×

The minimal habitat size for spreading in a weak competition system with two free boundaries. (English) Zbl 1319.35081

In this paper, the dynamics of a Lotka-Volterra type weak competition system \[ \begin{aligned} u_t &= d_1u_{xx}+r_1u(1-u-kv),\\ v_t &= d_2v_{xx} +r_2v(1-v-hu) \end{aligned} \] is investigated, where \(u_x(0, t) = v_x(0, t) = 0\), \(u \equiv 0\) for \(x\geq s(t)\) and \(v\equiv 0\) for \(x \geq \sigma(t)\), and \(s(t)\) and \(\sigma(t)\) satisfy certain equations. While the coefficients \(k, h\) reflect the competition between the two species \(u(x, t)\) and \(v(x, t)\), the restriction \(0 < k\), \(h < 1\) reflects the case of weak competition. This paper continues the author’s previous work with others (for \(0 < k < 1 < h\)) in the case of weak competition. With the classification of the dynamics into four cases, it establishes a spreading-vanishing quartering. Some criteria are provided for each case to happen, estimates for the spreading speed and the long-term behaviour of solutions are established, the notion of minimal habitat size is introduced for successful spreading, and a sharp criterion is also established for species spreading and vanishing.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35R35 Free boundary problems for PDEs
92B05 General biology and biomathematics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] 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
[2] Du, Y.; Guo, Z. M., Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary II, J. Differential Equations, 250, 4336-4366, (2011) · Zbl 1222.35096
[3] Du, Y.; Guo, Z. M.; Peng, R., A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265, 2089-2142, (2013) · Zbl 1282.35419
[4] Du, Y.; Lin, Z. G., Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42, 377-405, (2010) · Zbl 1219.35373
[5] Du, Y.; Lin, Z. G., The diffusive competition model with a free boundary: invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19, 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. (JEMS), (2015), to appear · Zbl 1331.35399
[7] Du, Y.; Ma, L., Logistic type equations on \(\mathbb{R}^N\) 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, (2014) · Zbl 1296.35219
[9] Guo, J.-S.; Wu, C.-H., On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24, 873-895, (2012) · Zbl 1263.35132
[10] Guo, J.-S.; Wu, C.-H., Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28, 1-27, (2015) · Zbl 1316.92066
[11] Hilhorst, D.; Iida, M.; Mimura, M.; Ninomiya, H., A competition-diffusion system approximation to the classical two-phase Stefan problem, Jpn. J. Ind. Appl. Math., 18, 161-180, (2001) · Zbl 0980.35178
[12] Kaneko, Y., Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction-diffusion equations, Nonlinear Anal. Real World Appl., 18, 121-140, (2014) · Zbl 1297.35292
[13] Kaneko, Y.; Yamada, Y., A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21, 467-492, (2011) · Zbl 1254.35248
[14] Lei, C. X.; Lin, Z. G.; Zhang, Q. Y., The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257, 145-166, (2014) · Zbl 1286.35274
[15] Lin, Z. G., A free boundary problem for a predator-prey model, Nonlinearity, 20, 1883-1892, (2007) · Zbl 1126.35111
[16] Mimura, M.; Yamada, Y.; Yotsutani, S., A free boundary problem in ecology, Japan J. Appl. Math., 2, 151-186, (1985) · Zbl 0593.92019
[17] Peng, R.; Zhao, X.-Q., The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst. Ser. A, 33, 2007-2031, (2013) · Zbl 1273.35327
[18] Wang, M. X., On some free boundary problems of the prey-predator model, J. Differential Equations, 256, 3365-3394, (2014) · Zbl 1317.35110
[19] Wang, M. X., The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258, 1252-1266, (2015) · Zbl 1319.35094
[20] M.X. Wang, J.F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint. · Zbl 1373.35164
[21] Wu, C.-H., Spreading speed and traveling waves for a two-species weak competition system with free boundary, Discrete Contin. Dyn. Syst. Ser. B, 18, 2441-2455, (2013) · Zbl 1282.35122
[22] Zhao, J. F.; Wang, M. X., Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26, 655-672, (2014) · Zbl 1304.35783
[23] Zhou, P.; Xiao, D. M., The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256, 1927-1954, (2014) · Zbl 1316.35156
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.