Fast propagation for reaction-diffusion cooperative systems. (English) Zbl 1388.35106
Summary: This paper deals with the spatial propagation for reaction-diffusion cooperative systems. It is well-known that the solution of a reaction-diffusion equation with monostable nonlinearity spreads at a finite speed when the initial condition decays to zero exponentially or faster, and propagates fast when the initial condition decays to zero more slowly than any exponentially decaying function. However, in reaction-diffusion cooperative systems, a new possibility happens in which one species propagates fast although its initial condition decays exponentially or faster. The fundamental reason is that the growth sources of one species come from the other species. Simply speaking, we find a new interesting phenomenon that the spatial propagation of one species is accelerated by the other species. This is a unique phenomenon in reaction-diffusion systems. We present a framework of fast propagation for reaction-diffusion cooperative systems.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35C07 | Traveling wave solutions |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92D25 | Population dynamics (general) |
References:
| [1] | Alfaro, M., Slowing Allee effect versus accelerating heavy tails in monostable reaction diffusion equations, Nonlinearity, 30, 687-702 (2017) · Zbl 1358.35056 |
| [2] | Alfaro, M.; Coville, J., Propagation phenomena in monostable integro-differential equations: acceleration or not?, J. Differential Equations, 263, 5727-5758 (2017) · Zbl 1409.35204 |
| [3] | Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 39, 33-76 (1978) · Zbl 0407.92014 |
| [4] | Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, (Goldstein, J. A., Partial Differential Equations and Related Topics. Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446 (1975), Springer: Springer Berlin), 5-49 · Zbl 0325.35050 |
| [5] | Booty, M. R.; Haberman, R.; Minzoni, A. A., The accommodation of traveling waves of Fisher’s type to the dynamics of the leading tail, SIAM J. Appl. Math., 53, 1009-1025 (1993) · Zbl 0790.35043 |
| [6] | Cabré, X.; Roquejoffre, J. M., The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320, 679-722 (2013) · Zbl 1307.35310 |
| [7] | Capasso, V.; Paveri-Fontana, S. L., A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. épidémiol. Santé Publique, 27, 32-121 (1979) |
| [8] | Coulon, A. C.; Roquejoffre, J. M., Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations, Comm. Partial Differential Equations, 37, 2029-2049 (2012) · Zbl 1263.35142 |
| [9] | Fang, J.; Zhao, X.-Q., Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dynam. Differential Equations, 21, 663-680 (2009) · Zbl 1180.35275 |
| [10] | Felmer, P.; Yangari, M., Fast propagation for fractional KPP equations with slowly decaying initial conditions, SIAM J. Math. Anal., 45, 662-678 (2013) · Zbl 1371.35324 |
| [11] | Finkelshtein, D.; Kondratiev, Y.; Tkachov, P., Accelerated front propagation for monostable equations with nonlocal diffusion · Zbl 1427.35120 |
| [12] | Garnier, J., Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43, 1955-1974 (2011) · Zbl 1232.47058 |
| [13] | Hamel, F.; Nadin, G., Spreading properties and complex dynamics for monostable reaction-diffusion equations, Comm. Partial Differential Equations, 37, 511-537 (2012) · Zbl 1254.35024 |
| [14] | Hamel, F.; Roques, L., Fast propagation for KPP equations with slowly decaying initial conditions, J. Differential Equations, 249, 1726-1745 (2010) · Zbl 1213.35100 |
| [15] | He, J.; Wu, Y.; Wu, Y., Large time behavior of solutions for degenerate p-degree Fisher equation with algebraic decaying initial data, J. Math. Anal. Appl., 448, 1-21 (2017) · Zbl 1355.35024 |
| [16] | Henderson, C., Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data, Nonlinearity, 29, 3215-3240 (2016) · Zbl 1353.35179 |
| [17] | Hsu, C.-H.; Yang, T.-S., Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26, 121-139 (2013) · Zbl 1264.35255 |
| [18] | Kolmogorov, A. N.; Petrovsky, I. G.; Piskunov, N. S., Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. État Moscou, Sér. Internat. A, 1, 1-26 (1937) · Zbl 0018.32106 |
| [19] | Lau, K.-S., On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differential Equations, 59, 44-70 (1985) · Zbl 0584.35091 |
| [20] | Lewis, M. A.; Li, B.; Weinberger, H. F., Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45, 219-233 (2002) · Zbl 1032.92031 |
| [21] | Li, B.; Lewis, M. A.; Weinberger, H. F., Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58, 323-338 (2009) · Zbl 1162.92030 |
| [22] | Li, B.; Weinberger, H. F.; Lewis, M. A., Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196, 82-98 (2005) · Zbl 1075.92043 |
| [23] | Liang, X.; Yi, Y.; Zhao, X.-Q., Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231, 57-77 (2006) · Zbl 1105.37017 |
| [24] | Liang, X.; Zhao, X.-Q., Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math.. Comm. Pure Appl. Math., Comm. Pure Appl. Math., 61, 137-138 (2008), Erratum · Zbl 1165.37332 |
| [25] | Liang, X.; Zhao, X.-Q., Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259, 857-903 (2010) · Zbl 1201.35068 |
| [26] | Lui, R., Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93, 269-295 (1989) · Zbl 0706.92014 |
| [27] | Mallordy, J. F.; Roquejoffre, J. M., A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26, 1-20 (1995) · Zbl 0813.35041 |
| [28] | McKean, H. P., Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math., 28, 323-331 (1975) · Zbl 0316.35053 |
| [29] | Sattinger, D. H., Stability of waves of nonlinear parabolic systems, Adv. Math., 22, 312-355 (1976) · Zbl 0344.35051 |
| [30] | Volpert, A. I.; Volpert, V. A.; Volpert, V. A., Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140 (1994), Amer. Math. Soc.: Amer. Math. Soc. Providence · Zbl 0994.35049 |
| [31] | Wang, H., Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonlinear Sci., 21, 747-783 (2011) · Zbl 1242.35147 |
| [32] | Weinberger, H. F., Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13, 353-396 (1982) · Zbl 0529.92010 |
| [33] | Weinberger, H. F.; Lewis, M. A.; Li, B., Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45, 183-218 (2002) · Zbl 1023.92040 |
| [34] | Wu, S.-L.; Hsu, C.-H., Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368, 6033-6062 (2016) · Zbl 1439.35280 |
| [35] | Zhao, X.-Q.; Wang, W., Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4, 1117-1128 (2004) · Zbl 1097.34022 |
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