Spreading of two competing species governed by a free boundary model in a shifting environment. (English) Zbl 1390.35377

Summary: We investigate the long-time spreading behavior of two competing species in a shifting environment. The evolution of the population densities of the species is governed by a one space dimension diffusive Lotka-Volterra competition system with two different free boundaries, describing a dynamical process of two competitors invading into the new habitat in the same direction. It is assumed that the unfavourable region of the environment moves into the otherwise favourable homogeneous environment with a given speed \(c > 0\) in the spreading direction of the species. By close examination of certain simple cases, we show that such a shifting environment could reverse the fates of the species. A complete classification of the long-time dynamical behavior of the system is obtained for such cases.


35Q92 PDEs in connection with biology, chemistry and other natural sciences
35R35 Free boundary problems for PDEs
92D25 Population dynamics (general)
35B40 Asymptotic behavior of solutions to PDEs
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[1] Berestycki, H.; Diekmann, O.; Nagelkerke, C.; Zegeling, P., Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71, 2, 399-429, (2009) · Zbl 1169.92043
[2] Bunting, G.; Du, Y.; Krakowski, K., Spreading speed revisited: analysis of a free boundary model, Netw. Heterog. Media, 7, 583-603, (2012) · Zbl 1302.35194
[3] Du, Y.; Guo, Z., The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253, 996-1035, (2012) · Zbl 1257.35110
[4] Du, Y.; Lin, Z., Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42, 1, 377-405, (2010) · Zbl 1219.35373
[5] Du, Y.; Lin, Z., The diffusive competition model with a free boundary: invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19, 10, 3105-3132, (2014) · Zbl 1310.35245
[6] Du, Y.; Lou, B., Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17, 10, 2673-2724, (2015) · Zbl 1331.35399
[7] Du, Y.; Ma, L., Logistic type equations on \(\mathbb{R}^{\mathbb{N}}\) by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64, 107-124, (2001) · Zbl 1018.35045
[8] Du, Y.; Matsuzawa, H.; Zhou, M., Sharp estimate of the spreading speed determined by nonlinear free boundary problems, SIAM J. Math. Anal., 46, 375-396, (2010) · Zbl 1296.35219
[9] Du, Y.; Wei, L.; Zhou, L., Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, (2018), in press (published online Aug. 2017), see also · Zbl 1408.35227
[10] Du, Y.; Wu, C.-H., Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries, submitted for publication · Zbl 1396.35028
[11] Guo, J.; Wu, C.-H., Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28, 1-27, (2015) · Zbl 1316.92066
[12] Lei, C.; Du, Y., Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model, Discrete Contin. Dyn. Syst. Ser. B, 22, 895-911, (2017) · Zbl 1360.35299
[13] Li, B.; Bewick, S.; Shang, J.; Fagan, W., Persistence and spread of a species with a shifting habitat edge, SIAM J. Appl. Math., 74, 5, 1397-1417, (2014) · Zbl 1345.92120
[14] Potapov, A.; Lewis, M., Climate and competition: the effect of moving range boundaries on habitat invisibility, Bull. Math. Biol., 66, 975-1008, (2004) · Zbl 1334.92454
[15] Sutherst, R., Climate change and invasive species: a conceptual framework, (Mooney, H. A.; Hobbs, R. J., Invasive Species in a Changing World, (2000), Island Press Washington, DC), 211-240
[16] Walther, G.; Post, E.; Convey, P.; Menzel, A.; Parmesan, C.; Beebee, T.; Fromentin, J.-M.; Hoegh-Guldberg, O.; Bairlein, F., Ecological responses to recent climate change, Nature, 416, 389-395, (2002)
[17] Wang, M. X.; Zhang, Y., Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159, 458-467, (2017) · Zbl 1371.35367
[18] Wang, M. X.; Zhang, Y., Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264, 3527-3558, (2018) · Zbl 1391.35191
[19] Wu, C.-H., The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259, 3, 873-897, (2015) · Zbl 1319.35081
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