Global dynamics and spatio-temporal patterns in a two-species chemotaxis system with two chemicals

Abstract

In this paper, we consider the signal-dependent diffusion and sensitivity in a chemotaxis–competition population system with two different signals in a two-dimensional bounded domain. We consider more general signal production functions and assume that the signal-dependent diffusion is a decreasing function which may be degenerate with respect to the density of the corresponding signal. We first obtain the global existence and uniform-in-time bound of classical solutions and show that the blow-up effect can be precluded for signal-dependent diffusion and sensitivity with certain properties. Then, by constructing Lyapunov functionals, we study the global attractivity of nonzero (boundary/positive) homogeneous steady states under three different strengths of competition. In particular, we obtain that the nonzero boundary constant steady states are globally asymptotically stable when they are globally attractive, which means no pattern formation occurs, while for interior constant steady state, its global attractivity can imply the global stability for some special signal production functions. Finally, numerical simulations show that for large signal sensitivity, different signal production functions can lead to various complex spatial–temporal patterns around the positive homogeneous steady state. In particular, for a given signal production mechanism, various patterns are observed for different population growth rates.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of People’s Republic of China (11671123, 11801089, 12071446), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan), and Jiangxi Provincial Natural Science Foundation (No. 20202BAB211003).

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Correspondence to Shangjiang Guo.

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Gao, J., Guo, S. Global dynamics and spatio-temporal patterns in a two-species chemotaxis system with two chemicals. Z. Angew. Math. Phys. 72, 25 (2021). https://doi.org/10.1007/s00033-020-01449-8

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Mathematics Subject Classification

  • 35B32
  • 35B36
  • 92D40
  • 92D25

Keywords

  • Chemotaxis
  • Signal-dependent
  • Competition model
  • Global existence
  • Lyapunov functional
  • Spatial–temporal patterns