Mathematical analysis and multiscale derivation of a nonlinear predator-prey cross-diffusion-fluid system with two chemicals. (English) Zbl 07900520
Summary: A nonlinear cross-diffusion-fluid system with chemicals terms describing the dynamics of predator-prey living in a Newtonian fluid is proposed in this paper. The existence of weak solution for the proposed macro-scale system is proved on the basis of the Schauder fixed-point theory, a priori estimates, and compactness arguments. The proposed system is derived from the underlining description delivered by a kinetic-fluid theory model by multiscale approach. Finally, we discuss the computational results for the proposed macro-scale system in two dimensional space.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35Q35 | PDEs in connection with fluid mechanics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
kinetic-fluid theory; chemical cross-diffusion-fluid; micro-macro decomposition; Schauder fixed-point theory; pattern formation; finite-volume method; finite-element methodSoftware:
FreeFem++References:
| [1] | Jüngel, A., (Diffusive and nondiffusive population models. Diffusive and nondiffusive population models, Model. simul. sci. eng. technol, 2010, Birkhäuser Boston Inc: Birkhäuser Boston Inc Boston MA), 397-425, Mathematical modeling of collective behavior in socio-economic and life sciences · Zbl 1398.92205 |
| [2] | Shigesada, N.; Kawasaki, K., Biological invasions: theory and practice, 1997, Oxford University Press |
| [3] | Anaya, V.; Bendahmane, M.; Sepúlveda, M., Numerical analysis for a three interacting species model with nonlocal and cross diffusion, ESAIM Math Model Numer Anal, 49, 1, 171-192, 2015 · Zbl 1314.65115 |
| [4] | Chen, X.; Daus, E.; Jüngel, A., Global existence analysis of cross-diffusion population systems for multiple species, Arch Ration Mech Anal, 227, 2, 715-747, 2018 · Zbl 1384.35135 |
| [5] | Grošelj, D.; Jenko, F.; Frey, E., How turbulence regulates biodiversity in systems with cyclic competition, Phys Rev E, 91, Article 033009 pp., 2015 |
| [6] | Klebanoff, A.; Hastings, A., Chaos in three species food chains, J Math Biol, 32, 427-451, 1994 · Zbl 0823.92030 |
| [7] | McCann, K.; Yodzis, P., Bifurcation structure of a three-species food chain model, Theor Popul Biol, 48, 93-125, 1995 · Zbl 0854.92022 |
| [8] | Bendahmane, M.; Karami, F.; Zagour, M., Kinetic-fluid derivation and mathematical analysis of the cross-diffusion-brinkman system, Math Methods Appl Sci, 41, 16, 6288-6311, 2018 · Zbl 1516.35325 |
| [9] | Atlas, A.; Bendahmane, M.; Karami, F.; Meskine, D.; Zagour, M., Kinetic-fluid derivation and mathematical analysis of a nonlocal cross-diffusion-fluid system, Appl Math Model, 82, 379-408, 2020 · Zbl 1481.76201 |
| [10] | Bellomo, N.; Liao, J.; Quaini, A.; Russo, L.; Siettos, C., Human behavioral crowds review, critical analysis and research perspectives, Math Models Methods Appl Sci, 33, 8, 1611-1659, 2023 · Zbl 1517.90024 |
| [11] | Bellomo, N.; Outada, N.; Soler, J.; Tao, Y.; Winkler, M., Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision, Math Models Methods Appl Sci, 32, 04, 713-792, 2022 · Zbl 1497.35039 |
| [12] | Negreanu, M.; Tello, J., Global existence and asymptotic behavior of solutions to a predator-prey chemotaxis system with two chemicals, J Math Anal Appl, 474, 2, 1116-1131, 2019 · Zbl 1416.35280 |
| [13] | Wan, C.; Zheng, P.; Shan, W., On a quasilinear fully parabolic predator-prey model with indirect pursuit-evasion interaction, J Evol Equ, 23, 4, 39, 2023, Paper 78 · Zbl 1531.35046 |
| [14] | Yan, X.-P.; Yang, T.-J.; Zhang, C.-H., Turing patterns induced by cross-diffusion in a Predator-Prey system with functional response of Holling-II type, Qual Theory Dyn Syst, 23, 4, 2024, Paper 168 · Zbl 1537.35054 |
| [15] | Tello, J.; Winkler, M., Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25, 5, 1413-1425, 2012 · Zbl 1260.92014 |
| [16] | Bai, X.; Winkler, M., Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ Math J, 65, 2, 553-583, 2016 · Zbl 1345.35117 |
| [17] | Black, T.; Lankeit, J.; Mizukami, M., On the weakly competitive case in a two-species chemotaxis model, IMA J Appl Math, 81, 5, 860-876, 2016 · Zbl 1404.35033 |
