American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Initial trace of positive solutions of a class of degenerate heat equation with absorption
  • DCDS Home
  • This Issue
  • Next Article
    Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality
May  2013, 33(5): 2007-2031. doi: 10.3934/dcds.2013.33.2007

The diffusive logistic model with a free boundary and seasonal succession

Rui Peng 1, and  Xiao-Qiang Zhao 2,

1. 

Department of Mathematics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China, and Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada
2. 

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

Received  December 2011 Revised  April 2012 Published  December 2012

This paper concerns a diffusive logistic equation with a free boundary and seasonal succession, which is formulated to investigate the spreading of a new or invasive species, where the free boundary represents the expanding front and the time periodicity accounts for the effect of the bad and good seasons. The condition to determine whether the species spatially spreads to infinity or vanishes at a finite space interval is derived, and when the spreading happens, the asymptotic spreading speed of the species is also given. The obtained results reveal the effect of seasonal succession on the dynamical behavior of the spreading of the single species.
Keywords: free boundary, vanishing., spreading, seasonal succession, Diffusive logistic equation.
Mathematics Subject Classification: Primary: 35K20, 35R35; Secondary: 35J60, 92B0.
Citation: Rui Peng, Xiao-Qiang Zhao. The diffusive logistic model with a free boundary and seasonal succession. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2007-2031. doi: 10.3934/dcds.2013.33.2007
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" Lecture Notes in Math., 446, Springer, Berlin, (1975), 5-49.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Networks and Heterogeneous Media, ().   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), 778-800. doi: 10.1137/S0036141099351693.  Google Scholar

[5]

D. L. DeAngelis, J. C. Trexler and D. D. Donalson, "Competition Dynamics in a Seasionally Varying Wetland," Chapter 1, 1-13, in "Spatial Ecology" (Eds. S. Cantrell, C. Cosner and S. Ruan), CRC Press, Chapman and Hall, (2009).  Google Scholar

[6]

Y. 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.  Google Scholar

[7]

Y. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, preprint, 2011. Google Scholar

[8]

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.  Google Scholar

[9]

P. J. DuBowy, Waterfowl communities and seasonal environments: Temporal variabolity in interspecific competition, Ecology, 69 (1988), 1439-1453. Google Scholar

[10]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130. doi: 10.1007/s00285-011-0408-6.  Google Scholar

[11]

S. S. Hu and A. J. Tessier, Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage, Ecology, 76 (1995), 2278-2294. Google Scholar

[12]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc. Providence, RI, 1968.  Google Scholar

[13]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[14]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.  Google Scholar

[15]

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.  Google Scholar

[16]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light, American Naturalist, 157 (2001), 170-187. Google Scholar

[17]

T. R. Malthus, "An Essay on the Principle of Population," 1798. Printed for J. Johnson in St. Pauls Church-Yard, 1998. Google Scholar

[18]

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

[19]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295. doi: 10.1007/s10231-008-0075-4.  Google Scholar

[20]

R. Peng and D. Wei, The periodic-parabolic logistic equation on $\R^N$, Discrete and Continuous Dyn. Syst. Series A, 32 (2012), 619-641.  Google Scholar

[21]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.  Google Scholar

[22]

X.-Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" Lecture Notes in Math., 446, Springer, Berlin, (1975), 5-49.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Networks and Heterogeneous Media, ().   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), 778-800. doi: 10.1137/S0036141099351693.  Google Scholar

[5]

D. L. DeAngelis, J. C. Trexler and D. D. Donalson, "Competition Dynamics in a Seasionally Varying Wetland," Chapter 1, 1-13, in "Spatial Ecology" (Eds. S. Cantrell, C. Cosner and S. Ruan), CRC Press, Chapman and Hall, (2009).  Google Scholar

[6]

Y. 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.  Google Scholar

[7]

Y. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, preprint, 2011. Google Scholar

[8]

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.  Google Scholar

[9]

P. J. DuBowy, Waterfowl communities and seasonal environments: Temporal variabolity in interspecific competition, Ecology, 69 (1988), 1439-1453. Google Scholar

[10]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130. doi: 10.1007/s00285-011-0408-6.  Google Scholar

[11]

S. S. Hu and A. J. Tessier, Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage, Ecology, 76 (1995), 2278-2294. Google Scholar

[12]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Amer. Math. Soc. Providence, RI, 1968.  Google Scholar

[13]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[14]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154.  Google Scholar

[15]

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.  Google Scholar

[16]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light, American Naturalist, 157 (2001), 170-187. Google Scholar

[17]

T. R. Malthus, "An Essay on the Principle of Population," 1798. Printed for J. Johnson in St. Pauls Church-Yard, 1998. Google Scholar

[18]

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

[19]

G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295. doi: 10.1007/s10231-008-0075-4.  Google Scholar

[20]

R. Peng and D. Wei, The periodic-parabolic logistic equation on $\R^N$, Discrete and Continuous Dyn. Syst. Series A, 32 (2012), 619-641.  Google Scholar

[21]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.  Google Scholar

[22]

X.-Q. Zhao, "Dynamical Systems in Population Biology," Springer-Verlag, New York, 2003.  Google Scholar

[1]

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, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154
[2]

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
[3]

Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591
[4]

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, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317
[5]

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, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121
[6]

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
[7]

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
[8]

Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1407-1424. doi: 10.3934/mbe.2017073
[9]

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, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256
[10]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[11]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213
[12]

Kazuaki Taira. A mathematical study of diffusive logistic equations with mixed type boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021166
[13]

Lin Niu, Yi Wang, Xizhuang Xie. Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2161-2172. doi: 10.3934/dcdsb.2021014
[14]

Wentao Meng, Yuanxi Yue, Manjun Ma. The minimal wave speed of the Lotka-Volterra competition model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021265
[15]

Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
[16]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[17]

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
[18]

Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255
[19]

Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175
[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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (195)
  • HTML views (0)
  • Cited by (61)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy