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 and Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13
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), 778-800. doi: 10.1137/S0036141099351693.

[2]

J. Crank, Free and Moving Boundary Problem, Clarendon Press, Oxford, 1984.

[3]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[4]

P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008.

[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), 145-166. doi: 10.1016/j.jde.2014.03.015.

[6]

E. N. Dancer and P. Hess, The symmetry of positive solutions of periodic-parabolic problems, J. Comput. Appl. Math., 52 (1994), 81-89. doi: 10.1016/0377-0427(94)90350-6.

[7]

Y. H. Du and L. Ma, Logistic type equations on $\mathbb{R}^N2$ by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.

[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), 377-405. doi: 10.1137/090771089.

[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), 4336-4366. doi: 10.1016/j.jde.2011.02.011.

[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), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105.

[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), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.

[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), 279-305. doi: 10.1016/j.anihpc.2013.11.004.

[13]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., (2014), in press, arXiv:1301.5373.

[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), 375-396. doi: 10.1137/130908063.

[15]

A. Friedman, Partial Differential Equations of Parabolic Type, Printice-Hall, Englewood Cliffs, N.J., 1964.

[16]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math., vol. 247, Longman Sci. Tech., Harlow, 1991.

[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), 1-20. doi: 10.3934/dcds.2008.21.1.

[18]

O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, AMS, Providence, RI, 1967.

[19]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutor. Math. Biosci. IV: Evol. Ecol. (eds. Avner Friedman), Springer, 1922 (2008), 171-205. doi: 10.1007/978-3-540-74331-6_5.

[20]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publ. Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[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), 377-398. doi: 10.1016/j.jmaa.2014.09.055.

[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), 873-895. doi: 10.1007/s10884-012-9267-0.

[23]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.

[24]

R. Peng and D. Wei, The periodic-parabolic logistic equation on $\mathbb{R}^N2$, Discrete Contin. Dyn. Syst., 32 (2012), 619-641.

[25]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007.

[26]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Jpn. J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[27]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.

[28]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[29]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.

[30]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.

[31]

M. X. Wang and J. F. Zhao, Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4.

[32]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013.

[33]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022.

[34]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint, arXiv:1312.7751.

[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), 250-263. doi: 10.1016/j.nonrwa.2013.10.003.

[36]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.

[37]

W. Z. Gan and P. Zhou, A revisit to the diffusive logistic model with free boundary condition, Discrete. Contin. Dyn. Syst. Ser. B, in press.

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), 778-800. doi: 10.1137/S0036141099351693.

[2]

J. Crank, Free and Moving Boundary Problem, Clarendon Press, Oxford, 1984.

[3]

Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892. doi: 10.1088/0951-7715/20/8/004.

[4]

P. Zhou and D. M. Xiao, The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), 1927-1954. doi: 10.1016/j.jde.2013.12.008.

[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), 145-166. doi: 10.1016/j.jde.2014.03.015.

[6]

E. N. Dancer and P. Hess, The symmetry of positive solutions of periodic-parabolic problems, J. Comput. Appl. Math., 52 (1994), 81-89. doi: 10.1016/0377-0427(94)90350-6.

[7]

Y. H. Du and L. Ma, Logistic type equations on $\mathbb{R}^N2$ by a squeezing method involving boundary blow-up solutions, J. Lond. Math. Soc., 64 (2001), 107-124. doi: 10.1017/S0024610701002289.

[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), 377-405. doi: 10.1137/090771089.

[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), 4336-4366. doi: 10.1016/j.jde.2011.02.011.

[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), 3105-3132. doi: 10.3934/dcdsb.2014.19.3105.

[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), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.

[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), 279-305. doi: 10.1016/j.anihpc.2013.11.004.

[13]

Y. H. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., (2014), in press, arXiv:1301.5373.

[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), 375-396. doi: 10.1137/130908063.

[15]

A. Friedman, Partial Differential Equations of Parabolic Type, Printice-Hall, Englewood Cliffs, N.J., 1964.

[16]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math., vol. 247, Longman Sci. Tech., Harlow, 1991.

[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), 1-20. doi: 10.3934/dcds.2008.21.1.

[18]

O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, AMS, Providence, RI, 1967.

[19]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutor. Math. Biosci. IV: Evol. Ecol. (eds. Avner Friedman), Springer, 1922 (2008), 171-205. doi: 10.1007/978-3-540-74331-6_5.

[20]

G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publ. Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.

[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), 377-398. doi: 10.1016/j.jmaa.2014.09.055.

[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), 873-895. doi: 10.1007/s10884-012-9267-0.

[23]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons Ltd, 2003. doi: 10.1002/0470871296.

[24]

R. Peng and D. Wei, The periodic-parabolic logistic equation on $\mathbb{R}^N2$, Discrete Contin. Dyn. Syst., 32 (2012), 619-641.

[25]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031. doi: 10.3934/dcds.2013.33.2007.

[26]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Jpn. J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[27]

M. Mimura, Y. Yamada and S. Yotsutani, Stability analysis for free boundary problems in ecology, Hiroshima Math. J., 16 (1986), 477-498.

[28]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[29]

Y. Kaneko and Y. Yamada, A free boundary problem for a reaction-diffusion equation appearing in ecology, Adv. Math. Sci. Appl., 21 (2011), 467-492.

[30]

L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.

[31]

M. X. Wang and J. F. Zhao, Free boundary problem for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672. doi: 10.1007/s10884-014-9363-4.

[32]

M. X. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394. doi: 10.1016/j.jde.2014.02.013.

[33]

M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266. doi: 10.1016/j.jde.2014.10.022.

[34]

M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, preprint, arXiv:1312.7751.

[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), 250-263. doi: 10.1016/j.nonrwa.2013.10.003.

[36]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603. doi: 10.3934/nhm.2012.7.583.

[37]

W. Z. Gan and P. Zhou, A revisit to the diffusive logistic model with free boundary condition, Discrete. Contin. Dyn. Syst. Ser. B, in press.

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