Skip to main content
Springer Nature Link
Log in

Global Behavior in a Two-Species Chemotaxis-Competition System with Signal-Dependent Sensitivities and Nonlinear Productions

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions

$$\begin{aligned} \left\{ \begin{array}{l} \begin{aligned} &{}u_t = \nabla \cdot \big (D_1(v)\nabla u-uS_1(v)\nabla v\big )+\mu _1u(1-u^{\alpha _1}-a_1w),&{} x\in \Omega ,\ t>0&{},\\ &{} v_t=\Delta v-v+b_1w^{\gamma _1}, &{} x\in \Omega ,\ t>0&{},\\ &{}w_t = \nabla \cdot \big (D_2(z)\nabla w-wS_2(z)\nabla z\big )+\mu _2w(1-w^{\alpha _2}-a_2u),&{} x\in \Omega ,\ t>0&{},\\ &{} z_t=\Delta z-z+b_2u^{\gamma _2}, &{} x\in \Omega ,\ t>0&{}\\ \end{aligned} \end{array} \right. \end{aligned}$$

in a bounded and smooth domain \(\Omega \subset \mathbb R^2\), where the parameters \(\mu _i, \alpha _i, a_i, b_i, \gamma _i\) \((i=1,2)\) are positive constants, and the functions \(D_1(v),S_1(v),D_2(z),S_2(z)\) fulfill the following hypotheses: \(\Diamond \) \(D_i(\psi ),S_i(\psi )\in C^2([0,\infty ))\), \(D_i(\psi ),S_i(\psi )>0\) for all \(\psi \ge 0\), \(D_i^{\prime }(\psi )<0\) and \(\underset{\psi \rightarrow \infty }{\lim } D_i(\psi )=0\); \(\Diamond \) \(\underset{\psi \rightarrow \infty }{\lim } \frac{S_i(\psi )}{D_i(\psi )}\) and \(\underset{\psi \rightarrow \infty }{\lim } \frac{D^{\prime }_i(\psi )}{D_i(\psi )}\) exist. We first confirm the global boundedness of the classical solution provided that the additional conditions \(2\gamma _1\le 1+\alpha _2\) and \(2\gamma _2\le 1+\alpha _1\) hold. Moreover, by constructing several suitable Lyapunov functionals, it is demonstrated that the global solution exponentially or algebraically converges to the constant stationary solutions and the corresponding convergence rates are determined under some specific stress conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26(3), 399–415 (1970)

    MathSciNet  Google Scholar 

  2. Horstmann, D., Wang, G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)

    MathSciNet  Google Scholar 

  3. Li, T., Frassu, S., Viglialoro, G.: Combining effects ensuring boundedness in an attraction-repulsion chemotaxis model with production and consumption. Z. Angew. Math. Phys. 74(3), 109 (2023)

    MathSciNet  Google Scholar 

  4. Lin, K., Mu, C.: Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete Contin. Dyn. Syst. 36(9), 5025–5046 (2016)

    MathSciNet  Google Scholar 

  5. Winkler, M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Comm. Partial Differ. Equ. 35(8), 1516–1537 (2010)

    MathSciNet  Google Scholar 

  6. Lankeit, J.: Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J. Differ. Equ. 258(4), 1158–1191 (2015)

    MathSciNet  Google Scholar 

  7. Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997)

    MathSciNet  Google Scholar 

  8. Tao, Y.: Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete Contin. Dyn. Syst. Ser. B 18(10), 2705–2722 (2013)

    MathSciNet  Google Scholar 

  9. Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248(12), 2889–2905 (2010)

    MathSciNet  Google Scholar 

  10. Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215(1), 52–107 (2005)

    MathSciNet  Google Scholar 

  11. Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system. J. Math. Pures Appl. (9) 100(5), 748–767 (2013)

    MathSciNet  Google Scholar 

  12. Fu, X., Tang, L.-H., Liu, C., Huang, J.-D., Hwa, T., Lenz, P.: Stripe formation in bacterial systems with density-suppressed motility. Phys. Rev. Lett. 108(19), 198102 (2012)

