Abstract
This paper is concerned with the doubly tactic model
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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References
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)
Cai, Y., Cao, Q., Wang, Z.A.: Asymptotic dynamics and spatial patterns of a ratio-dependent predator–prey system with prey–taxis. Appl. Anal. https://doi.org/10.1080/00036811.2020.1728259
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)
Cao, X.: Global radial renormalized solution to a producer–scrounger model with singular sensitivities. Math. Models Methods Appl. Sci. 30, 1119–1165 (2020)
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)
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)
Chakraborty A, A., Singh, M., Lucy, D., et al.: Predator-prey model with prey-taxis and diffusion. Math. Comput. Model. 46, 482–498 (2007)
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)
Guttal, V., Couzin, I.D.: Social interactions, information use, and the evolution of collective migration. Proc. Natl. Acad. Sci. USA 107, 16172–16177 (2010)
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)
Hoffman, W., Heinemann, D., Wiens, J.A.: The ecology of seabird feeding flocks in Alaska. Auk 98, 437–456 (1981)
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)
Jin, H., Wang, Z.A.: Global stability of prey-taxis systems. J. Differ. Equ. 262, 1257–1290 (2017)
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)
Lee, J.M., Hillen, T., Lewis, M.A.: Continuous traveling waves for prey-taxis. Bull. Math. Biol. 70, 654–676 (2008)
Lee, J.M., Hillen, T., Lewis, M.A.: Pattern formation in prey-taxis systems. J. Biol. Dyn. 3, 551–573 (2009)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Wang, J., Wang, M.: Global solution of a diffusive predator–prey model with prey-taxis. Comput. Math. Appl. 77, 2676–2694 (2019)
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)
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)
Winkler, M.: Aggregation versus global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)
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)
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)
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)
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This work is partially supported by NSFC (No.12071030).
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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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DOI: https://doi.org/10.1007/s00033-020-01448-9