November  2017, 22(9): 3295-3316. doi: 10.3934/dcdsb.2017138

A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

* Corresponding author

Received  September 2016 Revised  December 2016 Published  April 2017

This paper is concerned with a prey-predator model with sign-changing intrinsic growth rate in heterogeneous time-periodic environment, where the prey species lives in the whole space but the predator species lives in a region enclosed by a free boundary. It is shown that the results for the case of the non-periodic environment remain true in time-periodic environment. In fact, we first establish a similar spreading-vanishing dichotomy, which implies that if the predator species could spread successfully, then the two species will coexist, and this is certainly for the situation that the predation is relatively weak. Furthermore, some criteria are also obtained for spreading and vanishing. At last, some rough estimates of the asymptotic spreading speed are given if spreading occurs.

Citation: Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138
References:
[1]

G. BuntingY. 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, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons Ltd, 2003. doi: 10.1002/0470871296.

[3]

Q. ChenF. Li and F. Wang, A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 13-35.  doi: 10.3934/dcdsb.2016.21.13.

[4]

Q. ChenF. Li and F. Wang, The diffusive competition problem with a free boundary in heterogeneous time-periodic environment, J. Math. Anal. Appl., 433 (2016), 1594-1613.  doi: 10.1016/j.jmaa.2015.08.062.

[5]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.

[6]

Y. DuZ. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[7]

Y. Du and Z. 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 Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.

[9]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[10]

Y. DuM. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.

[11]

H. GuZ. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53.  doi: 10.1016/j.aml.2014.05.015.

[12]

H. GuZ. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117.  doi: 10.1090/S0002-9939-2014-12214-3.

[13]

H. GuB. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.

[14]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[15]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.

[16]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. , vol. 247, Longman Sci. Tech. , Harlow, 1991.

[17]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.

[18]

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

[19]

C. LeiZ. Lin and H. Wang, The free boundary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326-1341.  doi: 10.1016/j.jde.2012.10.021.

[20]

C. LeiZ. Lin and Q. 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.

[21]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.

[22]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.

[23]

J. Wang, The selection for dispersal: A diffusive competition model with a free boundary, Z. Angew. Math. Phys., 66 (2015), 2143-2160.  doi: 10.1007/s00033-015-0519-9.

[24]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.  doi: 10.1016/j.jmaa.2014.09.055.

[25]

M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.

[26]

M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.

[27]

M. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.

[28]

M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.  doi: 10.1016/j.jfa.2015.10.014.

[29]

M. Wang, Dynamics for a diffusive prey-predator model with different free boundaries, preprint, arXiv: 1511.06479v2.

[30]

M. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive predator-prey model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.

[31]

M. WangW. Sheng and Y. Zhang, Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary, J. Math. Anal. Appl., 441 (2016), 309-329.  doi: 10.1016/j.jmaa.2016.04.007.

[32]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys. , 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.

[33]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[34]

M. Wang and J. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, (2015), 1-23.  doi: 10.1007/s10884-015-9503-5.

[35]

C. H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.

[36]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.

[37]

P. Zhou and D. 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.

show all references

References:
[1]

G. BuntingY. 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, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons Ltd, 2003. doi: 10.1002/0470871296.

[3]

Q. ChenF. Li and F. Wang, A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 13-35.  doi: 10.3934/dcdsb.2016.21.13.

[4]

Q. ChenF. Li and F. Wang, The diffusive competition problem with a free boundary in heterogeneous time-periodic environment, J. Math. Anal. Appl., 433 (2016), 1594-1613.  doi: 10.1016/j.jmaa.2015.08.062.

[5]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, Ⅱ, J. Differential Equations, 250 (2011), 4336-4366.  doi: 10.1016/j.jde.2011.02.011.

[6]

Y. DuZ. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016.

[7]

Y. Du and Z. 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 Z. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105.

[9]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.

[10]

Y. DuM. Wang and M. Zhou, Semi-wave and spreading speed for the diffusive competition model with a free boundary, J. Math. Pures Appl., 107 (2017), 253-287.  doi: 10.1016/j.matpur.2016.06.005.

[11]

H. GuZ. Lin and B. Lou, Long time behavior of solutions of a diffusion-advection logistic model with free boundaries, Appl. Math. Lett., 37 (2014), 49-53.  doi: 10.1016/j.aml.2014.05.015.

[12]

H. GuZ. Lin and B. Lou, Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries, Proc. Amer. Math. Soc., 143 (2015), 1109-1117.  doi: 10.1090/S0002-9939-2014-12214-3.

