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Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response. (English) Zbl 1221.34152

Summary: Stochastically asymptotic stability in the large of a predator–prey system with Beddington–DeAngelis functional response with stochastic perturbation is considered. The result shows that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided the noise is sufficiently small. Some simulation figures are introduced to support the analytical findings.

MSC:

34D23 Global stability of solutions to ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
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[1] Holling, C.S., The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canad entomologist, 91, 293-320, (1959)
[2] Holling, C.S., Some characteristics of simple types of predation and parasitism, Canad entomologist, 91, 385-395, (1959)
[3] Hassell, M.P.; Varley, C.C., New inductive population model for insect parasites and its bearing on biological control, Nature, 223, 1133-1137, (1969)
[4] Beddington, J.R., Mutual interference between parasites or predators and its effect on searching efficiency, J animal ecol, 44, 331-341, (1975)
[5] DeAngelis, D.L.; Goldsten, R.A.; Neill, R., A model for trophic interaction, Ecology, 56, 881-892, (1975)
[6] Crowley, P.H.; Martin, E.K., Functional response and interference within and between year classes of a dragonfly population, J N am benthol soc, 8, 211-221, (1989)
[7] Jost, C.; Arditi, R., From pattern to process: identifying predator – prey interactions, Population ecology, 43, 229-243, (2001)
[8] Skalski, G.T.; Gilliam, J.F., Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82, 3083-3092, (2001)
[9] Cantrell, R.S.; Cosner, C., On the dynamics of predator-prey models with the beddington – deangelis functional response, J math anal appl, 257, 206-222, (2001) · Zbl 0991.34046
[10] Hwang, T.W., Global analysis of the predator – prey system with beddington – deangelis functional response, J math anal appl, 281, 395-401, (2003) · Zbl 1033.34052
[11] Hwang, T.W., Uniqueness of limit cycles of the predator – prey system with beddington – deangelis functional response, J math anal appl, 290, 113-122, (2004) · Zbl 1086.34028
[12] Fan, M.; Kuang, Y., Dynamics of a non-autonomous predator – prey system with the beddington – deangelis functional response, J math anal appl, 295, 15-39, (2004) · Zbl 1051.34033
[13] Liu, Z.; Yuan, R., Stability and bifurcation in a delayed predator – prey system with beddington – deangelis functional response, J math anal appl, 296, 521-537, (2004) · Zbl 1051.34060
[14] Liu, S.; Zhang, J., Coexistence and stability of predator – prey model with beddington – deangelis functional response and stage structure, J math anal appl, 342, 446-460, (2008) · Zbl 1146.34057
[15] Zhao, M.; Lv, S., Chaos in a three-species food chain model with a beddington – deangelis functional response, Chaos solitons fractals, 40, 2305-2316, (2009) · Zbl 1198.37139
[16] Zhao, M.; Zhang, L., Permanence and chaos in a host – parasitoid model with prolonged diapause for the host, Commun nonlinear sci numer simulat, 14, 4197-4203, (2009)
[17] Gakkhar, S.; Negi, K.; Sahani, S.K., Effects of seasonal growth on a ratio-dependent delayed prey – predator system, Commun nonlinear sci numer simulat, 14, 850-862, (2009) · Zbl 1221.34187
[18] Li, W.; Wang, L., Stability and bifurcation of a delayed three-level food chain model with beddington – deangelis functional response, Nonlinear anal real world appl, 10, 2471-2477, (2009) · Zbl 1163.34348
[19] Guo, G.; Wu, J., 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
[20] Nie H, Wu J. Coexistence of an unstirred chemostat model with Beddington-DeAngelis functional response and inhibitor. Nonlinear Anal Real World Appl, doi:10.1016/j.nonrwa.2010.01.010. · Zbl 1203.35128
[21] Gard, T.C., Persistence in stochastic food web models, Bull math biol, 46, 357-370, (1984) · Zbl 0533.92028
[22] Gard, T.C., Stability for multispecies population models in random environments, Nonlinear anal, 10, 1411-1419, (1986) · Zbl 0598.92017
[23] May, R.M., Stability and complexity in model ecosystems, (2001), Princeton University Press NJ
[24] Beddington, J.R.; MAY, R.M., Harvesting natural populations in a randomly fluctuating environment, Science, 197, 463-465, (1977)
[25] Mao, X., Stochastic stabilisation and destabilisation, Systems control lett, 23, 279-290, (1994)
[26] Braumann, C.A., Variable effort harvesting models in random environments: generalization to density-dependent noise intensities, Math biosci, 177& 178, 229-245, (2002) · Zbl 1003.92027
[27] Mao, X.; Yuan, C.; Zou, J., Stochastic differential delay equations of population dynamics, J math anal appl, 304, 296-320, (2005) · Zbl 1062.92055
[28] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J math anal appl, 334, 69-84, (2007) · Zbl 1113.92052
[29] Rudnicki, R.; Pichor, K., Influence of stochastic perturbation on prey – predator systems, Math biosci, 206, 108-119, (2007) · Zbl 1124.92055
[30] Li, X.; Mao, X., Population dynamical behavior of non-autonomous lotka – volterra competitive system with random perturbation, Discrete contin dyn syst, 24, 523-545, (2009) · Zbl 1161.92048
[31] Zhu, C.; Yin, G., On hybrid competitive lotka – volterra ecosystems, Nonlinear anal, 71, e1370-e1379, (2009) · Zbl 1238.34059
[32] Liu, M.; Wang, K., Survival analysis of stochastic single-species population models in polluted environments, Ecol model, 220, 1347-1357, (2009)
[33] Liu, M.; Wang, K., Persistence and extinction of a stochastic single-species model under regime switching in a polluted environment, J theor biol, 264, 934-944, (2010) · Zbl 1406.92673
[34] Mao, X., Stochastic differential equations and applications, (1997), Horwood Publishing Chichester · Zbl 0874.60050
[35] Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev, 43, 525-546, (2001) · Zbl 0979.65007
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