Article Contents

2020, Volume 25, Issue 5: 1649-1670. Doi: 10.3934/dcdsb.2019245

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A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure

1.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518000, China
3.
School of Mathematics and Statistics, Wuhan University, Wuhan 430000, China

^* Corresponding author: Mingxin Wang (mxwang@hit.edu.cn)
^* Corresponding author: Mingxin Wang (mxwang@hit.edu.cn)

Received: March 2019

Revised: June 2019

Early access: November 2019

Published: May 2020

This work is supported by NSFC Grants 11771110, 11971128

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Abstract

In this paper we consider a free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. In our model, the individuals of a new or invasive predatory species are classified as belonging to either the immature or mature case. Firstly, the global existence, uniqueness, regularity of the solution are derived. And then when vanishing happens, we get uniform estimates and the long time behavior of the solution. At last, a sharp criterion governing spreading and vanishing for the free boundary problem is studied by the upper and lower solution method.

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Mathematics Subject Classification: Primary: 35A01, 35K65, 35R35, 35B40, 92D25.

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References

[1]	N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225. doi: 10.1016/0022-0396(79)90088-3.
[2]	S. M. Baer, B. W. Kooi, Y. A. Kuznetsov and H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model, SIAM J. Appl. Math., 66 (2006), 1339-1365. doi: 10.1137/050627757.
[3]	H. Bunting, Y. H. 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.
[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]	Y. H. Du and Z. G. 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.
[6]	Y. H. Du, P. Y. H. Pang and M. X. Wang, Qualitative analysis of a prey-predator model with stage structure for the predator, SIAM J. Appl. Math., 69 (2008), 596-620. doi: 10.1137/070684173.
[7]	Y. H. Du, M. X. Wang and M. L. 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.
[8]	S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200. doi: 10.1007/s00285-004-0278-2.
[9]	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.
[10]	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.
[11]	M. Li and Z. G. Lin, The spreading fronts in a mutualistic model with advection, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2089-2105. doi: 10.3934/dcdsb.2015.20.2089.
[12]	S. Y. Liu, H. M. Huang and M. X. Wang, Asymptotic spreading of a diffusive competition model with different free boundaries, J. Differential Equations, 266 (2019), 4769-4799. doi: 10.1016/j.jde.2018.10.009.
[13]	S. Y. Liu, H. M. Huang and M. X. Wang, Spatial spreading of a logistic SI epidemic model with degenerate diffusion and double free boundaries, preprint.
[14]	F. Rothe, Uniform bounds from bounded $L^p$ functionals in reaction-diffusion equations, J. Differential Equations, 45 (1982), 207-233. doi: 10.1016/0022-0396(82)90067-5.
[15]	L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.
[16]	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.
[17]	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.
[18]	J. P. Wang and M. X. Wang, The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary, Math. Methods Appl. Sci., 41 (2018), 6741-6762. doi: 10.1002/mma.5189.
[19]	M. X. Wang, Existence and uniqueness of solutions of free boundary problems in heterogeneous environments, Discrete Contin. Dyn. Syst. Ser. B., 24 (2019), 415-421. doi: 10.3934/dcdsb.2018179.
[20]	M. X. 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.
[21]	M. X. Wang and Q. Y. Zhang, Dynamics for the diffusive Leslie-Gower model with double free boundaries, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 2591-2607. doi: 10.3934/dcds.2018109.
[22]	M. X. Wang and Y. Zhang, Dynamics for a diffusive prey-predator model with different free boundaries, J. Differential Equations, 264 (2018), 3527-3558. doi: 10.1016/j.jde.2017.11.027.
[23]	M. X. Wang and Y. Zhang, Note on a two-species competition-diffusion model with two free boundaries, Nonlinear Anal., 159 (2017), 458-467. doi: 10.1016/j.na.2017.01.005.
[24]	M. X. Wang and Y. Zhang, Two kinds of free boundary problems for the diffusive prey-predator model, Nonlinear Anal.: Real World Appl., 24 (2015), 73-82. doi: 10.1016/j.nonrwa.2015.01.004.
[25]	M. X. Wang and J. F. Zhao, A free boundary problem for a predator-prey model with double free boundaries, J. Dynam. Differential Equations, 29 (2017), 957-979. doi: 10.1007/s10884-015-9503-5.
[26]	M. X. Wang and J. F. 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.
[27]	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.
[28]	Q. Y. Zhang and M. X. Wang, Dynamics for the diffusive mutualist model with advection and different free boundaries, J. Math. Anal. Appl., 474 (2019), 1512-1535. doi: 10.1016/j.jmaa.2019.02.037.
[29]	J. F. Zhao, C. M. Song and H. T. Zhang, A diffusive stage-structured model with a free boundary, Bound. Value Probl., 138 (2018), 1-23. doi: 10.1186/s13661-018-1060-5.
[30]	J. F. Zhao and M. X. 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.
[31]	Y. G. Zhao and M. X. Wang, Free boundary problems for the diffusive competition system in higher dimension with sign-changing coefficients, IMA J. Appl. Math., 81 (2016), 255-280. doi: 10.1093/imamat/hxv035.
[32]	L. Zhou, S. Zhang and Z. H. Liu, An evolutional free-boundary problem of a reaction-diffusion-advection system, Proc. Royal Soc. Edinburgh Sect. A, 147 (2017), 615-648. doi: 10.1017/S0308210516000226.