    Google Scholar 

  13. Yoon, C., Kim, Y.-J.: Global existence and aggregation in a Keller–Segel model with Fokker–Planck diffusion. Acta Appl. Math. 149(1), 101–123 (2017)

    MathSciNet  Google Scholar 

  14. Tao, Y., Winkler, M.: Effects of signal-dependent motilities in a Keller–Segel-type reaction-diffusion system. Math. Models Methods Appl. Sci. 27(09), 1645–1683 (2017)

    MathSciNet  Google Scholar 

  15. Jin, H.-Y., Kim, Y.-J., Wang, Z.-A.: Boundedness, stabilization, and pattern formation driven by density-suppressed motility. SIAM J. Appl. Math. 78(3), 1632–1657 (2018)

    MathSciNet  Google Scholar 

  16. Jin, H.-Y.: Boundedness and large time behavior in a two-dimensional Keller–Segel–Navier–Stokes system with signal-dependent diffusion and sensitivity. Discrete Contin. Dyn. Syst. 38(7), 3595–3616 (2018)

    MathSciNet  Google Scholar 

  17. Winkler, M.: A result on parabolic gradient regularity in Orlicz spaces and application to absorption-induced blow-up prevention in a Keller–Segel-type cross-diffusion system. Int. Math. Res. Not. IMRN 2023(19), 16336–16393 (2023)

    MathSciNet  Google Scholar 

  18. Liu, D.-M., Tao, Y.-S.: Boundedness in a chemotaxis system with nonlinear signal production. Appl. Math. J. Chinese Univ. Ser. A 31(4), 379–388 (2016)

    MathSciNet  Google Scholar 

  19. Winkler, M.: A critical blow-up exponent in a chemotaxis system with nonlinear signal production. Nonlinearity 31(5), 2031–2056 (2018)

    MathSciNet  Google Scholar 

  20. Tao, Y., Winkler, M.: Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B 20(9), 3165–3183 (2015)

    MathSciNet  Google Scholar 

  21. Yu, H., Wang, W., Zheng, S.: Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals. Nonlinearity 31(2), 502–514 (2018)

    MathSciNet  Google Scholar 

  22. Li, X., Wang, Y.: Boundedness in a two-species chemotaxis parabolic system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B 22(7), 2717–2729 (2017)

    MathSciNet  Google Scholar 

  23. Zheng, J.: Boundedness in a two-species quasi-linear chemotaxis system with two chemicals. Topol. Methods Nonlinear Anal. 49(2), 463–480 (2017)

    MathSciNet  Google Scholar 

  24. Zhong, H.: Boundedness in a quasilinear two-species chemotaxis system with two chemicals in higher dimensions. J. Math. Anal. Appl. 500(1), 125130 (2021)

    MathSciNet  Google Scholar 

  25. 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), 111504 (2017)

    MathSciNet  Google Scholar 

  26. Zheng, P., Mu, C., Mi, Y.: Global stability in a two-competing-species chemotaxis system with two chemicals. Differ. Integral Equ. 31(7–8), 547–558 (2018)

    MathSciNet  Google Scholar 

  27. Zhang, Q.: Competitive exclusion for a two-species chemotaxis system with two chemicals. Appl. Math. Lett. 83, 27–32 (2018)

    MathSciNet  Google Scholar 

  28. Tu, X., Mu, C., Zheng, P., Lin, K.: Global dynamics in a two-species chemotaxis-competition system with two signals. Discrete Contin. Dyn. Syst. 38(7), 3617–3636 (2018)

    MathSciNet  Google Scholar 

  29. Black, T.: Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete Contin. Dyn. Syst. Ser. B 22(4), 1253–1272 (2017)

    MathSciNet  Google Scholar 

  30. Qiu, H., Guo, S.: Global existence and stability in a two-species chemotaxis system. Discrete Contin. Dyn. Syst. Ser. B 24(4), 1569–1587 (2019)

