NONE 20260512111045721 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Georges Chamoun Faculty of Engineering (ESIB) Saint Joseph University of Beirut B.P 11-514 Riad El Solh, Beyrouth 1107 2050 LEBANON This paper presents a rigorous mathematical analysis of a two-species chemotaxis-competition model on weighted networks, incorporating nonlinear degenerate chemotactic sensitivities and volume-filling effects. The aim is to understand long-term population dynamics in networked environments. Using a Lyapunov functional approach, we characterize the asymptotic behavior of solutions. Under weak competition, coexistence occurs with exponential convergence, while strong competition leads to the extinction of one species and algebraic convergence to a segregated state. These theoretical results are validated through numerical simulations that highlight the effects of competition parameters, network topology, and initial conditions. The proposed numerical scheme demonstrates convergence with exponential or algebraic decay of relative errors, in agreement with the analytical results. This study highlights the role of network structure in shaping chemotaxis-driven competitive dynamics. 05 12 2026 05 12 2026 67 8 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2026.25.8 https://wseas.com/journals/mathematics/2026/a165106-004(2026).pdf A. J. Lotka, Elements of physical biology. Williams and Wilkins, 1925. URL: https://archive.org/details/ elementsofphysic017171mbp/page/n5/ mode/2up (Accessed: 01 April 2026) V. Volterra, Variations and fluctuations of the number of individuals in animal species living together. ICES Journal of Marine Science, Vol. 3, No. 1, 1928, pp. 3-51. DOI: https: //doi.org/10.1093/icesjms/3.1.3 L. Shaikhet and A. Korobeinikov, Persistence and Stochastic Extinction in a LotkaVolterra Predator-Prey Stochastically Perturbed Model. Mathematics, Vol. 12, No. 10, 2024, 1588. DOI: https://doi.org/10.3390/math12101588 S. Samir, F. Ferraro, C. Grilletta, S. Azaele, and A. Maritan, Generalized Lotka-Volterra Systems with Time Correlated Stochastic Interactions. Physical Review Letters, Vol. 133, No. 16, 2024, 167101. DOI: https://doi.org/10.1103/PhysRevLett. 133.167101 D. Wrzosek, Volume filling effect in modelling chemotaxis. Mathematical Modelling of Natural Phenomena, Vol. 5, No. 1, 2010, pp. 123-147. URL: https://doi.org/10.1051/mmnp/20105106. Y. Li and Y.X. Li, Finite-time blow-up in higher-dimensional parabolic-parabolic Chemotaxis system for two species. Nonlinear Analysis: Theory, Methods & Applications, Vol. 109, 2014, pp. 7284. DOI: https: //doi.org/10.1016/j.na.2014.05.021 Q. Zhang, X. Liu, and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals. Journal of Mathematical Physics, Vol. 58, No. 11, 2017, 111504. DOI: https: //doi.org/10.1063/1.5011725 C. Liu and E. Pan, Global solutions to a two-species chemotaxis system with gradient dependent sensitivity. Discrete and Continuous Dynamical Systems Series B, Vol. 29, No. 10, 2024, pp. 4269-4288. DOI: https: //doi.org/10.3934/dcdsb.2024042 D. Tran, T. Tran, and V. Tran, On the nonlocal parabolic-elliptic Keller-Segel model in bounded domains. Communications on Pure and Applied Analysis, Vol. 23, No. 11, 2024, pp. 1730-1747. DOI: https://doi. org/10.3934/cpaa.2024064 G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals. Nonlinear Analysis: Real World Applications, Vol. 48, 2019, pp. 288325. DOI: https://doi.org/10.1016/j.nonrwa. 2019.01.017 L.Wang, C. Mu, X. Hu, and P. Zheng, Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems- B, Vol. 25, No. 1, 2020, pp. 191-221. DOI: https://doi.org/10.3934/dcdsb.2019178 G. Chamoun, M. Ibrahim, M. Saad, and R. Talhouk, Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model. Discrete and Continuous Dynamical Systems Series B, Vol. 25, No. 11, 2020, pp. 4165-4188. DOI:https://doi.org/10.3934/ dcdsb.2020092 G. Li and Y. Yao, Two-species competition model with chemotaxis: well-posedness, stability, and dynamics. Nonlinearity, Vol 35, No. 3, 2022, pp. 1329-1359. DOI: https:// doi.org/10.1088/1361-6544/ac4a8d S. Wang, S. Zhou, S. Shi, and W. Chen, Unconditionally bound preserving and energy dissipative schemes for a class of KellerSegel equations. SIAM Journal on Numerical Analysis, Vol. 58, No. 3, 2020, pp. 16741695. DOI: https://doi.org/10.1137/ 19M1246705 J. Shen and J. Xu, Fully decoupled and energy stable BDF schemes for KellerSegel equations. Journal of Computational Physics, Vol. 449, 2022, 110799. DOI: https://doi. org/10.1016/j.jcp.2021.110799 X. Huang and J. Shein, Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D. Acta Applicandae Mathematicae, Vol. 190, No. 10, 2024. DOI:https: //doi.org/10.1007/s10440-024-00647-0 Y. Lin, S. Liu and Y. Wu, Blow-up phenomenon to the semilinear heat equation for unbounded Laplacians on graphs. Revista Matemtica Complutense, Vol. 38, No. 1, 2025, pp. 281294. DOI: https://doi.org/ 10.1007/s13163-024-00497-2 S. Hou and X. Kong, Existence and asymptotic behaviors of solutions to Chern-Simons systems and equations on finite graphs. Calculus of Variations and Partial Differential Equations, Vol. 64, No. 77, 2025. DOI: https: //doi.org/10.1007/s00526-025-02940-8 G. Chamoun, Mathematical analysis of parabolic models with volume-filling effect in weighted networks. Journal of Dynamics and Differential Equations, Vol. 35, No. 3, 2023, pp. 2115-2137. DOI: https://doi.org/10. 1007/s10884-021-10049-7 H.-Y. Jin and T. Xiang, Convergence rates of solutions for a two-species chemotaxis-navier-stokes system with competitive kinetics. Discrete and continuous dynamical systems series B, Vol. 24, No. 4, 2019, pp. 1919-1942. DOI: https://doi.org/10.3934/dcdsb.2018249