• 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

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.
Citation: 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
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.  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.  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, (2009), 1.   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.  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.  doi: 10.1137/090771089.  Google Scholar

[9]

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

[10]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession,, J. Math. Biol., 64 (2012), 109.  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.   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, (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.  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.  doi: 10.1002/cpa.20154.  Google Scholar

[15]

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

[16]

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

[21]

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

[22]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", Springer-Verlag, (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, 446 (1975), 5.   Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.  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.  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, (2009), 1.   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.  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.  doi: 10.1137/090771089.  Google Scholar

[9]

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

[10]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession,, J. Math. Biol., 64 (2012), 109.  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.   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, (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.  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.  doi: 10.1002/cpa.20154.  Google Scholar

[15]

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

[16]

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

[21]

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

[22]

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

[1]

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

[2]

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

[3]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[5]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[6]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[7]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

[8]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194

[9]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[10]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[11]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[12]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[13]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453

[14]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[15]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[16]

Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029

[17]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[18]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[19]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

[20]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (135)
  • HTML views (0)
  • Cited by (57)

Other articles
by authors

[Back to Top]