A Multi-Species Keller-Segel Chemotaxis-Competition Model: Global Existence, Boundedness, and Mass Persistence

Abstract

This paper investigates the population dynamics of solutions to a parabolic-parabolic-elliptic type of multi-species Keller-Segel chemotaxis system under the Neumann boundary conditions in a smoothly bounded domain. It studies dynamical properties such as $L^\rho$-bounds, global existence, global boundedness, and combined mass persistence of solutions for the aforementioned system. Under certain specified parameter conditions, the paper shows that the system admits a unique global classical solution that remains uniformly bounded from above. Furthermore, it establishes that the entire population persists at all times; in other words, this study proves that any globally bounded classical solution maintains a positive lower mass bound.

Keywords

• Keller-Segel system
• chemotaxis
• global existence
• global boundedness
• mass persistence

References

1. E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology 26 (1970) 399-415.
2. E. F. Keller, L. A. Segel, Traveling bans of chemotactic bacteria: A theoretical analysis, Journal of Theoretical Biology 30 (1971) 377-380.
3. N. Bellomo, A. Bellouquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences 25 (09) (2015) 1663-1763.
4. T. Hillen, K. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology 58 (2009) 183-217.
5. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis, Jahresber Deutsch. Math.-Verein. 106 (2004) 51-69.
6. M. A. Herrero, E. Medina, J. J. L. Velzquez, Singularity patterns in a chemotaxis model, Mathematische Annalen 306 (1996) 583-623.
7. M. A. Herrero, J. J. L. Velzquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity 10 (6) (1997) 1739-1754.
8. T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, Journal of Inequalities and Applications 6 (2001) 37-55.

9. T. Nagai, T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Advances in Mathematical Sciences and Applications 8 (1998) 145-156.
10. J. I. Tello, M. Winkler, A chemotaxis system with logistic source, Common Partial Differential Equations 32 (2007) 849-877.
11. B. Hu, Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Applied Mathematics Letters 64 (2017) 1-7.
12. Y. Tao, M. Winkler, Persistence of mass in a chemotaxis system with logistic source, Journal of Differential Equations 259 (2015) 6142-6161.
13. H. I. Kurt, Global boundedness and mass persistence of solutions to a chemotaxis-competition system with logistic source, Süleyman Demirel University Journal of Natural and Applied Sciences 29 (1) (2025) 167-175.
14. M. A. J. Chaplain, J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Applied Mathematics Letters 57 (2016) 1-6.
15. T. B. Issa, W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM Journal on Applied Dynamical Systems 16 (2) (2017) 926-973.
16. H. I. Kurt, Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting, Journal of Differential Equations 416 (2) (2025) 1429-1461.
17. H. I. Kurt, Improvement of criteria for global boundedness in a minimal parabolic-elliptic chemotaxis system with singular sensitivity, Applied Mathematics Letters 167 (2025) 109570.
18. H. I. Kurt, W. Shen, Finite-time blow-up prevention by logistic source in chemotaxis models with singular sensitivity in any dimensional setting, SIAM Journal on Mathematical Analysis 53 (1) (2021) 973-1003.
19. H. I. Kurt, W. Shen, Chemotaxis models with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions, Nonlinear Analysis: Real World Applications 69 (2023) 103762.
20. H. I. Kurt, W. Shen, S. Xue, Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with ingular sensitivity and logistic source, Mathematical Models and Methods in Applied Sciences 34 (9) (2024) 1649-1700.
21. J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete and Continuous Dynamical Systems - Series B 20 (2015) 1499-1527.
22. J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, International Journal of Differential Equations 258 (2015) 1158-1191.
23. M. Le, H. I. Kurt, Global boundedness in a chemotaxis-growth system with weak singular sensitivity in any dimensional setting, Nonlinear Analysis: Real World Applications 86 (2025) 104392.
24. J. I. Tello, Mathematical analysis and stability of a chemotaxis problem with a logistic growth term, Mathematical Methods in the Applied Sciences 27 (2004) 1865-1880.
25. G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, Journal of Mathematical Analysis and Applications 439 (1) (2016) 197-212.
26. M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations 35 (8) (2010) 1516-1537.
27. J. I. Tello, M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity 25 (2012) 1413-1425.
28. T. Black, J. Lankeit, M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA Journal of Applied Mathematics 81 (2016) 860-876.
29. C. Stinner, J. I. Tello, M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology 68 (2014) 1607-1626.
30. M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Mathematical Methods in the Applied Sciences 41 (2018) 234-249.
31. K. Lin, C. Mu, H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Computers and Mathematics with Applications 75 (2018) 837-849.
32. L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, Journal of Mathematical Analysis and Applications 484 (2020) 123705.
33. X. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, The Indiana University Mathematics Journal 65 (2016) 553-583.
34. T. B. Issa, R.B. Salako, Asymptotic dynamics in a two-species chemotaxis model with non-local terms, Discrete and Continuous Dynamical Systems - B 22 (10) (2017) 3839-3874.
35. T. B. Issa, W. Shen, Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments, Journal of Dynamics and Differential Equations 31 (2019) 1839-1871.
36. H. I. Kurt, W. Shen, Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence, Journal of Differential Equations 355 (2023) 248-295.
37. H. I. Kurt, W. Shen, Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics, Discrete and Continuous Dynamical Systems 44 (4) (2024) 882-904.
38. K. Lin, C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete and Continuous Dynamical Systems - Series B 22 (2017) 2233-2260.
39. T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, Journal of Mathematical Analysis and Applications 459 (2018) 1172-1200.
40. L. Xie, On a fully parabolic chemotaxis system with nonlinear signal secretion, Nonlinear Analysis: Real World Applications 49 (2019) 24-44.
41. M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations 248 (2010) 2889-2905.
42. Y. Tao, M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM Journal on Mathematical Analysis 47 (6) (2015) 4229-4250.