×

Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis. (English) Zbl 1195.35171

Summary: This paper deals with a predator-prey model consisting of a \(2\times 2\) reaction-diffusion-taxis system recently proposed by B. E. Ainseba, M. Bendahmane and A. Noussair [Nonlinear Anal., Real World Appl. 9, 2086–2105 (2005; Zbl 1156.35404)]. The central point of this system is that the spatial-temporal variations of the predators’ velocity are directed by the prey gradient. The global existence and uniqueness of classical solutions to this system are proved by the contraction mapping principle, together with \(L^p\) estimates and Schauder estimates of parabolic equations. The crucial point of the proof is to deal with the prey-tactic term with a nonlinear tactic function.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K58 Semilinear parabolic equations
92D25 Population dynamics (general)
35K57 Reaction-diffusion equations

Citations:

Zbl 1156.35404
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ainseba, B.E.; Bendahmane, M.; Noussair, A., A reaction-diffusion system modeling predator-prey with prey-taxis, Nonlinear anal. RWA, 9, 2086-2105, (2008) · Zbl 1156.35404
[2] Keller, E.F.; Segel, L.A., Initiation of slime mold aggregation viewed as an instability, J. theoret. biol., 26, 399-415, (1970) · Zbl 1170.92306
[3] Horstmann, D.; Winkler, M., Boundedness vs. blow-up in a chemotaxis system, J. differential equations, 215, 52-107, (2005) · Zbl 1085.35065
[4] Cieślak, T.; Morales-Rodrigo, C., Quasilinear nonlinear non-uniformly parabolic-elliptic system modelling chemotaxis with volume filling effect: existence and uniqueness of global-in-time solutions, Topol. methods nonlinear anal., 29, 361-382, (2007) · Zbl 1128.92005
[5] Hillen, T.; Painter, K., Global existence for a parabolic chemotaxis model with prevention of overcrowding, Advin. appl. math., 26, 280-301, (2001) · Zbl 0998.92006
[6] Kowalczyk, R.; Szymańska, Z., On the global existence of solutions to an aggregation model, J. math. anal. appl., 343, 379-398, (2008) · Zbl 1143.35333
[7] Tao, Y., Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. math. anal. appl., 354, 60-69, (2009) · Zbl 1160.92027
[8] Tao, Y.; Wang, M., Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21, 2221-2238, (2008) · Zbl 1160.35431
[9] Tello, J.T.; Winkler, M., A chemotaxis system with logistic source, Comm. partial differential equations, 32, 849-877, (2007) · Zbl 1121.37068
[10] Friedman, A.; Lolas, G., Analysis of a mathematical model of tumor lymphangiogenesis, Math. models methods appl. sci., 1, 95-107, (2005) · Zbl 1060.92036
[11] Bendahmane, M., Analysis of a reaction-diffusion system modeling predator-prey with prey-taxis, Net. hetero. med., 3, 863-879, (2008) · Zbl 1160.35438
[12] Bendahmane, M.; Langlais, M.; Saad, M., On some anisotropic reaction-diffusion systems with \(L^1\)-data modeling the propagation of an epidemic disease, Nonlinear anal., 54, 617-636, (2003) · Zbl 1029.35114
[13] Bendahmane, M.; Karlsen, K.H., Nonlinear anisotropic elliptic and parabolic equations in \(\mathbb{R}^N\) with advection and lower order terms and locally integrable data, Potential anal., 22, 207-227, (2005) · Zbl 1087.35035
[14] Bendahmane, M.; Karlsen, K.H., Renormalized entropy solutions for quasilinear anisotropic degenerate parabolic equations, SIAM J. math. anal., 36, 2, 405-422, (2004) · Zbl 1090.35104
[15] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type, in: Amer. Math. Soc. Transl., vol. 23, Providence, RI, 1968
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.