May  2020, 25(5): 1649-1670. doi: 10.3934/dcdsb.2019245

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  May 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. 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.

[2]

S. M. BaerB. W. KooiY. 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. BuntingY. 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. DuP. 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. DuM. 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. LiuH. 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. ZhaoC. 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. ZhouS. 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.

show all references

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. BaerB. W. KooiY. 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. BuntingY. 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. DuP. 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. DuM. 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. LiuH. 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. ZhaoC. 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. ZhouS. 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.

[1]

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

[2]

Lei Yang, Lianzhang Bao. Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1083-1109. doi: 10.3934/dcdsb.2020154

[3]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763

[4]

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

[5]

Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240

[6]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199

[7]

Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086

[8]

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

[9]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213

[10]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191

[11]

Hongfei Xu, Jinfeng Wang, Xuelian Xu. Dynamics and pattern formation in a cross-diffusion model with stage structure for predators. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021237

[12]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173

[13]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

[14]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55

[15]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

[16]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617

[17]

Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255

[18]

Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979

[19]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355

[20]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (532)
  • HTML views (242)
  • Cited by (1)

Other articles
by authors

[Back to Top]