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Dynamics in two-predator and one-prey models with signal-dependent motility

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Abstract

This paper deals with the global boundedness and asymptotic stability of the solution of the two-predator and one-prey systems with density-dependent motion in a n-dimensional bounded domain with Neumann boundary conditions. In a previous paper, Qiu et al. (J Dyn Differ Equ, 1–25, 2021) proved the global existence and uniform boundedness of classical solution by limiting the conditions on motility functions and the coefficients of logistic source. By contrast, we relax the limitation conditions in Qiu et al. (2021) by constructing the weight function. Moreover, under diverse competition circumstances, the global stabilities of nonnegative spatially homogeneous equilibria for the special model are established.

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References

  1. Amann, H.: Dynamic theory of quasilinear parabolic equations. II. Reaction–diffusion systems. Differ. Integral Equ. 3(1), 13–75 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Amann, H.: Dynamic theory of quasilinear parabolic systems. III. Math. Z. 202(2), 219–250 (1989)

    MathSciNet  MATH  Google Scholar 

  3. Ahn I., Yoon C.: Global well-posedness and stability analysis of prey–predator model with indirect prey-taxis. J. Differ. Equ. 268(8) (2019)

  4. Ahn, J., Yoon, C.: Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing. Nonlinearity 32, 1327–1351 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Bai, X., Winkler, M.: Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. 65, 553–583 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Chesson, P.L., Murdoch, W.W.: Aggregation of risk: relationships among host-parasitoid models. Am. Nat. 127(5), 696–715 (1986)

    Google Scholar 

  7. Desvillettes, L., Kim, Y.J., Trescases, A., Yoon, C.: A logarithmic chemotaxis model featuring global existence and aggregation. Nonlin. Anal. Real World Appl. 50, 562–582 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Fuest, M.: Global solutions near homogeneous steady states in a multidimensional population model with both predator-and prey-taxis. SIAM J. Math. Anal. 52(6), 5865–5891 (2020)

    MathSciNet  MATH  Google Scholar 

  9. Fujie, K., Jiang, J.: Boundedness of classical solutions to a degenerate Keller–Segel type model with signal-dependent motilities. Acta Appl. Math. 176, 3 (2021)

    MathSciNet  MATH  Google Scholar 

  10. Fujie, K., Senba, T.: Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions. Nonlinear Anal.: Int. Multidiscip. J. 222, 112987 (2022)

    MathSciNet  MATH  Google Scholar 

  11. He, X., Zheng, S.: Global boundedness of solutions in a reaction–diffusion system of predator–prey model with prey-taxis. Appl. Math. Lett. 49, 73–77 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Hsiao, L., Xin, Z.P.: On the asymptotic behavior of solutions of a reacting-diffusing system: a two predators-one prey model. SIAM J. Math. Anal. 18(3), 647–669 (1987)

    MathSciNet  MATH  Google Scholar 

  13. Jie, J., Hao, W., Zheng, S.: Blow-up for a Three dimensional Keller–Segel model with consumption of chemoattractant. J. Differ. Equ. 264(8), 5432–5464 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Jin, H.Y., Wang, Z.A.: Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion. Eur. J. Appl. Math. 32(4), 652–682 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Jin, H.Y., Wang, Z.A.: Global stability of prey-taxis systems. J. Differ. Equ. 262(3), 1257–1290 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Kareiva, P., Odell, G.: Swarms of predators exhibit “preytaxis’’ if individual predators use area-restricted search. Am. Nat. 130(2), 233–270 (1987)

    Google Scholar 

  17. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)

    MathSciNet  MATH  Google Scholar 

  18. Kozono, H., Sugiyama, Y.: Global strong solution to the semi-linear Keller–Segel system of parabolic-parabolic type with small data in scale invariant spaces. J. Differ. Equ. 247, 1–32 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence, RI (1968)

    Google Scholar 

  20. Lankeit, J.: A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 39, 394–404 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Lankeit, J.: Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion. J. Differ. Equ. 262(7), 4052–4084 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Li, C., Wang, X., Shao, Y.: Steady states of a predator-prey model with prey-taxis. Nonlinear Anal. 97, 155–168 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Li, G., Winkler, M.: Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities. arXiv:2206.13327

  24. Li, G., Winkler, M.: Relaxation in a Keller–Segel-consumption system involving signal-dependent motilities. arXiv:2206.13292

  25. Lin, J.J., Wang, W., Zhao, C., Yang, T.H.: Global dynamics and traveling wave solutions of two predators-one prey models. Discrete Contin. Dyn. Syst. Ser. B. 20(4), 1135–1154 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Liu, D.: Global classical solution to a chemotaxis consumption model with singular sensitivity. Nonlinear Anal. Real World Appl. 41, 497–508 (2018)

  27. Liu, Y., Tao, Y.: Dynamics in a parabolic-elliptic two-species population competition model with cross-diffusion for one species. J. Math. Anal. Appl. 456(1), 1–15 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Loladze, I., Kuang, Y., Elser, J.J., Fagan, W.F.: Competition and stoichiometry: coexistence of two predators on one prey. Theor. Popul. Biol. 65(1), 1–15 (2004)

