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

Meng Zhao; Wan-Tong Li; Jia-Feng Cao

doi:10.3934/dcdsb.2017138

Article Contents

2017, Volume 22, Issue 9: 3295-3316. Doi: 10.3934/dcdsb.2017138

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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
* Corresponding author

Received: September 2016

Revised: December 2016

Published: October 2017

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Abstract

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.

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Mathematics Subject Classification: 35K51, 35R35, 35B40, 92B05.

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	G. Bunting , Y. 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.
	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley and Sons Ltd, 2003. doi: 10.1002/0470871296.
	Q. Chen , F. 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.
	Q. Chen , F. 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.
	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.
	Y. Du , Z. 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.
	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.
	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.
	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.
	Y. Du , M. 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.
	H. Gu , Z. 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.
	H. Gu , Z. 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.
	H. Gu , B. 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.
	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.
	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.
	P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. , vol. 247, Longman Sci. Tech. , Harlow, 1991.
	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.
	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. , Providence, RI, 1968.
	C. Lei , Z. 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.
	C. Lei , Z. 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.
	Z. Lin , A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007) , 1883-1892. doi: 10.1088/0951-7715/20/8/004.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	M. Wang, Dynamics for a diffusive prey-predator model with different free boundaries, preprint, arXiv: 1511.06479v2.
	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.
	M. Wang , W. 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.
	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.
	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.
	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.
	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.
	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.
	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.