Abstract
This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions
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.
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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.
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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
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DOI: https://doi.org/10.1007/s00245-024-10137-2