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 & 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. Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons Ltd, 2003. doi: 10.1002/0470871296. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons Ltd, 2003. doi: 10.1002/0470871296. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

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

[2]

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

[3]

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

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Yaying Dong, Shanbing Li, Yanling Li. Effects of dispersal for a predator-prey model in a heterogeneous environment. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2511-2528. doi: 10.3934/cpaa.2019114

[15]

Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048

[16]

Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046

[17]

Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172

[18]

Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823

[19]

Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127

[20]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (39)
  • HTML views (25)
  • Cited by (1)

Other articles
by authors

[Back to Top]