Asymptotic behavior in a doubly tactic resource consumption model with proliferation

Abstract

This paper is concerned with the doubly tactic model

$$\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta u-\chi _u\nabla \cdot (u\nabla w)+uw,&{} x\in \Omega , t>0, \\ v_t=\Delta v-\chi _v\nabla \cdot (v\nabla u)+vw, &{}x\in \Omega , t>0, \\ w_t=\Delta w-\lambda (u+v)w-\mu w,&{} x\in \Omega , t>0, \end{array}\right. \end{aligned}$$

in a smoothly bounded domain \(\Omega \subset {\mathbb {R}}^N\) (\(N\ge 1\)) with positive parameters \(\chi _u, \chi _v, \lambda \) and nonnegative parameter \(\mu \), for the spatiotemporal evolution of forager–exploiter groups u and v, which simultaneously consume a common nutrient w and proliferate. It is shown that for all suitably regular small initial data, the corresponding Neumann initial-boundary value problem possesses a globally classical solution, which approaches spatially homogeneous profiles at an exponential rate.

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Acknowledgements

This work is partially supported by NSFC (No.12071030).

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Correspondence to Yifu Wang.

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Li, J., Wang, Y. Asymptotic behavior in a doubly tactic resource consumption model with proliferation. Z. Angew. Math. Phys. 72, 21 (2021). https://doi.org/10.1007/s00033-020-01448-9

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Keywords

  • Forager–exploiter
  • Classical solution
  • Sequential taxis
  • Asymptotic behavior

Mathematics Subject Classification

  • 35K55
  • 35B45
  • 35B40
  • 35Q92
  • 92C17