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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)

Received  March 2019 Revised  June 2019 Published  November 2019

Fund Project: This work is supported by NSFC Grants 11771110, 11971128

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.

Citation: Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019245
##### 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.  Google Scholar [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.  Google Scholar [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.  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), 778-800.  doi: 10.1137/S0036141099351693.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [15] L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  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), 778-800.  doi: 10.1137/S0036141099351693.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [15] L. I. Rubinstein, The Stefan Problem, American Mathematical Society, Providence, RI, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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