Abstract
The current paper is devoted to the study of the spreading speeds of two species competition diffusion-advection systems in time almost periodic and space periodic media. We first show that there is a finite spreading speed interval for such diffusion-advection systems. The principal Lyapunov exponent and the principal Floquent bundle theory have been applied to study the spreading speed of time almost periodic and space periodic systems. Under some sufficient conditions, we prove that the spreading speed interval of such systems in any direction is a singleton in the partially spatially homogeneous case and the general case, respectively.
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Bao, X., Li, W.-T.: Propagation phenomena for partially degenerate nonlocal dispersal models in time and space periodic habitats. Nonlinear Anal., Real World Appl. 51, 102975 (2020)
Bao, X., Wang, Z.-C.: Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system. J. Differ. Equ. 255, 2402–2435 (2013)
Bao, X., Li, W.-T., Shen, W., Wang, Z.-C.: Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems. J. Differ. Equ. 265, 3048–3091 (2018)
Bao, X., Shen, W., Shen, Z.: Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Commun. Pure Appl. Anal. 18, 361–396 (2019)
Bao, X., Li, W.-T., Wang, Z.-C.: Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Commun. Pure Appl. Anal. 19, 253–277 (2020)
Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55, 949–1032 (2002)
Besicovitch, A.S.: Almost Periodic Functions. Cambridge University Press, Cambridge (1932)
Cao, F., Shen, W.: Spreading speeds and transition fronts of lattice KPP equations in time heterogeneous media. Discrete Contin. Dyn. Syst. 37, 4697–4727 (2017)
Fang, J., Yu, X., Zhao, X.-Q.: Traveling waves and spreading speeds for time-space periodic monotone systems. J. Funct. Anal. 272, 4222–4262 (2017)
Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Math., vol. 377. Springer, Berlin (1974)
Fischer, A.: Approximation of almost periodic functions by periodic ones. Czechoslov. Math. J. 48, 193–205 (1998)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Guo, J.-S., Liang, X.: The minimal speed of traveling fronts for Lotka-Volterra competition system. J. Dyn. Differ. Equ. 23, 353–363 (2011)
Han, B.S., Wang, Z.C., Du, Z.: Traveling waves for nonlocal Lotka-Volterra competition systems. Discrete Contin. Dyn. Syst., Ser. B 25, 1959–1983 (2020)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math., vol. 840. Springer, Berlin (1981)
Hosono, Y.: The minimal spread of traveling fronts for a diffusive Lotka-Volterra competition model. Bull. Math. Biol. 66, 435–448 (1998)
Huang, W.: Problem on minimum wave speed for a Lotka-Volterra reaction diffusion competition model. J. Dyn. Differ. Equ. 22, 285–297 (2010)
Huang, W., Han, M.: Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model. J. Differ. Equ. 251, 1549–1561 (2011)
Huang, J., Shen, W.: Speeds of spread and propagation for KPP models in time almost and space periodic media. SIAM J. Appl. Dyn. Syst. 8, 790–821 (2009)
Kan-on, Y.: Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. SIAM J. Math. Anal. 26, 340–363 (1995)
Kan-on, Y.: Fisher wave fronts for the Lotka-Volterra competition model with diffusion. Nonlinear Anal. 28, 145–164 (1997)
Kong, L., Rawal, N., Shen, W.: Spreading speeds and linear determinacy for two species competition systems with nonlocal dispersal in periodic habitats. Math. Model. Nat. Phenom. 10, 113–141 (2015)
Lewis, M., Weinberger, H., Li, B.: Spreading speed and linear determinacy for two species competition models. J. Math. Biol. 45, 219–233 (2002)
Li, B., Weinberger, H.F., Lewis, M.: Spreading speeds as slowest wave speeds for cooperative system. Math. Biosci. 196, 82–98 (2005)
Li, W.-T., Lin, G., Ruan, S.: Existence of traveling wave solutions in delayed reaction diffusion systems with applications to diffusion competition system. Nonlinearity 19, 1253–1273 (2006)
Liang, X., Zhao, X.-Q.: Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007)
