Han, Er-Dong; Guo, Peng Dynamical behaviors of a diffusive predator-prey system with Beddington-DeAngelis functional response. (English) Zbl 1305.34072 Int. J. Biomath. 7, No. 3, Article ID 1450033, 20 p. (2014). Summary: We present a diffusive predator-prey system with Beddington-DeAngelis functional response, where the prey species can disperse between the two patches, and there is competition between the two predators. Sufficient conditions for the permanence and extinction of system are established based on the upper and lower solutions method and comparison theory. Furthermore, the global asymptotic stability of positive solutions is obtained by constructing a suitable Lyapunov function. By using the continuation theorem in coincidence degree theory, we show the periodicity of positive solutions. Finally, we illustrate global asymptotic stability of the model by a simulation. MSC: 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D20 Stability of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations Keywords:Beddington-DeAngelis functional response; diffusion; permanence; extinction; periodic solution; asymptotic stability PDFBibTeX XMLCite \textit{E.-D. Han} and \textit{P. Guo}, Int. J. Biomath. 7, No. 3, Article ID 1450033, 20 p. (2014; Zbl 1305.34072) Full Text: DOI References: [1] Bainov D. D., Pitman Monographs and Surveys in Pure and Applied Mathematics 66, in: Impulsive Differential Equations: Periodic Solutions and Applications (1993) [2] DOI: 10.2307/3866 · doi:10.2307/3866 [3] DOI: 10.1111/j.1939-7445.2001.tb00062.x · Zbl 1005.92035 · doi:10.1111/j.1939-7445.2001.tb00062.x [4] DOI: 10.1006/jmaa.2000.7343 · Zbl 0991.34046 · doi:10.1006/jmaa.2000.7343 [5] Cantrell R. S., Spatial Ecology Via Reaction–Diffusion Equations (2003) · Zbl 1059.92051 [6] DOI: 10.1016/j.jmaa.2005.10.011 · Zbl 1102.34033 · doi:10.1016/j.jmaa.2005.10.011 [7] DOI: 10.2307/1936298 · doi:10.2307/1936298 [8] Gaines R. E., Coincidence Degree and Nonlinear Differential Equations (1977) · Zbl 0339.47031 · doi:10.1007/BFb0089537 [9] DOI: 10.1016/j.jmaa.2011.11.044 · Zbl 1234.35284 · doi:10.1016/j.jmaa.2011.11.044 [10] DOI: 10.1016/S0022-247X(02)00395-5 · Zbl 1033.34052 · doi:10.1016/S0022-247X(02)00395-5 [11] DOI: 10.1016/j.jmaa.2003.09.073 · Zbl 1086.34028 · doi:10.1016/j.jmaa.2003.09.073 [12] DOI: 10.1007/s10144-001-8187-3 · doi:10.1007/s10144-001-8187-3 [13] DOI: 10.1098/rspb.2000.1186 · doi:10.1098/rspb.2000.1186 [14] DOI: 10.1007/s00442-008-1225-5 · doi:10.1007/s00442-008-1225-5 [15] Krawcewicz W., Theory of Degrees, with Applications to Bifurcations and Differential Equations (1997) · Zbl 0882.58001 [16] Li Y., Acta Math. Sinica 39 pp 790– (1996) [17] DOI: 10.1006/jmaa.1997.5576 · Zbl 0894.34075 · doi:10.1006/jmaa.1997.5576 [18] DOI: 10.1090/S0002-9939-99-05210-7 · Zbl 0917.34057 · doi:10.1090/S0002-9939-99-05210-7 [19] DOI: 10.1016/j.jmaa.2010.08.029 · Zbl 1213.34097 · doi:10.1016/j.jmaa.2010.08.029 [20] DOI: 10.1016/j.cnsns.2010.12.026 · Zbl 1219.92064 · doi:10.1016/j.cnsns.2010.12.026 [21] DOI: 10.1016/S0096-3003(02)00035-8 · Zbl 1026.34082 · doi:10.1016/S0096-3003(02)00035-8 [22] DOI: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2 · doi:10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2 [23] Ton T. V., Acta Math. Acad. Paedagog. Nyházi. 25 pp 45– (2009) [24] DOI: 10.1016/j.mcm.2008.05.052 · Zbl 1187.34052 · doi:10.1016/j.mcm.2008.05.052 [25] DOI: 10.1016/j.nonrwa.2009.09.019 · Zbl 1197.35042 · doi:10.1016/j.nonrwa.2009.09.019 [26] DOI: 10.1016/j.nonrwa.2010.08.015 · Zbl 1206.35239 · doi:10.1016/j.nonrwa.2010.08.015 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.