[13]

H. GuB. Lou and M. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002.

[14]

J. S. Guo and C. H. Wu, On a free boundary problem for a two-species weak competition system, J. Dynam. Differential Equations, 24 (2012), 873-895.  doi: 10.1007/s10884-012-9267-0.

[15]

J. S. Guo and C. H. Wu, Dynamics for a two-species competition-diffusion model with two free boundaries, Nonlinearity, 28 (2015), 1-27.  doi: 10.1088/0951-7715/28/1/1.

[16]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. , vol. 247, Longman Sci. Tech. , Harlow, 1991.

[17]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76.  doi: 10.1016/j.jmaa.2015.02.051.

[18]

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

[19]

C. LeiZ. Lin and H. Wang, The free boundary problem describing information diffusion in online social networks, J. Differential Equations, 254 (2013), 1326-1341.  doi: 10.1016/j.jde.2012.10.021.

[20]

C. LeiZ. Lin and Q. 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.

[21]

Z. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004.

[22]

R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.

[23]

J. Wang, The selection for dispersal: A diffusive competition model with a free boundary, Z. Angew. Math. Phys., 66 (2015), 2143-2160.  doi: 10.1007/s00033-015-0519-9.

[24]

J. Wang and L. Zhang, Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment, J. Math. Anal. Appl., 423 (2015), 377-398.  doi: 10.1016/j.jmaa.2014.09.055.

[25]

M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations, 256 (2014), 3365-3394.  doi: 10.1016/j.jde.2014.02.013.

[26]

M. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022.

[27]

M. Wang, Spreading and vanishing in the diffusive prey-predator model with a free boundary, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 311-327.  doi: 10.1016/j.cnsns.2014.11.016.

[28]

M. Wang, A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment, J. Funct. Anal., 270 (2016), 483-508.  doi: 10.1016/j.jfa.2015.10.014.

[29]

M. Wang, Dynamics for a diffusive prey-predator model with different free boundaries, preprint, arXiv: 1511.06479v2.

[30]

M. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive predator-prey model, Nonlinear Anal. Real World Appl., 24 (2015), 73-82.  doi: 10.1016/j.nonrwa.2015.01.004.

[31]

M. WangW. Sheng and Y. Zhang, Spreading and vanishing in a diffusive prey-predator model with variable intrinsic growth rate and free boundary, J. Math. Anal. Appl., 441 (2016), 309-329.  doi: 10.1016/j.jmaa.2016.04.007.

[32]

M. Wang and Y. Zhang, The time-periodic diffusive competition models with a free boundary and sign-changing growth rates, Z. Angew. Math. Phys. , 67 (2016), Art. 132, 24 pp. doi: 10.1007/s00033-016-0729-9.

[33]

M. Wang and J. Zhao, Free boundary problems for a Lotka-Volterra competition system, J. Dynam. Differential Equations, 26 (2014), 655-672.  doi: 10.1007/s10884-014-9363-4.

[34]

M. Wang and J. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, (2015), 1-23.  doi: 10.1007/s10884-015-9503-5.

[35]

C. H. Wu, The minimal habitat size for spreading in a weak competition system with two free boundaries, J. Differential Equations, 259 (2015), 873-897.  doi: 10.1016/j.jde.2015.02.021.

[36]

J. Zhao and M. Wang, A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment, Nonlinear Anal. Real World Appl., 16 (2014), 250-263.  doi: 10.1016/j.nonrwa.2013.10.003.

[37]

P. Zhou and D. 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.

[1]

Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133

[2]

Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245

[3]

Qiaoling Chen, Fengquan Li, Feng Wang. A diffusive logistic problem with a free boundary in time-periodic environment: Favorable habitat or unfavorable habitat. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 13-35. doi: 10.3934/dcdsb.2016.21.13

[4]

Ke Wang, Qi Wang, Feng Yu. Stationary and time-periodic patterns of two-predator and one-prey systems with prey-taxis. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 505-543. doi: 10.3934/dcds.2017021

[5]

Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2189-2219. doi: 10.3934/dcdsb.2021129

[6]

Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737

[7]

Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317

[8]

Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121

[9]

Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133

[10]

R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423

[11]

Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180

[12]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256

[13]

Rafael Ortega. Variations on Lyapunov's stability criterion and periodic prey-predator systems. Electronic Research Archive, 2021, 29 (6) : 3995-4008. doi: 10.3934/era.2021069

[14]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[15]

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173

[16]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

[17]

Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061

[18]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398

[19]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks and Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583

[20]

J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (221)
  • HTML views (106)
  • Cited by (3)

Other articles
by authors

[Back to Top]