| [18] | Stinner, C.; Tello, J.; Winkler, M., Competitive exclusion in a two-species chemotaxis model, J Math Biol, 68, 7, 1607-1626, 2014 · Zbl 1319.92050 |
| [19] | Issa, T.; Shen, W., Uniqueness and stability of coexistence states in two species models with/without chemotaxis on bounded heterogeneous environments, J Dynam Differential Equations, 31, 4, 2305-2338, 2019 · Zbl 1439.35492 |
| [20] | Issa, T.; Shen, W., Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J Appl Dyn Syst, 16, 2, 926-973, 2017 · Zbl 1368.35017 |
| [21] | Cruz, E.; Negreanu, M.; Tello, J., Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals, Z Angew Math Phys, 69, 4, 20, 2018, Paper 107 · Zbl 1400.35025 |
| [22] | Zhang, Q.; Liu, X.; Yang, X., Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J Math Phys, 58, 11, Article 111504 pp., 2017 · Zbl 1387.92014 |
| [23] | Yadav, O.; Jiwari, R., A finite element approach for analysis and computational modelling of coupled reaction diffusion models, Numer Methods Partial Differential Equations, 35, 830-850, 2019 · Zbl 1418.65138 |
| [24] | Gambino, G.; Lombardo, M.; Sammartino, M., A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion, Appl Numer Math, 59, 5, 1059-1074, 2009 · Zbl 1165.65385 |
| [25] | Benito, J.; García, A.; Gavete, L.; Negreanu, M.; Ureña, F.; Vargas, A., Convergence and numerical simulations of prey-predator interactions via a meshless method, Appl Numer Math, 161, 333-347, 2021 · Zbl 1460.65098 |
| [26] | Bellomo, N.; Bellouquid, A.; Chouhad, N., From a multiscale derivation of nonlinear cross-diffusion models to Keller-Segel models in a Navier-Stokes fluid, Math Models Methods Appl Sci, 26, 2041-2069, 2016 · Zbl 1353.35038 |
| [27] | Zagour, M., Multiscale derivation of a time-dependent SEIRD reaction-diffusion system for COVID-19, 285-306, 2022, Springer International Publishing: Springer International Publishing Cham · Zbl 1504.92167 |
| [28] | Jin, S., Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM J Sci Comput, 21, 2, 441-454, 1999 · Zbl 0947.82008 |
| [29] | Klar, A., Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, SIAM J Sci Comput, 19, 6, 2032-2050, 1998 · Zbl 0918.65090 |
| [30] | Ladyzhenskaya S, UN. Linear and quasi-linear equations of parabolic type. Transl. AMS, vol. 23, Providence; 1980. |
| [31] | Temam R. Navier-stokes tquations: theory and numerical analysis. · Zbl 0383.35057 |
| [32] | Alikakos, N., \( l^p\) Bounds of solutions of reaction-diffusion equations, Comm Partial Differential Equations, 4, 827-868, 1979 · Zbl 0421.35009 |
| [33] | Simon J. Compact sets in the space \(l^p ( 0 , t ; b ). 4 (146) (1987) 65-96\). · Zbl 0629.46031 |
| [34] | Eymard, R.; Gallout, T.; Herbin, R., Finite volume methods, (Handbook of numerical analysis. Handbook of numerical analysis, Biomathematics, Vol. VII, 2000, North-Holland: North-Holland Amsterdam) · Zbl 0981.65095 |
| [35] | Andreianov, B.; Bendahmane, M.; Ruiz-Baier, R., Analysis of a finite volume method for a cross-diffusion model in population dynamics, Math Models Methods Appl Sci, 21, 2, 307-344, 2011 · Zbl 1228.65178 |
| [36] | Hecht, F., New development in freefem++, J Numer Math, 20, 251-265, 2013 · Zbl 1266.68090 |
| [37] | Bürger, R.; Ordoñez, R.; Sepúlveda, M.; Villada, L., Numerical analysis of a three-species chemotaxis model, Comput Math Appl, 80, 1, 183-203, 2020 · Zbl 1452.92007 |
| [38] | Bendahmane, M.; Nzeti, H.; Tagoudjeu, J.; Zagour, M., Stochastic reaction-diffusion system modeling predator-prey interactions with prey-taxis and noises, Chaos, 33, 7, 26, 2023, Paper 073103 · Zbl 1541.35274 |
| [39] | Zagour, M., Toward multiscale derivation of behavioral dynamics: Comment to “what is life? active particles tools towards behavioral dynamics in social-biology and economics” , by b. bellomo, m. esfahanian, v. secchini, and p. terna, Phys Life Rev, 46, 273-274, 2023 |
| [40] | Bendahmane, M.; Tagoudjeu, J.; Zagour, M., Odd-even based asymptotic preserving scheme for a 2D stochastic kinetic-fluid model, J Comput Phys, 471, 25, 2022, Paper 111649 · Zbl 07605609 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