    MathSciNet  Google Scholar 

  31. Wang, L., Zhang, J., Mu, C., Hu, X.: Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst. Ser. B 25(1), 191–221 (2020)

    MathSciNet  Google Scholar 

  32. Jin, H.-Y., Liu, Z., Shi, S., Xu, J.: Boundedness and stabilization in a two-species chemotaxis-competition system with signal-dependent diffusion and sensitivity. J. Differ. Equ. 267(1), 494–524 (2019)

    MathSciNet  Google Scholar 

  33. Miao, L., Fu, S.: Global behavior of a two-species predator-prey chemotaxis model with signal-dependent diffusion and sensitivity. Discrete Contin. Dyn. Syst. Ser. B 28(8), 4344–4365 (2023)

    MathSciNet  Google Scholar 

  34. Tello, J.I., Winkler, M.: Stabilization in a two-species chemotaxis system with a logistic source. Nonlinearity 25(5), 1413–1425 (2012)

    MathSciNet  Google Scholar 

  35. Wang, D., Zeng, F., Jiang, M.: Global existence and boundedness of solutions to a two-species chemotaxis-competition system with singular sensitivity and indirect signal production. Z. Angew. Math. Phys. 74(1), 33 (2023)

    MathSciNet  Google Scholar 

  36. Xiang, Y., Zheng, P., Xing, J.: Boundedness and stabilization in a two-species chemotaxis-competition system with indirect signal production. J. Math. Anal. Appl. 507(2), 125825 (2022)

    MathSciNet  Google Scholar 

  37. Xiang, Y., Zheng, P.: On a two-species chemotaxis-competition system with indirect signal consumption. Z. Angew. Math. Phys. 73(2), 50 (2022)

    MathSciNet  Google Scholar 

  38. Liu, A., Dai, B.: Boundedness and stabilization in a two-species chemotaxis system with two chemicals. J. Math. Anal. Appl. 506(1), 125609 (2022)

    MathSciNet  Google Scholar 

  39. Zheng, P., Mu, C.: Global boundedness in a two-competing-species chemotaxis system with two chemicals. Acta Appl. Math. 148(1), 157–177 (2017)

    MathSciNet  Google Scholar 

  40. Tao, Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381(2), 521–529 (2011)

    MathSciNet  Google Scholar 

  41. Kowalczyk, R., Szymańska, Z.: On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343(1), 379–398 (2008)

    MathSciNet  Google Scholar 

  42. Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252(1), 692–715 (2012)

    MathSciNet  Google Scholar 

  43. 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)

    MathSciNet  Google Scholar 

  44. Fu, S., Miao, L.: Global existence and asymptotic stability in a predator-prey chemotaxis model. Nonlinear Anal. Real World Appl. 54, 103079 (2020)

    MathSciNet  Google Scholar 

  45. Porzio, M.M., Vespri, V.: Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103(1), 146–178 (1993)

    Google Scholar 

  46. Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. In: Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)

    Google Scholar 

Download references

Acknowledgements

The authors express their sincere gratitude to the editors and two anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.

Funding

This research is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), and the Operational Programme Integrated Infrastructure (OPII) for the project 313011BWH2: “InoCHF-Research and development in the field of innovative technologies in the management of patients with CHF”, co-financed by the European Regional Development Fund.

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, Shandong, People’s Republic of China

    Zhan Jiao & Tongxing Li

  2. Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, Košice, 040 01, Slovakia

    Irena Jadlovská

Authors
  1. Zhan Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irena Jadlovská
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tongxing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tongxing Li.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiao, Z., Jadlovská, I. & Li, T. Global Behavior in a Two-Species Chemotaxis-Competition System with Signal-Dependent Sensitivities and Nonlinear Productions. Appl Math Optim 90, 11 (2024). https://doi.org/10.1007/s00245-024-10137-2

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-024-10137-2

Keywords

Mathematics Subject Classification

Search

Navigation