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Article

Yousefnezhad, Mohsen ; Mohammadi, Seyyed Abbas ; Bozorgnia, Farid
A free boundary problem for a predator-prey model with nonlinear prey-taxis. (English). Applications of Mathematics, vol. 63 (2018), issue 2, pp. 125-147
MSC: 35K51, 35K55, 35K57, 35K59, 35R35, 92B05, 92D25 | MR 3795243 | Zbl 06890302 | DOI: 10.21136/AM.2018.0227-17
Keywords:
prey-predator model; prey-taxis; free boundary; classical solutions; global existence
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.
References:
[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. DOI 10.1016/j.nonrwa.2007.06.017 | MR 2441768 | Zbl 1156.35404
[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. DOI 10.1016/j.amc.2013.03.074 | MR 3047783 | Zbl 1291.65257
[3] Bunting, G., Du, Y., Krakowski, K.: Spreading speed revisited: analysis of a free boundary model. Netw. Heterog. Media 7 (2012), 583-603. DOI 10.3934/nhm.2012.7.583 | MR 3004677 | Zbl 1302.35194
[4] Chen, X., Friedman, A.: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 32 (2000), 778-800. DOI 10.1137/S0036141099351693 | MR 1814738 | Zbl 0972.35193
[5] Du, Y., Guo, Z.: The Stefan problem for the Fisher-KPP equation. J. Differ. Equations 253 (2012), 996-1035. DOI 10.1016/j.jde.2012.04.014 | MR 2922661 | Zbl 1257.35110
[6] Friedman, A.: Variational Principles and Free-Boundary Problems. Pure and Applied Mathematics. A Wiley-Interscience Publication, John Wiley & Sons, New York (1982). MR 0679313 | Zbl 0564.49002
[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. DOI 10.1016/j.aml.2015.04.017 | MR 3361698 | Zbl 06587041
[8] Hillen, T., Painter, K. J.: A user's guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009), 183-217. DOI 10.1007/s00285-008-0201-3 | MR 2448428 | Zbl 1161.92003
[9] Hsu, S.-B., Huang, T.-W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55 (1995), 763-783. DOI 10.1137/S0036139993253201 | MR 1331585 | Zbl 0832.34035
[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). MR 0241822 | Zbl 0174.15403
[11] Levin, S. A.: A more functional response to predator-prey stability. Amer. Natur. 111 (1977), 381-383. DOI 10.1086/283170
[12] Li, A.-W.: Impact of noise on pattern formation in a predator-prey model. Nonlinear Dyn. 66 (2011), 689-694. DOI 10.1007/s11071-010-9941-x | MR 2859594
[13] Li, L., Jin, Z.: Pattern dynamics of a spatial predator-prey model with noise. Nonlinear Dyn. 67 (2012), 1737-1744. DOI 10.1007/s11071-011-0101-8 | MR 2877412
[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. DOI 10.1016/j.na.2013.11.022 | MR 3146380 | Zbl 1287.35018
[15] Lin, Z.: A free boundary problem for a predator-prey model. Nonlinearity 20 (2007), 1883-1892. DOI 10.1088/0951-7715/20/8/004 | MR 2343682 | Zbl 1126.35111
[16] Markowich, P. A.: Applied Partial Differential Equations: A Visual Approach. Springer, Berlin (2007). DOI 10.1007/978-3-540-34646-3 | MR 2309862 | Zbl 1109.35001
[17] Mimura, M., Kawasaki, K.: Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9 (1980), 49-64. DOI 10.1007/BF00276035 | MR 0648845 | Zbl 0425.92010
[18] Mimura, M., Murray, J. D.: On a diffusive prey-predator model which exhibits patchiness. J. Theor. Biol. 75 (1978), 249-252. DOI 10.1016/0022-5193(78)90332-6 | MR 0518476
[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. DOI 10.1016/j.jde.2016.08.033 | MR 3552561 | Zbl 1351.35070
[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. DOI 10.1137/S0036139995288976 | MR 1462051 | Zbl 0990.35128
[21] Painter, K. J., Hillen, T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10 (2002), 501-543. MR 2052525 | Zbl 1057.92013
[22] Sun, G.-Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85 (2016), 1-12. DOI 10.1007/s11071-016-2671-y | MR 3510594
[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. DOI 10.1016/j.nonrwa.2009.05.005 | MR 2646615 | Zbl 1195.35171
[24] Xiao, D., Ruan, S.: Global dynamics of a ratio-dependent predator-prey system. J. Math. Biol. 43 (2001), 268-290. DOI 10.1007/s002850100097 | MR 1868217 | Zbl 1007.34031
[25] Wang, M.: On some free boundary problems of the prey-predator model. J. Differ. Equations 256 (2014), 3365-3394. DOI 10.1016/j.jde.2014.02.013 | MR 3177899 | Zbl 1317.35110
[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. DOI 10.1016/j.cnsns.2014.11.016 | MR 3312368 | Zbl 1354.92074
[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. DOI 10.1016/S0252-9602(15)30078-3 | MR 3432748 | Zbl 1363.35185
[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. DOI 10.1016/j.nonrwa.2013.10.003 | MR 3123816 | Zbl 1296.35220
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