Asymptotic behavior in a doubly tactic resource consumption model with proliferation


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Black, T.: Global generalized solutions to a forager-exploiter model with superlinear degradation and their eventual regularity properties. Math. Models Methods Appl. Sci. 30, 1075–1117 (2020)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cai, Y., Cao, Q., Wang, Z.A.: Asymptotic dynamics and spatial patterns of a ratio-dependent predator–prey system with prey–taxis. Appl. Anal.

  3. 3.

    Cao, X.: Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces. Discrete Contin. Dyn. Syst. Ser. B 35, 1891–1904 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cao, X.: Global radial renormalized solution to a producer–scrounger model with singular sensitivities. Math. Models Methods Appl. Sci. 30, 1119–1165 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cao, X., Tao, Y.: Boundedness and stabilization enforced by mild saturation of taxis in a producer–scrounger model. Nonlinear Anal. Real World Appl. 57, 103189 (2021)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cao, X., Lankeit, J.: Global classical small-data solutions for a 3D chemotaxis Navier–Stokes system involving matrix-valued sensitivities. Calc. Var. PDE 55, 55–107 (2016)

    Article  Google Scholar 

  7. 7.

    Chakraborty A, A., Singh, M., Lucy, D., et al.: Predator-prey model with prey-taxis and diffusion. Math. Comput. Model. 46, 482–498 (2007)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Eftimie, R., De Vries, G., Lewis, M.A.: Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci. USA 104, 6974–6980 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Guttal, V., Couzin, I.D.: Social interactions, information use, and the evolution of collective migration. Proc. Natl. Acad. Sci. USA 107, 16172–16177 (2010)

    Article  Google Scholar 

  10. 10.

    Hieber, M., Pruss, J.: Heat kernels and maximal \(L^p-L^q\) estimates for parabolic evolution equations. Commun. Partial Differ. Equ. 22, 1647–1669 (1997)

    Article  Google Scholar 

  11. 11.

    Hoffman, W., Heinemann, D., Wiens, J.A.: The ecology of seabird feeding flocks in Alaska. Auk 98, 437–456 (1981)

    Google Scholar 

  12. 12.

    Ishida, S., Seki, K., Yokota, T.: Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains. J. Differ. Equ. 256, 2993–3010 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jin, H., Wang, Z.A.: Global stability of prey-taxis systems. J. Differ. Equ. 262, 1257–1290 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Krzyżanowski, P., Winkler, M., Wrzosek, D.: Migration-driven benefit in a two-species nutrient taxis system. Nonlinear Anal. Real World Appl. 48, 94–116 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lee, J.M., Hillen, T., Lewis, M.A.: Continuous traveling waves for prey-taxis. Bull. Math. Biol. 70, 654–676 (2008)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lee, J.M., Hillen, T., Lewis, M.A.: Pattern formation in prey-taxis systems. J. Biol. Dyn. 3, 551–573 (2009)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lewis, M.A.: Spatial coupling of plant and herbivore dynamics: the contribution of herbivore dispersal to transient and persistent waves of damage. Theor. Popul. Biol. 45, 277–312 (1994)

    Article  Google Scholar 

  18. 18.

    Li, J., Pang, P.Y.H., Wang, Y.: Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization. Nonlinearity 32, 2815–2847 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Liu, Y.: Global existence and boundedness of classical solutions to a forager-exploiter model with volume-filling effects. Nonlinear Anal. Real World Appl. 50, 519–531 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Liu, Y., Zhuang, Y.: Boundedness in a high-dimensional forager-exploiter model with nonlinear resource consumption by two species. Z. Angew. Math. Phys. 71, 151 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Myowin, H., Pang, Y.H., Wang, Y.: Asymptotic behavior of classical solutions of a three-dimensional Keller–Segel–Navier–Stokes system modeling coral fertilization. Z. Angew. Math. Phys. 71, 90 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Short, M.B., D’Orsogna, M.R., Pasour, V.B., Tita, G.E., Brantingham, P.J., Bertozzi, A.L., Chayes, L.B.: A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 18, 1249–1267 (2008)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Tania, N., Vanderlei, B., Heath, J.P., Edelstein-Keshet, L.: Role of social interactions in dynamic patterns of resource patches and forager aggregation. Proc. Natl. Acad. Sci. USA 109, 11228–11233 (2012)

    Article  Google Scholar 

  24. 24.

    Tao, Y., Winkler, M.: Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252, 2520–2543 (2012)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Tao, Y., Winkler, M.: Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food. Math. Models Methods Appl. Sci. 29, 2151–2182 (2019)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Wang, J., Wang, M.: Global bounded solution of the higher-dimensional forager-exploiter model with/without growth sources. Math. Models Methods Appl. Sci. 30, 1297–1323 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wang, J., Wang, M.: Global solution of a diffusive predator–prey model with prey-taxis. Comput. Math. Appl. 77, 2676–2694 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Wang, X., Wang, W., Zhang, G.: Global bifurcation of solutions for a predator–prey model with prey-taxis. Math. Methods Appl. Sci. 38, 431–443 (2015)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Winkler, M.: Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions. Math. Models Methods Appl. Sci. 29, 373–418 (2019)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Winkler, M.: Aggregation versus global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)

    Article  Google Scholar 

  31. 31.

    Wu, S., Wang, J., Shi, J.: Dynamics and pattern formation of a diffusive predator–prey model with predator-taxis. Math. Models Methods Appl. Sci. 28, 2275–2312 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Xiang, T.: Global dynamics for a diffusive predator–prey model with prey-taxis and classical Lotka–Volterra kinetics. Nonlinear Anal. Real World Appl. 39, 278–299 (2018)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Yang, C., Cao, X., Jiang, Z., Zheng, S.: Boundedness in a quasilinear fully parabolic Keller–Segel system of higher dimension with logistic source. J. Math. Anal. Appl. 430, 585–591 (2015)

    MathSciNet  Article  Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Yifu Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, J., Wang, Y. Asymptotic behavior in a doubly tactic resource consumption model with proliferation. Z. Angew. Math. Phys. 72, 21 (2021).

Download citation


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

Mathematics Subject Classification

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