# American Institute of Mathematical Sciences

January  2016, 21(1): 13-35. doi: 10.3934/dcdsb.2016.21.13

## A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China 2 School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024

Received  December 2014 Revised  September 2015 Published  November 2015

We study a diffusive logistic equation with a free boundary in time-periodic environment. To understand the effect of the diffusion rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$ , and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d, h_0, \mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha-\gamma, h_{0}, T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha-\gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha-\gamma, h_0, T)>0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the diffusion rate is small or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.
Citation: Qiaoling Chen, Fengquan Li, Feng Wang. A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13
##### References:
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Anal. Appl., 423 (2015), 377.  doi: 10.1016/j.jmaa.2014.09.055.  Google Scholar [22] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system,, J. Dynam. Differential Equations, 24 (2012), 873.  doi: 10.1007/s10884-012-9267-0.  Google Scholar [23] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons Ltd, (2003).  doi: 10.1002/0470871296.  Google Scholar [24] R. Peng and D. Wei, The periodic-parabolic logistic equation on $\mathbbR^N$,, Discrete Contin. Dyn. Syst., 32 (2012), 619.   Google Scholar [25] R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete Contin. Dyn. Syst., 33 (2013), 2007.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar [26] M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Jpn. J. Appl. Math., 2 (1985), 151.  doi: 10.1007/BF03167042.  Google Scholar [27] M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology,, Hiroshima Math. J., 16 (1986), 477.   Google Scholar [28] M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction diffusion equations,, Hiroshima Math. J., 17 (1987), 241.   Google Scholar [29] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology,, Adv. Math. Sci. Appl., 21 (2011), 467.   Google Scholar [30] L. I. Rubinstein, The Stefan Problem,, American Mathematical Society, (1971).   Google Scholar [31] M. X. Wang and J. F. Zhao, Free boundary problem for a Lotka-Volterra competition system,, J. Dynam. Differential Equations, 26 (2014), 655.  doi: 10.1007/s10884-014-9363-4.  Google Scholar [32] M. X. Wang, On some free boundary problems of the prey-predator model,, J. Differential Equations, 256 (2014), 3365.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar [33] M. X. 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show all references

##### References:
 [1] X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing,, SIAM J. Math. Anal., 32 (2000), 778.  doi: 10.1137/S0036141099351693.  Google Scholar [2] J. Crank, Free and Moving Boundary Problem,, Clarendon Press, (1984).   Google Scholar [3] Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.  doi: 10.1088/0951-7715/20/8/004.  Google Scholar [4] P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment,, J. Differential Equations, 256 (2014), 1927.  doi: 10.1016/j.jde.2013.12.008.  Google Scholar [5] C. X. Lei, Z. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat,, J. Differential Equations, 257 (2014), 145.  doi: 10.1016/j.jde.2014.03.015.  Google Scholar [6] E. N. Dancer and P. Hess, The symmetry of positive solutions of periodic-parabolic problems,, J. Comput. Appl. Math., 52 (1994), 81.  doi: 10.1016/0377-0427(94)90350-6.  Google Scholar [7] Y. H. Du and L. Ma, Logistic type equations on $\mathbbR^N$ by a squeezing method involving boundary blow-up solutions,, J. Lond. Math. Soc., 64 (2001), 107.  doi: 10.1017/S0024610701002289.  Google Scholar [8] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377.  doi: 10.1137/090771089.  Google Scholar [9] Y. H. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary II,, J. Differential Equations, 250 (2011), 4336.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [10] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105.  doi: 10.3934/dcdsb.2014.19.3105.  Google Scholar [11] Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, J. Funct. Anal., 265 (2013), 2089.  doi: 10.1016/j.jfa.2013.07.016.  Google Scholar [12] Y. H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model,, Ann. Inst. H. Poincare Anal. NonLineaire, 32 (2015), 279.  doi: 10.1016/j.anihpc.2013.11.004.  Google Scholar [13] Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., (2014).   Google Scholar [14] Y. H. Du, H. Matsuzawa and M. L. Zhou, Sharp estimate of the spreading speed determined by nonlinear free boundary problems,, SIAM J. Math. Anal., 46 (2014), 375.  doi: 10.1137/130908063.  Google Scholar [15] A. Friedman, Partial Differential Equations of Parabolic Type,, Printice-Hall, (1964).   Google Scholar [16] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity,, Pitman Res. Notes Math., (1991).   Google Scholar [17] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model,, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1.  doi: 10.3934/dcds.2008.21.1.  Google Scholar [18] O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, Translations of Mathematical Monographs, (1967).   Google Scholar [19] Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics,, in Tutor. Math. Biosci. IV: Evol. Ecol. (eds. Avner Friedman), 1922 (2008), 171.  doi: 10.1007/978-3-540-74331-6_5.  Google Scholar [20] G. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publ. Co., (1996).  doi: 10.1142/3302.  Google Scholar [21] J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment,, J. Math. Anal. Appl., 423 (2015), 377.  doi: 10.1016/j.jmaa.2014.09.055.  Google Scholar [22] J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system,, J. Dynam. Differential Equations, 24 (2012), 873.  doi: 10.1007/s10884-012-9267-0.  Google Scholar [23] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons Ltd, (2003).  doi: 10.1002/0470871296.  Google Scholar [24] R. Peng and D. Wei, The periodic-parabolic logistic equation on $\mathbbR^N$,, Discrete Contin. Dyn. Syst., 32 (2012), 619.   Google Scholar [25] R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete Contin. Dyn. Syst., 33 (2013), 2007.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar [26] M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Jpn. J. Appl. Math., 2 (1985), 151.  doi: 10.1007/BF03167042.  Google Scholar [27] M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology,, Hiroshima Math. J., 16 (1986), 477.   Google Scholar [28] M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction diffusion equations,, Hiroshima Math. J., 17 (1987), 241.   Google Scholar [29] Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology,, Adv. Math. Sci. Appl., 21 (2011), 467.   Google Scholar [30] L. I. Rubinstein, The Stefan Problem,, American Mathematical Society, (1971).   Google Scholar [31] M. X. Wang and J. F. Zhao, Free boundary problem for a Lotka-Volterra competition system,, J. Dynam. Differential Equations, 26 (2014), 655.  doi: 10.1007/s10884-014-9363-4.  Google Scholar [32] M. X. Wang, On some free boundary problems of the prey-predator model,, J. Differential Equations, 256 (2014), 3365.  doi: 10.1016/j.jde.2014.02.013.  Google Scholar [33] M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient,, J. Differential Equations, 258 (2015), 1252.  doi: 10.1016/j.jde.2014.10.022.  Google Scholar [34] M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint,, , ().   Google Scholar [35] J. F. Zhao and M. X. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment,, Nonlinear Anal. Real World Appl., 16 (2014), 250.  doi: 10.1016/j.nonrwa.2013.10.003.  Google Scholar [36] G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Netw. Heterog. Media, 7 (2012), 583.  doi: 10.3934/nhm.2012.7.583.  Google Scholar [37] W. Z. Gan and P. Zhou, A revisit to the diffusive logistic model with free boundary condition,, Discrete. Contin. Dyn. Syst. Ser. B, ().   Google Scholar
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