complexity analysis and mathematical tools towards the modelling of living systems. Phys. Life. Rev. 6(3), 144–175.CrossRefGoogle ScholarPubMed
Bertotti, M. L. & Delitala, M. (2008) Conservation laws and asymptotic behavior of a model of social dynamics. Nonlinear Anal. Real World Appl. 9(1), 183–196.CrossRefGoogle Scholar
Bianca, C. (2014) How do mutative events modify moments evolution in thermostatted kinetic models? Commun. Nonlinear Sci. 19(7), 2155–2159.CrossRefGoogle Scholar
Bianca, C. & Mogno, C. (2018) Modelling pedestrian dynamics into a metro station by thermostatted kinetic theory methods. Math. Comput. Model. Dyn. Syst. 24(2), 207–235.CrossRefGoogle Scholar
Berryman, A. A. (1992) The orgins and evolution of predator-prey theory. Ecology 73(5), 1530–1535.CrossRefGoogle Scholar
Brauer, F., Castillo-Chavez, C. & Castillo-Chavez, C. (2012) Mathematical Models in Population Biology and Epidemiology, Vol. 2, Springer, New York.CrossRefGoogle Scholar
Freedman, H. I. (1980) Deterministic Mathematical Models in Population Ecology, Vol. 57, Marcel Dekker Incorporated, New York.Google Scholar
Wangersky, P. J. (1978) Lotka-volterra population models. Annu. Rev. Ecol. Syst. 9(1), 189–218.CrossRefGoogle Scholar
Bellomo, N. & Preziosi, L. (2000) Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Model. 32(3-4), 413–452.CrossRefGoogle Scholar
Asfaw, M. D., Kassa, S. M., Lungu, E. M. & Bewket, W. (2019) Effects of temperature and rainfall in plant–herbivore interactions at different altitude. Ecol. Model. 406, 50–59.CrossRefGoogle Scholar
Das, A. & Samanta, G. P. (2018) Stochastic prey–predator model with additional food for predator. Phys. A Stat. Mech. Appl. 512, 121–141.CrossRefGoogle Scholar
Das, A. & Samanta, G. P. (2020) A prey–predator model with refuge for prey and additional food for predator in a fluctuating environment. Phys. A Stat. Mech. Appl. 538, 122844.CrossRefGoogle Scholar
Menale, M. & Carbonaro, B. (2020) The mathematical analysis towards the dependence on the initial data for a discrete thermostatted kinetic framework for biological systems composed of interacting entities. AIMS Biophys. 7(3), 204–218.CrossRefGoogle Scholar
Carbonaro, B. & Menale, M. (2021) Towards the dependence on parameters for the solution of the thermostatted kinetic framework. Axioms 10(2), 59.CrossRefGoogle Scholar
Bianca, C., Carbonaro, B. & Menale, M. (2020) On the cauchy problem of vectorial thermostatted kinetic frameworks. Symmetry 12(4), 517.CrossRefGoogle Scholar
Burini, D., Chouhad, N. & Bellomo, N. (2023) Waiting for a mathematical theory of living systems from a critical review to research perspectives. Symmetry 15(2), 351.10.3390/sym15020351CrossRefGoogle Scholar
Benhabib, J. & Nishimura, K. (2012). The Hopf bifurcation and existence and stability of closed orbits in multisector models of optimal economic growth. In: Nonlinear Dynamics in Equilibrium Models, Springer, Berlin/Heidelberg, pp. 51–73.CrossRefGoogle Scholar
Buonomo, B., Marca, R. D., d’Onofrio, A. & Groppi, M. (2022) A behavioural modelling approach to assess the impact of COVID-19 vaccine hesitancy. J. Theor. Biol. 534, 110973.CrossRefGoogle ScholarPubMed
Huang, C., Zhang, H., Cao, J. & Hu, H. (2019) Stability and Hopf bifurcation of a delayed prey–predator model with disease in the predator. Int. J. Bifurcat. Chaos 29(07), 1950091.CrossRefGoogle Scholar
Sen, M., Banerjee, M. & Morozov, A. (2012) Bifurcation analysis of a ratio-dependent prey–predator model with the allee effect. Ecol. Complex. 11, 12–27.CrossRefGoogle Scholar
Zhang, X. & Zhang, Q.-l. (2009) Bifurcation analysis and control of a class of hybrid biological economic models. Nonlinear Anal. Hybrid Syst. 3(4), 578–587.CrossRefGoogle Scholar