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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 and 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.

[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.

[3]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media, to appear.

[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]

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).

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

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.

[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.

[3]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Networks and Heterogeneous Media, to appear.

[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]

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).

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

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