Bir Lojistik Kemotaksis-Rekabet Sisteminin Çözümlerinin Küresel Sınırlılığı ve Kitlesel Kalıcılığı

Yıl 2025, Cilt: 29 Sayı: 1, 167 - 175, 25.04.2025

Halil İbrahim Kurt

https://doi.org/10.19113/sdufenbed.1627078

Cited By: 3

Öz

Bu makale, düzgün sınırlı bir alanda lojistik kinetik içeren parabolik eliptik tipte bir kemotaksi-rekabet sistemine ait global varlık, global sınırlılık ve kütlesel kalıcılığı gibi çözümlerin popülasyon dinamiklerini incelemektedir. Tello ve Winkler'in ilk olarak yukarıda belirtilen sistemin küresel varlığını ve sınırlılığını incelemiştir. Daha sonra Tao ve Winkler, verilen sistemin çözümlerin kütlesel kalıcılığı gibi dinamik özelliklerini araştırmışlardır. Bu çalışmada, bilinen bazı sonuçlar geliştirilmiştir ve uygun koşullar altında sistemin küresel olarak var ve sınırlı olan tek bir klasik çözüme sahip olduğunu göstermiştir. Buna ilave olarak, popülasyonun bir bütün olarak asla yok olmadığı gösterilmiştir.

Anahtar Kelimeler

Lineer hassasiyet , Lojistik kaynak , Global varlık , Global sınırlılık , Kitlesel kalıcılık

Global Boundedness and Mass Persistence of Solutions to A Chemotaxis-Competition System with Logistic Source

Öz

This article examines the population dynamics of solutions such as global existence, global boundedness, and mass persistence, to a parabolic elliptic type of chemotaxis-competition system including logistics kinetics in a smooth bounded domain. Tello and Winkler were the first to investigate the global existence and global boundedness of the system mentioned above. Then Tao and Winkler examined qualitative properties of the given system such as the mass persistence of solutions. This study improves some known results and reveals that under some suitable conditions, there exists a classical solution to the system described above that is globally bounded. In addition, it is shown that the population as a whole is never extinct.

Anahtar Kelimeler

Regular sensitivity , Logistic source , Global existence , Global boundedness , Mass persistence

Teşekkür

I appriciate all your time and consideration. Best Regards, Halil İbrahim Kurt

Kaynakça

• [1] Keller, E.F., Segel, L. A. Segel. 1970. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology, vol. 26, pp. 399-415.
• [2] Keller, E.F., Segel, L. A. Segel. 1971. Traveling bans of chemotactic bacteria: a theoretical analysis. Journal of Theoretical Biology, vol. 30, pp. 377-380.
• [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, vol. 25, pp. 1663-1763.
• [4] Hillen, T., Painter, K. 2009. A user’s guide to PDE models for chemotaxis. Journal of Mathematical Biology, vol. 58, pp. 183-217.
• [5] Horstmann, D. 2004. From 1970 until present: the Keller-Segel model in chemotaxis. Jahresber DMV, vol. 106, pp. 51-69.
• [6] Herrero, M. A., Medina, E., Velzquez, J. J. L. 1996. Singularity patterns in a chemotaxis model. Mathematische Annalen, vol. 306, pp. 583-623.
• [7] Herrero , M. A., Velzquez, J. J. L. 1997. Finite-time aggregation into a single point in a reactiondiffusion system. Nonlinearity, vol. 10, pp. 1739-1754.
• [8] Nagai, T. 2001. Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. Journal of Inequalities and Applications, vol 6, pp. 37-55.
• [9] 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, vol. 8, pp. 145-156.
• [10] Tello, J. I., Winkler, M. 2007. A chemotaxis system with logistic source. Common Partial Differential Equations, vol. 32, pp. 849-877.
• [11] Chaplain , M. A. J., Tello, J. I. 2016. On the stability of homogeneous steady states of a chemotaxis system with logistic growth term. Applied Mathematics Letters, vol. 57, pp. 1-6.
• [12] Hu, B., Tao, Y. 2017. Boundedness in a parabolic elliptic chemotaxis-growth system under a critical parameter condition. Applied Mathematics Letters, vol. 64, pp. 1-7.
• [13] Issa, T. B., Shen, W. 2017. Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources. SIAM Journal on Applied Dynamical Systems, vol. 16, no. 2, pp. 926-973.
• [14] Lankeit, J. 2015. Chemotaxis can prevent thresholds on population density. Discrete and Continuous Dynamical Systems - B, vol. 20, pp. 1499-1527.
• [15] Lankeit, J. 2015. Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. Journal of Differential Equations, vol. 258, pp. 1158-1191.
• [16] Tello, J. I. 2004. Mathematical analysis and stability of a chemotaxis problem with a logistic growth term. Mathematical Models and Methods in Applied Sciences, vol. 27, pp. 1865-1880.
• [17] Viglialoro, G. 2016. Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source. Journal of Mathematical Analysis and Applications, vol. 439, no. 1, pp. 197-212.
• [18] Winkler, M. 2010. Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Common Partial Differential Equations, vol. 35, no. 8, pp. 1516-1537.
• [19] 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, vol. 109, pp. 56-71.
• [20] 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, vol. 53, no. 1, 973-1003.
• [21] Zhao, X. 2023. Boundedness in a logistic chemotaxis system with weakly singular sensitivity in dimension two, Nonlinearity, vol 36, pp. 3909-3938.
• [22] Kurt, H. I. 2025. Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting. Journal of Differential Equations, vol. 416, no. 2, pp. 1429-1461.
• [23] Cao, J., Wang, W., Yu. H. 2016. Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity, Journal of Mathematical Analysis and Applications, vol. 436, no. 1, pp. 382-392.
• [24] 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, vol. 38, no. 6, pp. 1212-1224.
• [25] 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, vol. 69, pp. 103762-27.
• [26] Kurt, H. I. Improvement of criteria for global boundedness in a minimal parabolic-elliptic chemotaxis system with singular sensitivity, Preprint.
• [27] 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, vol. 34, no. 9, pp. 1649-1700.
• [28] 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, vol. 542, pp. 128803.
• [29] Le, M., Kurt, H.I. Global boundedness in a chemotaxis-growth system with weak singular sensitivity in any dimensional setting, Preprint..
• [30] Tao, Y., Winkler, M. 2015. Persistence of mass in a chemotaxis system with logistic source. Journal of Differential Equations, vol. 259 , pp. 6142-6161.
• [31] Henry, D. 1977. Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin, Heidelberg, New York.
• [32] Winkler, M. 2010. Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. Journal of Differential Equations, vol. 248, pp. 2889-2905.
• [33] Tao, Y., Winkler, M. 2015. Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion. SIAM Journal on Mathematical Analysis vol. 47, no. 6.