Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis. (English) Zbl 1381.35188
Summary: This paper studies a reaction-diffusion system of a predator-prey model with Holling type II functional response and prey-taxis, proposed by B. Ainseba et al. [Nonlinear Anal., Real World Appl. 9, No. 5, 2086–2105 (2008; Zbl 1156.35404)], where the prey-taxis means a direct movement of the predator in response to a variation of the prey (which results in the aggregation of the predator). The global existence of classical solutions was established by Y. Tao [Nonlinear Anal., Real World Appl. 11, No. 3, 2056–2064 (2010; Zbl 1195.35171)]. In this paper we prove furthermore that the global classical solutions are globally bounded, by means of the Gagliardo-Nirenberg inequality, the \(L^{p}-L^{q}\) estimates for the Neumann heat semigroup, and the \(L^{p}\) estimates with Moser’s iteration of parabolic equations.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92D25 | Population dynamics (general) |
| 35A08 | Fundamental solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K58 | Semilinear parabolic equations |
| 37N25 | Dynamical systems in biology |
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] | Tao, Y. S., Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonlinear Anal. RWA, 11, 2056-2064 (2010) · Zbl 1195.35171 |
| [3] | Li, C. L.; Wang, X. H.; Shao, Y. F., Steady states of a predator-prey mdol with prey-taxis, Nonlinear Anal., 97, 155-168 (2014) · Zbl 1287.35018 |
| [4] | Ko, W.; Ryu, K., A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appl., 344, 217-230 (2008) · Zbl 1144.35029 |
| [5] | Ko, W.; Ryu, K., A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. RWA, 10, 2558-2573 (2009) · Zbl 1163.35339 |
| [6] | Cao, X. R., Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35, 5 (2015) · Zbl 1515.35047 |
| [7] | Winkler, M., Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248, 2889-2905 (2010) · Zbl 1190.92004 |
| [8] | Tao, Y. S.; Winkler, M., Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252, 692-715 (2012) · Zbl 1382.35127 |
| [9] | Painter, K. J.; Hillen, T., Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10, 501-543 (2002) · Zbl 1057.92013 |
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