| Title:
|
A free boundary problem for a predator-prey model with nonlinear prey-taxis (English) |
| Author:
|
Yousefnezhad, Mohsen |
| Author:
|
Mohammadi, Seyyed Abbas |
| Author:
|
Bozorgnia, Farid |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2018 |
| Pages:
|
125-147 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper deals with a reaction-diffusion system modeling a free boundary problem of the predator-prey type with prey-taxis over a one-dimensional habitat. The free boundary represents the spreading front of the predator species. The global existence and uniqueness of classical solutions to this system are established by the contraction mapping principle. With an eye on the biological interpretations, numerical simulations are provided which give a real insight into the behavior of the free boundary and the stability of the solutions. (English) |
| Keyword:
|
prey-predator model |
| Keyword:
|
prey-taxis |
| Keyword:
|
free boundary |
| Keyword:
|
classical solutions |
| Keyword:
|
global existence |
| MSC:
|
35K51 |
| MSC:
|
35K55 |
| MSC:
|
35K57 |
| MSC:
|
35K59 |
| MSC:
|
35R35 |
| MSC:
|
92B05 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 06890302 |
| idMR:
|
MR3795243 |
| DOI:
|
10.21136/AM.2018.0227-17 |
| . |
| Date available:
|
2018-05-09T08:53:27Z |
| Last updated:
|
2020-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147186 |
| . |
| Reference:
|
[1] Ainseba, B. E., Bendahmane, M., Noussair, A.: A reaction-diffusion system modeling predator-prey with prey-taxis.Nonlinear Anal., Real World Appl. 9 (2008), 2086-2105. Zbl 1156.35404, MR 2441768, 10.1016/j.nonrwa.2007.06.017 |
| Reference:
|
[2] Bozorgnia, F., Arakelyan, A.: Numerical algorithms for a variational problem of the spatial segregation of reaction-diffusion systems.Appl. Math. Comput. 219 (2013), 8863-8875. Zbl 1291.65257, MR 3047783, 10.1016/j.amc.2013.03.074 |
| Reference:
|
[3] Bunting, G., Du, Y., Krakowski, K.: Spreading speed revisited: analysis of a free boundary model.Netw. Heterog. Media 7 (2012), 583-603. Zbl 1302.35194, MR 3004677, 10.3934/nhm.2012.7.583 |
| Reference:
|
[4] Chen, X., Friedman, A.: A free boundary problem arising in a model of wound healing.SIAM J. Math. Anal. 32 (2000), 778-800. Zbl 0972.35193, MR 1814738, 10.1137/S0036141099351693 |
| Reference:
|
[5] Du, Y., Guo, Z.: The Stefan problem for the Fisher-KPP equation.J. Differ. Equations 253 (2012), 996-1035. Zbl 1257.35110, MR 2922661, 10.1016/j.jde.2012.04.014 |
| Reference:
|
[6] Friedman, A.: Variational Principles and Free-Boundary Problems.Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, New York (1982). Zbl 0564.49002, MR 0679313 |
| Reference:
|
[7] He, X., Zheng, S.: Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis.Appl. Math. Lett. 49 (2015), 73-77. Zbl 06587041, MR 3361698, 10.1016/j.aml.2015.04.017 |
| Reference:
|
[8] Hillen, T., Painter, K. J.: A user's guide to PDE models for chemotaxis.J. Math. Biol. 58 (2009), 183-217. Zbl 1161.92003, MR 2448428, 10.1007/s00285-008-0201-3 |
| Reference:
|
[9] Hsu, S.-B., Huang, T.-W.: Global stability for a class of predator-prey systems.SIAM J. Appl. Math. 55 (1995), 763-783. Zbl 0832.34035, MR 1331585, 10.1137/S0036139993253201 |
| Reference:
|
[10] Ladyzhenskaya, O. A., Solonnikov, V. A., Ural'tseva, N. N.: Linear and Quasi-Linear Equations of Parabolic Type.Translations of Mathematical Monographs 23, American Mathematical Society, Providence (1968). Zbl 0174.15403, MR 0241822 |
| Reference:
|