Liang, X., Zhao, X.-Q.: Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259, 857–903 (2012)
Lim, T., Zlatos, A.: Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion. Trans. Am. Math. Soc. 368, 8615–8631 (2016)
Lui, R.: Biological growth and spread modeled by system of recursions. I. Mathematical theory. Math. Biosci. 93, 269–295 (1989)
Mierczyński, J., Shen, W.: Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations. J. Differ. Equ. 191, 175–205 (2003)
Nadin, G., Rossi, L.: Propagation phenomena for time heterogeneous KPP reaction-diffusion equations. J. Math. Pures Appl. 98, 633–653 (2012)
Nadin, G., Rossi, L.: Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients. Anal. PDE 8, 1351–1377 (2015)
Nolen, J., Xin, J.: Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete Contin. Dyn. Syst. 13, 1217–1234 (2005)
Nolen, J., Rudd, M., Xin, J.: Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. Dyn. Partial Differ. Equ. 2, 1–24 (2005)
Poláčik, P., Jereščák, I.: Exponential separation and invariant bundles for maps in ordered Banaches space with applications to parabolic equation. J. Dyn. Differ. Equ. 5(2), 279–303 (1993)
Rossi, L., Ryzhik, L.: Transition waves for a class of space-time dependent monostable equations. Commun. Math. Sci. 12, 879–900 (2014)
Sell, G.R.: Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold Company, New York (1971)
Shen, W.: Traveling waves in diffusive random media. J. Dyn. Differ. Equ. 16, 1011–1060 (2004)
Shen, W.: Spreading and generalized propagating speeds of discrete KPP models in time varying environments. Front. Math. China 4, 523–562 (2009)
Shen, W.: Variational principle for spatial spreading speed and generalized wave solutions in time almost periodic and space periodic KPP model. Trans. Am. Math. Soc. 362, 5125–5168 (2010)
Shen, W.: Existence of generalized traveling wave in time recurrent and space periodic monostable equations. J. Appl. Anal. Comput. 1, 69–93 (2011)
Shen, W.: Existence, uniqueness, and stability of generalized traveling waves in time dependent of monostable equations. J. Dyn. Differ. Equ. 23, 1–44 (2011)
Shen, W.: Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence. Nonlinearity 30(9), 3466–3491 (2017)
Tao, T., Zhu, B., Zlatos, A.: Transition fronts for inhomogeneous monostable reaction diffusion equations via linearization at zero. Nonlinearity 12 (2014). https://doi.org/10.1088/0951-7715/27/9/2409
Weinberger, H.F., Lewis, M., Li, B.: Analysis of linear determinacy for speed in cooperative models. J. Math. Biol. 45, 183–218 (2002)
Yi, Y.: Almost automorphic and almost periodic dynamics in skew-product semiflows, Part I. Almost automorphy and almost periodicity. Mem. Am. Math. Soc. 136(647) (1998). https://doi.org/10.1090/memo/0647
Yu, X., Zhao, X.-Q.: Propagation phenomena for a reaction advection diffusion competition model in a periodic habitat. J. Dyn. Differ. Equ. 29(1), 41–66 (2017)
Zhang, L., Li, W.T., Wang, Z.C., Sun, Y.J.: Entire solutions for nonlocal dispersal equations with bistable nonlinearity: asymmetric case. Acta Math. Sin. Engl. Ser. 35, 1771–1794 (2019)
Zhao, G., Ruan, S.: Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion. J. Math. Pures Appl. 95, 627–671 (2011)
Zhao, G., Ruan, S.: Time periodic traveling wave solutions for periodic advection reaction diffusion systems. J. Differ. Equ. 257, 1078–1147 (2014)
Acknowledgements
The author would like to thank the referee for valuable comments and suggestions which improved the presentation of this manuscript. Xiongxiong Bao was partially supported by NSF of China (11701041), Natural Science Basic Research Plan in Shaanxi Province of China (2020JM-223) and the Fundamental Research Funds for the Central Universities (300102129201), CHD.
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Bao, X. Spreading Speeds for Two Species Competition Systems in Time Almost Periodic and Space Periodic Media. Acta Appl Math 171, 11 (2021). https://doi.org/10.1007/s10440-020-00376-0
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DOI: https://doi.org/10.1007/s10440-020-00376-0