Tek Tür, İki Kimyasal ve Tekillik İçeren Lojistik Bir Kemotaksi Sisteminde Süreklilik ve Sınırlılık

Abstract

Kemotaksi sistemleri, biyolojik popülasyonların kimyasal gradyanlara tepkisini tanımlar; ancak tekil duyarlılıklar, matematiksel kararlılığı zorlaştıran patlama (blow-up) olgularına yol açabilmektedir. Bu çalışma, tek tür ve iki etkileşen kimyasal madde içeren, lojistik büyüme terimiyle desteklenmiş parabolik–eliptik–eliptik tipte bir kemotaksi modelini homojen Neumann sınır koşulları altında ele almaktadır. Patlamayı önleyen mekanizmaların belirlenmesi gerekliliğinden hareketle, klasik çözümlerin sürekliliği ve küresel sınırlılığı için yeterli koşullar titizlikle ortaya konmuştur. Özellikle, lojistik sönüm etkisi yeterince güçlü olduğunda sistemin zamana bağlı olarak pozitifliğini koruyan benzersiz ve küresel olarak sınırlı bir çözüme sahip olduğu kanıtlanmıştır. Bu bulgular, lojistik düzenlemenin tekil duyarlılığa rağmen kemotaktik birikimi nasıl dengelediğini ortaya koyarak önceki teorik sonuçları geliştirmektedir.

Keywords

• Kemotaksi
• çok türlü sistem
• tekil duyarlılık
• küresel sınırlılık
• süreklilik

Persistence and boundedness in a logistics chemotaxis system including one-species, two-chemicals, and singularity

Abstract

Chemotaxis systems describe how biological populations respond to chemical gradients, yet singular sensitivities often lead to blow-up phenomena that challenge mathematical stability. This study addresses a parabolic–elliptic–elliptic chemotaxis model involving one species and two interacting chemicals with a logistic growth term under homogeneous Neumann boundary conditions. Motivated by the need to identify mechanisms preventing blow-up, we establish rigorous conditions ensuring persistence and global boundedness of classical solutions. Specifically, it is proven that when the logistic damping effect is sufficiently strong, the system admits a unique global solution that remains positive and uniformly bounded over time. These findings advance previous theoretical results by clarifying how logistic regulation stabilizes chemotactic aggregation even under singular sensitivity.

Keywords

• Chemotaxis
• multi-species system
• singular sensitivity
• global boundedness
• persistence

References

1. [1] Keller, E. F., Segel, L. A. 1970. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology, 26, 399-415.
2. [2] Keller, E. F., Segel, L. A. 1971. Traveling bans of chemotactic bacteria: a theoretical analysis. Journal of Theoretical Biology, 30, 377-380.
3. [3] Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M. 2015. Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences 25, 1663-1763.
4. [4] Hillen, T., Painter, K. 2009. A user’s guide to PDE models for chemotaxis, Journal of Theoretical Biology 58, 183-217.
5. [5] Horstmann, D. 2004. From 1970 until present: the Keller-Segel model in chemotaxis, Jahresber DMV, 106, 51-69.
6. [6] Nagai, T., Senba, T. 1998. Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Advances in Mathematical Sciences and Applications, 8, 145-156.
7. [7] Fujie, K., Winkler, M., Yokota, T. 2015. Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal dependent sensitivity, Mathematical Methods in the Applied Sciences, 38 (6), 1212-1224.
8. [8] Fujie, K., Senba, T. 2016. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete and Continuous Dynamical Systems B, 21(1), 81-102.

9. [9] Kurt, H. I. 2025. Improvement of criteria for global boundedness in a minimal parabolic-elliptic chemotaxis system with singular sensitivity, Applied Mathematics Letters, 167, 109570.
10. [10] Biler, P. 1999. Global solutions to some parabolic-elliptic systems of chemotaxis, Advanced Mathematics and Applications, 9, 347-359.
11. [11] Black, T. 2020. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity, Discrete and Continuous Dynamical Systems S, 13 , 119-137.
12. [12] Fujie, K., Winkler, M., Yokota, T. 2014. Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Analysis, 109, 56-71.
13. [13] Kurt, H. I., Shen, W. 2021. 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), 973-1003.
14. [14] Zhao, X. 2023. Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two, Nonlinearity, 36, 3909-3938.
15. [15] Le, M. 2025. Boundedness in a chemotaxis system with weakly singular sensitivity in dimension two with arbitrary sub-quadratic degradation sources, Journal of Mathematical Analysis and Applications, 542, 128803.
16. [16] Kurt, H. I. 2025. Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting, Journal of Differential Equations, 416(2), 1429-1461.
17. [17] Le, M., Kurt, H. I. 2025. Global boundedness in a chemotaxis-growth system with weak singular sensitivity in any dimensional setting, Nonlinear Analysis: Real World Applications, 86, 104392.
18. [18] Kurt, H.I., Shen, W., Xue, S. 2024. Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source, Mathematical Models and Methods in Applied Sciences, 34(9), 1649-1700.
19. [19] Kurt, H. I., Shen, W. 2023. Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistence, Journal of Differential Equations, 355, 248-295.
20. [20] Kurt, H. I., Shen, W. 2024. Stabilization in two-species chemotaxis systems with singular sensitivity and Lotka-Volterra competitive kinetics, Discrete and Continuous Dynamical Systems, 44(4), 882-904.
21. [21] Le, M., Kurt, H. I. 2026. Persistence of positive classical solutions in a logistic chemotaxis system with weak singular sensitivity, Discrete and Continuous Dynamical Systems, Series B,
22. [22] Le, M., Kurt, H. I., Yaprak, R. 2026. Analytical and numerical analysis of boundedness in a two-species Keller-Segel model with weak nonlinear sensitivity, Communications in nonlinear science and numerical simulation, 154, 109563.
23. [23] Le, M., Bao, L., Kurt, H. I. 2025. Can logistic damping prevent blow-up in weak singular sensitivity chemotaxis systems with nonlinear boundary conditions?, Zeitschrift für angewandte Mathematik und Physik, 76 (364).
24. [24] Zhang, J., Mu, C., Tu, X. 2023. Finite-time blow-up of solution for a chemotaxis model with singular sensitivity and logistic source, Zeitschrift für angewandte Mathematik und Physik, 74, 229.
25. [25] Zhao, X. 2020. Boundedness to a logistic chemotaxis system with singular sensitivity, 1, arXiv: 2003.03016.
26. [26] Zhao, X. 2024. Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, Czechoslovak Mathematical Journal, 74(1), 127-151.
27. [27] Zhao, X., Zheng, S. 2017. Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Zeitschrift fur Angewandte Mathematik und Physik, 68, 2.
28. [28] Kurt, H. I., Shen, W. 2023. Chemotaxis models with singular sensitivity and logistic source: Boundedness, persistence, absorbing set, and entire solutions, Nonlinear Analysis: Real World Applications, 69, 103762-27.
29. [29] Ekici, M. 2025. Global Existence of Solutions to a One-Species and Two-Chemicals Chemotaxis System with Singular Sensitivity and Logistic Source, Journal of the Institute of Science and Technology, 15(3), 1080-1088.
30. [30] Kurt, H. I. 2025. Global Boundedness and Mass Persistence in a Multi-Species Chemotaxis System, Journal of the Institute of Science and Technology, 15(3), 1100-1109.