Skip to main content
Have a personal or library account? Click to login
Directional migration and competition in fluid media: Global existence, uniqueness, and robust simulation of chemotaxis and tumor growth dynamics Cover

Directional migration and competition in fluid media: Global existence, uniqueness, and robust simulation of chemotaxis and tumor growth dynamics

By:   
Open Access
|Jun 2026

References

  1. Lotka A.J., Elements of Physical Biology, Williams and Wilkins, United States of America, 1925.
  2. Volterra V., Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science, 3(1), 3–51, 1928.
  3. Mottoni P.D., Rothe F., Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM Journal on Applied Mathematics, 37(3), 648–663, 1979.
  4. Matano H., Mimura M., Pattern formation in competition-diffusion systems in nonconvex domains, Publications of the Research Institute for Mathematical Sciences, 19(3), 1049–1079, 1983.
  5. Mimura M., Ei S.I., Fang Q., Effect of domain-shape on coexistence problems in a competition-diffusion system, Journal of Mathematical Biology, 29(3), 219–237, 1991.
  6. Keller E.F., Segel L.A., Model for chemotaxis, Journal of Theoretical Biology, 30(2), 225–234, 1971.
  7. Chamoun G., Ibrahim M., Saad M., Talhouk R., Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model, Discrete and Continuous Dynamical Systems Series B, 25(11), 4165–4188, 2020.
  8. Chamoun G., Mathematical analysis of parabolic models with volume-filling effect in weighted networks, Journal of Dynamics and Differential Equations, 35(3), 2115–2137, 2023.
  9. Tello J.I., Wrzosek D., Predator-prey model with diffusion and indirect prey-taxis, Mathematical Models and Methods in Applied Sciences, 26(11), 2129–2162, 2016.
  10. Tsyganov M.A., Brindley J., Holden A.V., Biktashev V.N., Quasisoliton interaction of pursuit-evasion waves in a predator-prey system, Physical Review Letters, 91(21), 218102, 2003.
  11. Bendahmane M., Langlais M., A reaction-diffusion system with cross-diffusion modelling the spread of an epidemic disease, Journal of Evolution Equations, 10(4), 883–904, 2010.
  12. Berres S., Ruiz-Baier R., A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion, Nonlinear Analysis: Real World Applications, 12(5), 2888–2903, 2011.
  13. Özdemir N., Uçar E., Investigating an immune system-cancer mathematical model with Mittag-Leffler kernel, AIMS Mathematics, 5(2), 1519–1531, 2020.
  14. Uçar E., Özdemir N., New fractional cancer mathematical model via IL-10 cytokine and anti-PD-L1 inhibitor, Fractal and Fractional, 7(2), 151, 2023.
  15. Atangana A., Koca I., Witte’s conditions for uniqueness of solutions to a class of Fractal-Fractional ordinary differential equations, An International Journal of Optimization and Control: Theories & Applications, 14(4), 322–335, 2024.
  16. Pearce I.G., Chaplain M.A.J., Schofield P.G., Anderson A.R.A., Hubbard S.F., Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, Journal of Mathematical Biology, 55(3), 365–388, 2007.
  17. Tang X., Tao Y., Analysis of a chemotaxis model for multi-species host-parasitoid interactions, Applied Mathematical Sciences, 2(25), 1239–1252, 2008.
  18. Stinner C., Tello J.I., Winkler M., Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68(7), 1607–1626, 2014.
  19. Tello J.I., Wrzosek D., Inter-species competition and chemorepulsion, Journal of Mathematical Analysis and Applications, 459(2), 1233–1250, 2018.
  20. Mizukami M., Yokota T., Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, Journal of Differential Equations, 261(5), 2650–2669, 2016.
  21. Duarte-Rodríguez A., Rodríguez-Bellido M.Á., Rueda-Gómez D.A., Villamizar-Roa É.J., Numerical analysis for a chemotaxis-Navier-Stokes system, ESAIM: Mathematical Modelling and Numerical Analysis, 55(2), 417–445, 2021.
  22. Tuval I., Cisneros L., Dombrowski C., Wolgemuth C.W., Kessler J.O., Goldstein R.E., Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102(7), 2277–2282, 2005.
  23. Winkler M., How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Transactions of the American Mathematical Society, 369(5), 3067–3125, 2017.
  24. Lankeit J., Long-term behaviour in a chemotaxis-fluid system with logistic source, Mathematical Models and Methods in Applied Sciences, 26(11), 2071–2109, 2016.
  25. Chamoun G., Saad M., Talhouk R., Numerical analysis of a chemotaxis-swimming bacteria model on a general triangular mesh, Applied Numerical Mathematics, 127, 324–348, 2018.
  26. Holling C.S., The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, The Canadian Entomologist, 91(5), 293–320, 1959.
  27. Holling C.S., Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91(7), 385–398, 1959.
  28. Murray J.D., Mathematical Biology II:Spatial Models and Biomedical Applications, Springer, Germany, 2011.
  29. Tao Y., Winkler M., Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete and Continuous Dynamical Systems Series B, 20(9), 3165–3183, 2015.
  30. Jin H.Y., Xang T., Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes system with competitive kinetics, Discrete and Continuous Dynamical Systems Series B, 24(4), 1919–1942, 2019.
  31. Cao X., Kurima S., Mizukami M., Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Mathematical Methods in the Applied Sciences, 41(8), 3138–3154, 2018.
  32. Hirata M., Kurima S., Mizukami M., Yokota T., Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, Journal of Differential Equations, 263(1), 470–490, 2017.
  33. Hirata M., Kurima S., Mizukami M., Yokota, T., Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system, International Conference on Differential Equations and Applications, 24–28 July 2017, Bratislava, Slovakia.
  34. Li G., Yao Y., Two-species competition model with chemotaxis: well-posedness, stability and dynamics, Nonlinearity, 35(3),1329, 2022.
  35. Boyer F., Fabrie P., Éléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, (Master Thesis), Université Bordeaux 1, France, 2003.
  36. Ladyzhenskaya O.A., Solonnikov V.A., Ural’tseva N.N., Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, United States of America, 1968.
  37. Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Fluid, Gordon and Breach, United States of America, 1963.
  38. Chamoun G., Finite volume analysis of the two competing-species chemotaxis models with general diffusive functions, WSEAS Transactions on Biology and Biomedicine, 22, 232–247, 2025.
  39. Temam R., Navier-Stokes Equations and Nonlinear Functional Analysis (2nd Ed.), American Mathematical Society, United States of America, 2000.
  40. Eymard R., Hilhorst D., Vohralik M., A combined finite volume-nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems, Numerische Mathematik, 105(1), 73–131, 2006.
Language: English
Submitted on: Sep 28, 2025
Accepted on: Mar 30, 2026
Published on: Jun 2, 2026
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year

© 2026 Georges Chamoun, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

AHEAD OF PRINT