American Institute of Mathematical Sciences

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

A revisit to the diffusive logistic model with free boundary condition

 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].
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.

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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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