# 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 & 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.  doi: 10.3934/nhm.2012.7.583.  Google Scholar [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.  doi: 10.1017/S030821050001876X.  Google Scholar [3] R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity,, Rocky Mount. J. Math., 26 (1996), 1.  doi: 10.1216/rmjm/1181072101.  Google Scholar [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.  doi: 10.1137/S0036141099351693.  Google Scholar [5] S. B. Cui, Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors,, J. Funct. Anal., 245 (2007), 1.  doi: 10.1016/j.jfa.2006.12.020.  Google Scholar [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.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [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.  doi: 10.1137/090771089.  Google Scholar [8] Y. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., 17 (2015), 2673.  doi: 10.4171/JEMS/568.  Google Scholar [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.  doi: 10.1137/130908063.  Google Scholar [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.  doi: 10.1137/130908063.  Google Scholar [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.   Google Scholar [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.  doi: 10.1090/S0002-9939-00-05705-1.  Google Scholar [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.  doi: 10.1016/j.nonrwa.2013.02.003.  Google Scholar [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.  doi: 10.1007/s11425-013-4652-7.  Google Scholar [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.  doi: 10.1016/j.jde.2014.03.015.  Google Scholar [16] 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 [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.  doi: 10.3934/dcdsb.2013.18.2355.  Google Scholar [18] L. I. Rubinstein, The Stefan Problem,, American Mathematical Society, (1971).   Google Scholar [19] F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth,, Ecology, 44 (1963), 651.  doi: 10.2307/1933011.  Google Scholar [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.  doi: 10.1016/j.na.2010.11.047.  Google Scholar [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.  doi: 10.1016/j.jfa.2012.01.018.  Google Scholar [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.  doi: 10.1007/s11425-012-4443-6.  Google Scholar [23] 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

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.  doi: 10.3934/nhm.2012.7.583.  Google Scholar [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.  doi: 10.1017/S030821050001876X.  Google Scholar [3] R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity,, Rocky Mount. J. Math., 26 (1996), 1.  doi: 10.1216/rmjm/1181072101.  Google Scholar [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.  doi: 10.1137/S0036141099351693.  Google Scholar [5] S. B. Cui, Well-posedness of a multidimensional free boundary problem modelling the growth of nonnecrotic tumors,, J. Funct. Anal., 245 (2007), 1.  doi: 10.1016/j.jfa.2006.12.020.  Google Scholar [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.  doi: 10.1016/j.jde.2011.02.011.  Google Scholar [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.  doi: 10.1137/090771089.  Google Scholar [8] Y. Du and B. D. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., 17 (2015), 2673.  doi: 10.4171/JEMS/568.  Google Scholar [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.  doi: 10.1137/130908063.  Google Scholar [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.  doi: 10.1137/130908063.  Google Scholar [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.   Google Scholar [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.  doi: 10.1090/S0002-9939-00-05705-1.  Google Scholar [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.  doi: 10.1016/j.nonrwa.2013.02.003.  Google Scholar [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.  doi: 10.1007/s11425-013-4652-7.  Google Scholar [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.  doi: 10.1016/j.jde.2014.03.015.  Google Scholar [16] 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 [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.  doi: 10.3934/dcdsb.2013.18.2355.  Google Scholar [18] L. I. Rubinstein, The Stefan Problem,, American Mathematical Society, (1971).   Google Scholar [19] F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth,, Ecology, 44 (1963), 651.  doi: 10.2307/1933011.  Google Scholar [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.  doi: 10.1016/j.na.2010.11.047.  Google Scholar [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.  doi: 10.1016/j.jfa.2012.01.018.  Google Scholar [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.  doi: 10.1007/s11425-012-4443-6.  Google Scholar [23] 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
 [1] Rui Peng, Xiao-Qiang Zhao. The diffusive logistic model with a free boundary and seasonal succession. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2007-2031. doi: 10.3934/dcds.2013.33.2007 [2] Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020154 [3] Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317 [4] Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583 [5] Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020256 [6] Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020121 [7] Alan E. Lindsay, Michael J. Ward. An asymptotic analysis of the persistence threshold for the diffusive logistic model in spatial environments with localized patches. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1139-1179. doi: 10.3934/dcdsb.2010.14.1139 [8] 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 [9] Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020387 [10] Yihong Du, Zhigui Lin. The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3105-3132. doi: 10.3934/dcdsb.2014.19.3105 [11] Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011 [12] Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213 [13] Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713 [14] Xuejun Pan, Hongying Shu, Yuming Chen. Dirichlet problem for a diffusive logistic population model with two delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3139-3155. doi: 10.3934/dcdss.2020134 [15] Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182 [16] Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199 [17] Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067 [18] Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 [19] Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 [20] Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

2019 Impact Factor: 1.27

## Metrics

• PDF downloads (74)
• HTML views (0)
• Cited by (4)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]