Propagation dynamics for time-periodic and partially degenerate reaction-diffusion systems. (English) Zbl 1492.35088
In this paper, the authors study the propagation dynamics for partially degenerate reaction-diffusion systems with monostable and time-periodic nonlinearity. In the case where the considered system is cooperative, they establish the existence and exponential stability of periodic traveling fronts. In the case where the considered system is noncooperative, the authors also obtain the existence of the minimal wave speed of periodic traveling waves and its coincidence with the spreading speed. Due to the lack of compactness and standard parabolic estimates for the partially degenerate diffusion system, the authors prove the existence result by appealing to the asymptotic fixed point theorem with the help of some properties of the Kuratowski measure of noncompactness. In particular, this paper provides some new ideas to construct upper and lower solutions for the noncooperative system.
Reviewer: Jia-Bing Wang (Wuhan)
MSC:
| 35C07 | Traveling wave solutions |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K57 | Reaction-diffusion equations |
| 34K30 | Functional-differential equations in abstract spaces |
| 92D25 | Population dynamics (general) |
| 92D30 | Epidemiology |
Keywords:
periodic traveling waves; partially degenerate systems; noncooperative nonlinearity; asymptotic speed of spreadReferences:
| [1] | N. D. Alikakos, P. W. Bates, and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), pp. 2777-2805. · Zbl 0929.35067 |
| [2] | S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual, and P. Rohani, Seasonality and the dynamics of infectious diseases, Ecol. Lett., 9 (2006), pp. 467-684. |
| [3] | D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), pp. 33-76. · Zbl 0407.92014 |
| [4] | J. Banas, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 21 (1980), pp. 131-143. · Zbl 0438.47051 |
| [5] | X. Bao, W.-T. Li, W. Shen, and Z.-C. Wang, Spreading speeds and linear determinacy of time dependent diffusive cooperative/competitive systems, J. Differential Equations, 265 (2018), pp. 3048-3091. · Zbl 1395.35011 |
| [6] | X. Bao and Z.-C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013), pp. 2402-2435. · Zbl 1371.35017 |
| [7] | H. Berestycki, F. Hamel, and L. Roques, Analysis of the periodically fragmented environment model: II-Biological invasions and pulsating traveling fronts, J. Math. Pures Appl., 84 (2005), pp. 1101-1146. · Zbl 1083.92036 |
| [8] | V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bactetial and viral diseases, J. Math. Biol., 13 (1981), pp. 173-184. · Zbl 0468.92016 |
| [9] | X. Chen, J.-S. Guo, and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), pp. 189-236. · Zbl 1152.37033 |
| [10] | K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. · Zbl 0559.47040 |
| [11] | W. Ding, Y. Du, and X. Liang, Spreading in space-time periodic media governed by a monostable equation with free boundaries, Part 2: Spreading speed, AIHP Anal. Non Lineaire, 36 (2019), pp. 1539-1573. · Zbl 1421.35191 |
| [12] | J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), pp. 3678-3704. · Zbl 1315.35051 |
| [13] | J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc. (JEMS), 17 (2015), pp. 2243-2288. · Zbl 1352.37072 |
| [14] | P. C. Fife and M. Tang, Comparison principles for reaction-diffusion systems: Irregular comparison functions and applications to questions of stability and speed of propagation of disturbances, J. Differential Equations, 40 (1981), pp. 168-185. · Zbl 0431.35008 |
| [15] | J. K. Hale and O. Lopes, Fixed point theorems and dissipative processes, J. Differential Equations, 13 (1973), pp. 391-402. · Zbl 0256.34069 |
| [16] | F. Hamel, Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity, J. Math. Pures Appl., 89 (2008), pp. 355-399. · Zbl 1171.35061 |
| [17] | F. Hamel and L. Roques, Uniqueness and stability properties of monostable pulsating fronts, J. Eur. Math. Soc. (JEMS), 13 (2011), pp. 345-390. · Zbl 1219.35035 |
| [18] | Y. Jin and X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), pp. 2496-2516. · Zbl 1179.34031 |
| [19] | W.-T. Li, J. Wang, and X.-Q. Zhao, Propagation dynamics in a time periodic nonlocal dispersal model with stage structure, J. Dynam. Differential Equations, 32 (2020), pp. 1027-1064. · Zbl 1442.35211 |
| [20] | X. Liang, Y. Yi, and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), pp. 57-77. · Zbl 1105.37017 |
| [21] | X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 61 (2008), pp. 1-40. · Zbl 1106.76008 |
| [22] | F. Lutscher and G. Seo, The effect of temporal variability on persistence conditions in rivers, J. Theoret. Biol., 283 (2011), pp. 53-59. · Zbl 1397.92586 |
| [23] | G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, River Edge, NJ, 1996. · Zbl 0884.35001 |
| [24] | R. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Boston, 1995. · Zbl 0816.35001 |
| [25] | R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), pp. 1-44. · Zbl 0722.35046 |
| [26] | R. D. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc., 171 (1972), pp. 349-375. · Zbl 0256.47040 |
| [27] | W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), pp. 1-44. · Zbl 1223.35103 |
| [28] | H. L. Smith, Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995. · Zbl 0821.34003 |
| [29] | F.-B. Wang, A periodic reaction-diffusion model with a quiescent stage, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), pp. 283-295. · Zbl 1235.35165 |
| [30] | H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21 (2011), pp. 747-783. · Zbl 1242.35147 |
| [31] | H. F. Weinberger, On spreading speeds and traveling waves for growth and migration in periodic habitat, J. Math. Biol., 45 (2002), pp. 511-548. · Zbl 1058.92036 |
| [32] | S.-L. Wu and C.-H. Hsu, Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity, Adv. Nonlinear Anal., 9 (2020), pp. 923-957. · Zbl 1427.35131 |
| [33] | S.-L. Wu and M. Huang, Time periodic traveling waves for a periodic nonlocal dispersal model with delay, Proc. Amer. Math. Soc., 148 (2020), pp. 4405-4421. · Zbl 1445.35107 |
| [34] | X. Yu and X.-Q. Zhao, A periodic reaction-advection-diffusion model for a stream population, J. Differential Equations, 258 (2015), pp. 3037-3062. · Zbl 1335.35132 |
| [35] | L. Zhang, Z.-C. Wang, and X.-Q. Zhao, Time peridic traveling waves for a periodic and diffusive SIR epidemic model, J. Dynam. Differential Equations, 30 (2018), pp. 379-403. · Zbl 1384.35044 |
| [36] | L. Zhang, Z.-C. Wang, and X.-Q. Zhao, Propagation dynamics of a periodic and delayed reactiondiffusion model without quasi-monotonicity, Trans. Amer. Math. Soc., 472 (2019), pp. 1751-1782. · Zbl 1419.35117 |
| [37] | L. Zhang, Z.-C. Wang, and X.-Q. Zhao, Time periodic traveling wave solutions for a Kermark-McKendrick epidemic model with diffusion and seasonality, J. Evol. Equ., 20 (2019), pp. 1029-1059. · Zbl 1447.35104 |
| [38] | G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), pp. 627-671. · Zbl 1227.35125 |
| [39] | G. Zhao and S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014), pp. 1078-1147. · Zbl 1304.35199 |
| [40] | X.-Q. Zhao, Dynamical Systems in Population Biology, 2nd ed., Springer, New York, 2017. · Zbl 1393.37003 |
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