[11] Levin, S. A.: A more functional response to predator-prey stability.Amer. Natur. 111 (1977), 381-383. 10.1086/283170 |
| Reference:
|
[12] Li, A.-W.: Impact of noise on pattern formation in a predator-prey model.Nonlinear Dyn. 66 (2011), 689-694. MR 2859594, 10.1007/s11071-010-9941-x |
| Reference:
|
[13] Li, L., Jin, Z.: Pattern dynamics of a spatial predator-prey model with noise.Nonlinear Dyn. 67 (2012), 1737-1744. MR 2877412, 10.1007/s11071-011-0101-8 |
| Reference:
|
[14] Li, C., Wang, X., Shao, Y.: Steady states of a predator-prey model with prey-taxis.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 97 (2014), 155-168. Zbl 1287.35018, MR 3146380, 10.1016/j.na.2013.11.022 |
| Reference:
|
[15] Lin, Z.: A free boundary problem for a predator-prey model.Nonlinearity 20 (2007), 1883-1892. Zbl 1126.35111, MR 2343682, 10.1088/0951-7715/20/8/004 |
| Reference:
|
[16] Markowich, P. A.: Applied Partial Differential Equations: A Visual Approach.Springer, Berlin (2007). Zbl 1109.35001, MR 2309862, 10.1007/978-3-540-34646-3 |
| Reference:
|
[17] Mimura, M., Kawasaki, K.: Spatial segregation in competitive interaction-diffusion equations.J. Math. Biol. 9 (1980), 49-64. Zbl 0425.92010, MR 0648845, 10.1007/BF00276035 |
| Reference:
|
[18] Mimura, M., Murray, J. D.: On a diffusive prey-predator model which exhibits patchiness.J. Theor. Biol. 75 (1978), 249-252. MR 0518476, 10.1016/0022-5193(78)90332-6 |
| Reference:
|
[19] Monobe, H., Wu, C.-H.: On a free boundary problem for a reaction-diffusion-advection logistic model in heterogeneous environment.J. Diff. Equations 261 (2016), 6144-6177. Zbl 1351.35070, MR 3552561, 10.1016/j.jde.2016.08.033 |
| Reference:
|
[20] Othmer, H. G., Stevens, A.: Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks.SIAM J. Appl. Math. 57 (1997), 1044-1081. Zbl 0990.35128, MR 1462051, 10.1137/S0036139995288976 |
| Reference:
|
[21] Painter, K. J., Hillen, T.: Volume-filling and quorum-sensing in models for chemosensitive movement.Can. Appl. Math. Q. 10 (2002), 501-543. Zbl 1057.92013, MR 2052525 |
| Reference:
|
[22] Sun, G.-Q.: Mathematical modeling of population dynamics with Allee effect.Nonlinear Dyn. 85 (2016), 1-12. MR 3510594, 10.1007/s11071-016-2671-y |
| Reference:
|
[23] Tao, Y.: Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis.Nonlinear Anal., Real World Appl. 11 (2010), 2056-2064. Zbl 1195.35171, MR 2646615, 10.1016/j.nonrwa.2009.05.005 |
| Reference:
|
[24] Xiao, D., Ruan, S.: Global dynamics of a ratio-dependent predator-prey system.J. Math. Biol. 43 (2001), 268-290. Zbl 1007.34031, MR 1868217, 10.1007/s002850100097 |
| Reference:
|
[25] Wang, M.: On some free boundary problems of the prey-predator model.J. Differ. Equations 256 (2014), 3365-3394. Zbl 1317.35110, MR 3177899, 10.1016/j.jde.2014.02.013 |
| Reference:
|
[26] Wang, M.: Spreading and vanishing in the diffusive prey-predator model with a free boundary.Commun. Nonlinear Sci. Numer. Simul. 23 (2015), 311-327. Zbl 1354.92074, MR 3312368, 10.1016/j.cnsns.2014.11.016 |
| Reference:
|
[27] Yousefnezhad, M., Mohammadi, S. A.: Stability of a predator-prey system with prey taxis in a general class of functional responses.Acta Math. Sci., Ser. B, Engl. Ed. 36 (2016), 62-72. Zbl 1363.35185, MR 3432748, 10.1016/S0252-9602(15)30078-3 |
| Reference:
|
[28] Zhao, J., Wang, M.: A free boundary problem of a predator-prey model with higher dimension and heterogeneous environment.Nonlinear Anal., Real World Appl. 16 (2014), 250-263. Zbl 1296.35220, MR 3123816, 10.1016/j.nonrwa.2013.10.003 |
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