Minimal wave speed of a competition integrodifference system. (English) Zbl 1395.35068
Summary: In this paper, we study the minimal wave speed of a competitive system. By constructing upper and lower solutions, we confirm the existence of travelling wave solution at the critical wave speed. This completes earlier results found in the literature. Our conclusion implies that the asymptotic decay behaviour of solutions at the critical wave speed is different from that of solutions at larger wave speeds.
MSC:
| 35C07 | Traveling wave solutions |
| 37C65 | Monotone flows as dynamical systems |
| 39A20 | Multiplicative and other generalized difference equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B51 | Comparison principles in context of PDEs |
References:
| [1] | Carrère, C., spreading speeds for a two-species competition-diffusion system, J. Differ. Equ., 264, 2133-2156, (2018) · Zbl 1432.35112 |
| [2] | Diekmann, O., thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 69, 109-130, (1978) · Zbl 0415.92020 |
| [3] | Diekmann, O., run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differ. Equ., 33, 58-73, (1979) · Zbl 0377.45007 |
| [4] | Lewis, M.A.; Li, B.; Weinberger, H.F., spreading speed and linear determinacy for two-species competiotion models, J. Math. Biol., 45, 219-233, (2002) · Zbl 1032.92031 |
| [5] | Li, K.; Huang, J.; Li, X.; He, Y., asymptotic behavior and uniqueness of traveling wave fronts in a competitive recursion system, Z. Angew. Math. Phys., 67, (2016) · Zbl 1375.45006 |
| [6] | Liang, X.; Zhao, X.Q., asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60, 1-40, (2007) · Zbl 1106.76008 |
| [7] | Lin, G., spreading speeds of a Lotka-Volterra predator-prey system: the role of the predator, Nonlinear Anal., 74, 2448-2461, (2011) · Zbl 1211.35138 |
| [8] | Lin, G., traveling wave solutions for integro-difference systems, J. Differ. Equ., 258, 2908-2940, (2015) · Zbl 1309.45007 |
| [9] | Lin, G., minimal wave speed of competitive diffusive systems with time delays, Appl. Math. Lett., 76, 164-169, (2018) · Zbl 1524.35317 |
| [10] | Lin, G.; Li, W.T., asymptotic spreading of competition diffusion systems: the role of interspecific competitions, Eur. J. Appl. Math., 23, 669-689, (2012) · Zbl 1256.35179 |
| [11] | Lin, G.; Li, W.T., traveling wave solutions of a competitive recursion, Discrete Contin. Dyn. Syst. Ser. B, 17, 173-189, (2012) · Zbl 1258.37069 |
| [12] | Lin, G.; Li, W.T.; Ruan, S., spreading speeds and traveling waves in competitive recursion systems, J. Math. Biol., 62, 162-201, (2011) · Zbl 1232.92072 |
| [13] | Lui, R., biological growth and spread modeled by systems of recursions. I. mathematical theory, Math. Biosci., 93, 269-295, (1989) · Zbl 0706.92014 |
| [14] | Lui, R., biological growth and spread modeled by systems of recursions. II. biological theory, Math. Biosci., 107, 255-287, (1991) |
| [15] | Murray, J.D., Mathematical Biology, II. Spatial Models and Biomedical Applications, 18, (2003), Springer-Verlag, New York · Zbl 1006.92002 |
| [16] | Pan, S., asymptotic spreading in a Lotka-Volterra predator-prey system, J. Math. Anal. Appl., 407, 230-236, (2013) · Zbl 1306.92049 |
| [17] | Pan, S., invasion speed of a predator-prey system, Appl. Math. Lett., 74, 46-51, (2017) · Zbl 1419.37084 |
| [18] | Shigesada, N.; Kawasaki, K., Biological Invasions: Theory and Practice, (1997), Oxford University Press, Oxford |
| [19] | Smoller, J., Shock Waves and Reaction-Diffusion Equations, (1994), Springer-Verlag, New York · Zbl 0807.35002 |
| [20] | Tang, M.M.; Fife, P., propagating fronts for competing species equations with diffusion, Arch. Rational Mech. Anal., 73, 69-77, (1980) · Zbl 0456.92019 |
| [21] | Volpert, A.I.; Volpert, V.A.; Volpert, V.A., Traveling Wave Solutions of Parabolic Systems, 140, (1994), AMS, Providence, RI · Zbl 0835.35048 |
| [22] | 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 |
| [23] | Ye, Q.; Li, Z.; Wang, M.; Wu, Y., Introduction to Reaction Diffusion Equations, (2011), Science Press, Beijing |
| [24] | Zhang, Y.; Zhao, X.Q., bistable travelling waves in competitive recursion systems, J. Differ. Equ., 252, 2630-2647, (2012) · Zbl 1229.92083 |
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