Weak traveling wave solutions to a two-species competitive chemotaxis system incorporating Holling-Ⅱ functional responses

Shuyang Yu; Yujuan Jiao; Ziji Dang; Shuyang Yu; Yujuan Jiao; Ziji Dang

doi:10.3934/math.2026908

AIMS Mathematics

2026, Volume 11, Issue 7: 22432-22455. doi: 10.3934/math.2026908

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Research article

Weak traveling wave solutions to a two-species competitive chemotaxis system incorporating Holling-Ⅱ functional responses

College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730124, China

Received: 06 May 2026 Revised: 06 July 2026 Accepted: 20 July 2026 Published: 27 July 2026
MSC : 34C37, 35C07

This work focuses on analyzing a two-species competitive chemotaxis model with Holling-Ⅱ functional responses. First, the stability of the positive equilibrium point $ E^* $ is analyzed by applying the continuous Fourier transform and the Routh-Hurwitz criterion. Second, proof by contradiction and the generalized principal eigenvalue argument are employed to prove the non-existence of weak traveling wave solutions for sufficiently small wave speeds $ c $. Finally, the existence of weak traveling wave solutions is proved via the upper and lower solutions method and Schauder's fixed-point theorem for all sufficiently large $ c $.
- two-species competition model,
- chemotaxis,
- traveling wave solution,
- upper and lower solutions,
- Schauder's fixed-point theorem
Citation: Shuyang Yu, Yujuan Jiao, Ziji Dang. Weak traveling wave solutions to a two-species competitive chemotaxis system incorporating Holling-Ⅱ functional responses[J]. AIMS Mathematics, 2026, 11(7): 22432-22455. doi: 10.3934/math.2026908

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Abstract

This work focuses on analyzing a two-species competitive chemotaxis model with Holling-Ⅱ functional responses. First, the stability of the positive equilibrium point $ E^* $ is analyzed by applying the continuous Fourier transform and the Routh-Hurwitz criterion. Second, proof by contradiction and the generalized principal eigenvalue argument are employed to prove the non-existence of weak traveling wave solutions for sufficiently small wave speeds $ c $. Finally, the existence of weak traveling wave solutions is proved via the upper and lower solutions method and Schauder's fixed-point theorem for all sufficiently large $ c $.

References

[1]	Y. Wang, S. Guo, Traveling wave solutions for a two-species competitive Keller-Segel chemotaxis system, Nonlinear Anal.-Real, 73 (2023), 103900. https://doi.org/10.1016/j.nonrwa.2023.103900 doi: 10.1016/j.nonrwa.2023.103900
[2]	E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399–415. https://doi.org/10.1016/0022-5193(70)90092-5 doi: 10.1016/0022-5193(70)90092-5
[3]	E. Keller, L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. https://doi.org/10.1016/0022-5193(71)90051-8 doi: 10.1016/0022-5193(71)90051-8
[4]	Z. Feng, J. Jia, S. Zhou, Stability of steady-state solutions of a class of Keller-Segel models with mixed boundary conditions, Math. Method. Appl. Sci., 47 (2024), 9476–9492. https://doi.org/10.1002/mma.10079 doi: 10.1002/mma.10079
[5]	T. Li, Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differ. Equations, 250 (2011), 1310–1333. https://doi.org/10.1016/j.jde.2010.09.020 doi: 10.1016/j.jde.2010.09.020
[6]	X. Pan, C. Mu, W. Tao, Global dynamics and spatiotemporal patterns of a two-species chemotaxis system with chemical signaling loop and Lotka-Volterra competition, Stud. Appl. Math., 153 (2024), e12746. https://doi.org/10.1111/sapm.12746 doi: 10.1111/sapm.12746
[7]	T. B. Issa, R. B. Salako, W. Shen, Traveling wave solutions for two species competitive chemotaxis systems, Nonlinear Anal., 212 (2021), 112480. https://doi.org/10.1016/j.na.2021.112480 doi: 10.1016/j.na.2021.112480
[8]	D. Li, N. Tan, H. Qiu, Traveling wave solutions in a modified Leslie-Gower model with diffusion and chemotaxis, Z. Angew. Math. Phys., 75 (2024), 165. https://doi.org/10.1007/s00033-024-02308-6 doi: 10.1007/s00033-024-02308-6
[9]	D. Li, S. Guo, Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect, J. Math. Anal. Appl., 467 (2018), 1080–1099. https://doi.org/10.1016/j.jmaa.2018.07.050 doi: 10.1016/j.jmaa.2018.07.050
[10]	T. Li, Z. Wang, Traveling wave solutions of a singular Keller-Segel system with logistic source, Math. Biosci. Eng., 19 (2022), 8107–8131. https://doi.org/10.3934/mbe.2022379 doi: 10.3934/mbe.2022379
[11]	C. Song, Z. Wang, Traveling wave solutions for a density-suppressed motility model with strong Allee effect, J. Math. Anal. Appl., 555 (2026), 130063–130063. https://doi.org/10.1016/j.jmaa.2025.130063 doi: 10.1016/j.jmaa.2025.130063
[12]	A. R. Muok, F. A. Olsthoorn, A. Briegel, Unpacking alternative features of the bacterial chemotaxis system, Annu. Rev. Microbiol., 78 (2024), 169–189. https://doi.org/10.1146/annurev-micro-032421-110850 doi: 10.1146/annurev-micro-032421-110850
[13]	S. Ai, Z. Du, Traveling wave solutions for a Keller-Segel system with nonlinear chemical gradient, J. Math. Anal. Appl., 545 (2025), 129128. https://doi.org/10.1016/j.jmaa.2024.129128 doi: 10.1016/j.jmaa.2024.129128
[14]	J. Li, Z. Wang, Traveling wave solutions to the density-suppressed motility model, J. Differ. Equations, 301 (2021), 1–36. https://doi.org/10.1016/j.jde.2021.07.038 doi: 10.1016/j.jde.2021.07.038
[15]	R. B. Salako, W. Shen, Existence of traveling wave solutions of parabolic-parabolic chemotaxis systems, Nonlinear Anal.-Real, 42 (2018), 93–119. https://doi.org/10.1016/j.nonrwa.2017.12.004 doi: 10.1016/j.nonrwa.2017.12.004

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