American Institute of Mathematical Sciences

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May  2016, 21(3): 837-847. doi: 10.3934/dcdsb.2016.21.837

A revisit to the diffusive logistic model with free boundary condition

Wenzhen Gan 1, and  Peng Zhou 2,

1. 

School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China
2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7

Received  February 2015 Revised  July 2015 Published  January 2016

This short paper revisits a free boundary problem which is used to describe the spreading of a new or invasive species. Our main goal is to understand how the underlying long-time dynamical behaviors response to the initial data. To this end, we parameterize the initial function as $u_0=\sigma\phi^*$, where $\sigma$ is regarded as a variable parameter and $\phi^*$ is a given function. Our main result suggests that when the diffusion rate is small, the species can persist in the long run (called spreading) for any $\sigma>0$; while if the diffusion rate is large, the species will go to extinction finally (called vanishing) for small $\sigma>0$. Maybe of more interest is that for some intermediate diffusion rates, there appears a sharp threshold value $\sigma^*\in(0, \infty)$ such that vanishing happens provided $0<\sigma\leq\sigma^*$ and spreading happens provided $\sigma>\sigma^*$. This result can be seen as an improvement of Theorem 1.2 in [8].
Keywords: spreading, sharp threshold., vanishing, free boundary, Diffusive logistic model.
Mathematics Subject Classification: Primary: 35K57, 35K61; Secondary: 35R35, 92D2.
Citation: Wenzhen Gan, Peng Zhou. A revisit to the diffusive logistic model with free boundary condition. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 837-847. doi: 10.3934/dcdsb.2016.21.837
References:
[1]

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.

[2]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments, Proc. Roy. Soc. Edinburgh., 112 (1989), 293-318. doi: 10.1017/S030821050001876X.

[3]

R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mount. J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101.

[4]

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.

[5]

S. B. Cui, Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors, J. Funct. Anal., 245 (2007), 1-18. doi: 10.1016/j.jfa.2006.12.020.

[6]

Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, Journal of Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.

[7]

Y. 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.

[8]

Y. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724, arXiv:1301.5373 doi: 10.4171/JEMS/568.

[9]

Y. 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.

[10]

Y. Du, H. Matsuzawa and M. L. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787. doi: 10.1137/130908063.

[11]

M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound, 3 (2001), 337-344.

[12]

H. Ghidouche, P. Souplet and D. Tarzia, Decay of global solutions, stability and blow-up for a reaction-diffusion problem with free boundary, Proc. Am. Math. Soc., 129 (2001), 781-792. doi: 10.1090/S0002-9939-00-05705-1.

[13]

K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.

[14]

C. X. Lei, K. Kim and Z. G. Lin, The spreading frontiers of avian-human influenza described by the free boundary, Sci. China Math., 57 (2014), 971-990. doi: 10.1007/s11425-013-4652-7.

[15]

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.

[16]

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.

[17]

Z. G. Lin, Y. N. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete. Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376. doi: 10.3934/dcdsb.2013.18.2355.

[18]

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

[19]

F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology, 44 (1963), 651-663. doi: 10.2307/1933011.

[20]

P. Zhou, J. Bao and Z. G. Lin, Global existence and blowup of a localized problem with free boundary, Nonlinear Anal., 74 (2011), 2523-2533. doi: 10.1016/j.na.2010.11.047.

[21]

P. Zhou and Z. G. Lin, Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262 (2012), 3409-3429. doi: 10.1016/j.jfa.2012.01.018.

[22]

P. Zhou and Z. G. Lin, Global fast and slow solutions of a localized problem with free boundary, Sci. China Math., 55 (2012), 1937-1950. doi: 10.1007/s11425-012-4443-6.

[23]

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.

show all references

References:
[1]

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.

[2]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in a disrupted environments, Proc. Roy. Soc. Edinburgh., 112 (1989), 293-318. doi: 10.1017/S030821050001876X.

[3]

R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mount. J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101.

[4]

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.

[5]

S. B. Cui, Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors, J. Funct. Anal., 245 (2007), 1-18. doi: 10.1016/j.jfa.2006.12.020.

[6]

Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, Journal of Differential Equations, 250 (2011), 4336-4366. doi: 10.1016/j.jde.2011.02.011.

[7]

Y. 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.

[8]

Y. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724, arXiv:1301.5373 doi: 10.4171/JEMS/568.

[9]

Y. 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.

[10]

Y. Du, H. Matsuzawa and M. L. Zhou, Spreading speed and profile for nonlinear Stefan problems in high space dimensions, J. Math. Pures Appl., 103 (2015), 741-787. doi: 10.1137/130908063.

[11]

M. Fila and P. Souplet, Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem, Interfaces Free Bound, 3 (2001), 337-344.

[12]

H. Ghidouche, P. Souplet and D. Tarzia, Decay of global solutions, stability and blow-up for a reaction-diffusion problem with free boundary, Proc. Am. Math. Soc., 129 (2001), 781-792. doi: 10.1090/S0002-9939-00-05705-1.

[13]

K. I. Kim, Z. G. Lin and Q. Y. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications., 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003.

[14]

C. X. Lei, K. Kim and Z. G. Lin, The spreading frontiers of avian-human influenza described by the free boundary, Sci. China Math., 57 (2014), 971-990. doi: 10.1007/s11425-013-4652-7.

[15]

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.

[16]

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.

[17]

Z. G. Lin, Y. N. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete. Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376. doi: 10.3934/dcdsb.2013.18.2355.

[18]

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

[19]

F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology, 44 (1963), 651-663. doi: 10.2307/1933011.

[20]

P. Zhou, J. Bao and Z. G. Lin, Global existence and blowup of a localized problem with free boundary, Nonlinear Anal., 74 (2011), 2523-2533. doi: 10.1016/j.na.2010.11.047.

[21]

P. Zhou and Z. G. Lin, Global existence and blowup of a nonlocal problem in space with free boundary, J. Funct. Anal., 262 (2012), 3409-3429. doi: 10.1016/j.jfa.2012.01.018.

[22]

P. Zhou and Z. G. Lin, Global fast and slow solutions of a localized problem with free boundary, Sci. China Math., 55 (2012), 1937-1950. doi: 10.1007/s11425-012-4443-6.

[23]

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.

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