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The effect of mutual interference between predators on a predator-prey model with diffusion. (English) Zbl 1234.35284
Summary: We consider a diffusive predator-prey model with Beddington-DeAngelis functional response under homogeneous Dirichlet boundary conditions. The effect of large k which represents the extent of mutual interference between predators is extensively studied. By making use of the fixed point index theory, we obtain a complete understanding of the existence, uniqueness and stability of positive steady-states when k is sufficiently large. Moreover, we present some numerical simulations that supplement the analytic results in one dimension.
MSC:
35Q92PDEs in connection with biology and other natural sciences
35B09Positive solutions of PDE
References:
[1]Beddington, J. R.: Mutual interference between parasites or predators and its effect on searching efficiency, J. animal ecol. 44, No. 1, 331-340 (1975)
[2]Deangelis, D. L.; Goldstein, R. A.; O’neill, R. V.: A model for tropic interaction, Ecology 56, No. 2, 881-892 (1975)
[3]Dimitrov, D. T.; Kojouharov, H. V.: Complete mathematical analysis of predator-prey models with linear prey growth and beddington-deangelis functional response, Appl. math. Comput. 162, No. 2, 523-538 (2005) · Zbl 1057.92050 · doi:10.1016/j.amc.2003.12.106
[4]Blat, J.; Brown, K. J.: Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. roy. Soc. Edinburgh sect. A 97, 21-34 (1984) · Zbl 0554.92012 · doi:10.1017/S0308210500031802
[5]Li, L.: Coexistence theorems of steady states for predator-prey interacting systems, Trans. amer. Math. soc. 305, 143-166 (1988) · Zbl 0655.35021 · doi:10.2307/2001045
[6]Pao, C. V.: On nonlinear parabolic and elliptic equations, (1992)
[7]Blat, J.; Brown, K. J.: Global bifurcation of positive solutions in some systems of elliptic equations, SIAM J. Math. anal. 17, No. 6, 1339-1353 (1986) · Zbl 0613.35008 · doi:10.1137/0517094
[8]Casal, A.; Eilbeck, J. C.; López-Gómez, J.: Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Differential integral equations 7, No. 2, 411-439 (1994) · Zbl 0823.35050
[9]Du, Y. -H.; Lou, Y.: Some uniqueness and exact multiplicity results for a predator-prey model, Trans. amer. Math. soc. 349, No. 6, 2443-2475 (1997) · Zbl 0965.35041 · doi:10.1090/S0002-9947-97-01842-4
[10]Du, Y. -H.; Lou, Y.: S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. differential equations 144, No. 2, 390-440 (1998) · Zbl 0970.35030 · doi:10.1006/jdeq.1997.3394
[11]Du, Y. -H.; Lou, Y.: Qualitative behavior of positive solutions of a predator-prey model: effects of saturation, Proc. roy. Soc. Edinburgh sect. A 131, No. 2, 321-349 (2001) · Zbl 0980.35028 · doi:10.1017/S0308210500000895
[12]Chen, W. -Y.; Wang, M. -X.: Qualitative analysis of predator-prey models with beddington-deangelis functional response and diffusion, Math. comput. Modelling 42, 31-44 (2005) · Zbl 1087.35053 · doi:10.1016/j.mcm.2005.05.013
[13]Guo, G. -H.; Wu, J. -H.: Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear anal. 72, 1632-1646 (2010) · Zbl 1180.35528 · doi:10.1016/j.na.2009.09.003
[14]Guo, G. -H.; Wu, J. -H.: Existence and uniqueness of positive solution for a predator-prey model with diffusion, Acta math. Sci. ser. A 31, No. 1, 196-205 (2011)
[15]Dancer, E. N.: On the indices of fixed points of mappings in cones and applications, J. math. Anal. appl. 91, 131-151 (1983) · Zbl 0512.47045 · doi:10.1016/0022-247X(83)90098-7
[16]G.-H. Guo, J.-H. Wu, The multiplicity and uniqueness for a predator-prey-mutualist model with diffusion, Proc. Lond. Math. Soc. (3), in press.
[17]Guo, G. -H.; Li, Y. -L.: Global bifurcation and stability for the predator-prey model with B-D functional response, Acta math. Appl. sin. 31, No. 2, 320-330 (2008) · Zbl 1174.35061