    MATH  Google Scholar 

  29. Mu, C., Wang, L., Zheng, P., Zhang, Q.: Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system. Nonlinear Anal. Real World Appl. 14, 1634–1642 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Nagai, T.: Global existence and blow-up of solutions to a chemotaxis system. Nonlinear Anal. 47, 777–787 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives. Springer, New York (2001)

    MATH  Google Scholar 

  32. Pang, P.Y.H., Wang, M.: Strategy and stationary pattern in a three-species predator–prey model. J. Differ. Equ. 200(2), 245–273 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Porzio, M.M., Vespri, V.: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103(1), 146–178 (1993)

    MATH  Google Scholar 

  34. Qiu, S., Mu, C., Tu, X.: Dynamics for a three-species predator–prey model with density-dependent motilities. J. Dyn. Differ. Equ. 1–25 (2021)

  35. Sapoukhina, N., Tyutyunov, Y., Arditi, R.: The role of prey taxis in biological control: a spatial theoretical model. Am. Nat. 162(1), 61–76 (2003)

    Google Scholar 

  36. Stinner, C., Winkler, M.: Global weak solutions in a chemotaxis system with large singular sensitivity. Nonlin. Anal. RWA 12(6) (2011)

  37. Tao, Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381(2), 521–529 (2011)

    MathSciNet  MATH  Google Scholar 

  38. Tao, Y., Winkler, M.: A fully cross-diffusive two-component evolution system: existence and qualitative analysis via entropy-consistent thin-film-type approximation. J. Funct. Anal. 281(4), 109069 (2021)

    MathSciNet  MATH  Google Scholar 

  39. Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252(1), 692–715 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Tao, Y., Winkler, M.: Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system. Math. Mod. Meth. Appl. Sci. 27, 1645–1683 (2017)

    MathSciNet  MATH  Google Scholar 

  41. 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(3), 2520–2543 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Tello, J.I., Wrzosek, D.: Predator–prey model with diffusion and indirect prey-taxis. Math. Models Meth. Appl. Sci. 26(11), 2129–2162 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Ton, T.V., Hieu, N.T.: Dynamics of species in a model with two predators and one prey. Nonlinear Anal. 74(14), 4868–4881 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Tsyganov, M.A., Brindley, J., Holden, A.V., Biktashev, V.N.: Quasi-soliton interaction of pursuit-evasion waves in a predator–prey system. Phys. Rev. Lett. 91, 218102 (2003)

    Google Scholar 

  45. Wang, J., Wang, M.: Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis. Z. Angew. Math. Phys. 69(3), 1–24 (2018)

    MathSciNet  MATH  Google Scholar 

  46. Wang, J., Wang, M.: Global solution of a diffusive predator-prey model with prey-taxis. Comput. Math. Appl. 77(10), 2676–2694 (2019)

    MathSciNet  MATH  Google Scholar 

  47. Wang, Q., Song, Y., Shao, L.: Nonconstant positive steady states and pattern formation of 1D prey-taxis systems. J. Nonlinear Sci. 27(1), 71–97 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity. Math. Nachr. 283(2), 1664–1673 (2011)

    MathSciNet  MATH  Google Scholar 

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

  50. Winkler, M.: Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model. Bull. Math. Sci. 2250012 (2022)

  51. Winkler, M.: A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion. arXiv:2209.12724

  52. Winkler, M.: Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation. J. Differ. Equ. 263(8), 4826–4869 (2017)

    MathSciNet  MATH  Google Scholar 

  53. Winkler, M.: Chemotactic cross-diffusion in complex frameworks. Math. Mod. Meth. Appl. Sci. 26(11) (2016)

  54. Winkler, M.: Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption. J. Differ. Equ. 264, 2310–2350 (2018)

    MathSciNet  MATH  Google Scholar 

  55. Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Arch. Ration. Mech. An. 211(2), 455–487 (2014)

    MathSciNet  MATH  Google Scholar 

  56. Winkler, M.: The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties. Math. Models Methods Appl. Sci. 26, 987–1024 (2016)

    MathSciNet  MATH  Google Scholar 

  57. Wu, S., Shi, J., Wu, B.: Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis. J. Differ. Equ. 260(7), 5847–5874 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  59. Xu, H., Wang, L.: Global existence and asymptotic stability of solutions to a forager-exploiter model with logistic source. Z. Angew. Math. Phys. 74(4) (2023)

  60. Yan, J., Li, Y.: Global generalized solutions to a Keller–Segel system with nonlinear diffusion and singular sensitivity. Nonlinear Anal. 176, 288–302 (2018)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to referees for their valuable comments, which greatly improved the exposition of the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 1771062) and Chongqing Natural Science Foundation (Grant No. cstc2017jcyjXB0037).

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  1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Duo Zhang & Xuegang Hu

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Zhang, D., Hu, X. Dynamics in two-predator and one-prey models with signal-dependent motility. Z. Angew. Math. Phys. 74, 75 (2023). https://doi.org/10.1007/s00033-023-01967-1

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