zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global stability of stage-structured predator-prey models with Beddington-DeAngelis functional response. (English) Zbl 1219.92064
Summary: Two stage-structured predator-prey systems with Beddington-DeAngelis functional response are proposed. The first one is deterministic. The second one takes random perturbations into account. For each system, sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

MSC:
92D40Ecology
34D23Global stability of ODE
65C20Models (numerical methods)
WorldCat.org
Full Text: DOI
References:
[1] Skalski, G. T.; Gilliam, J. F.: Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology 82, 3083-3092 (2001)
[2] Hassell, M. P.; Varley, C. C.: New inductive population model for insect parasites and its bearing on biological control, Nature 223, 1133-1137 (1969)
[3] Beddington, J. R.: Mutual interference between parasites or predators and its effect on searching efficiency, J animal ecol 44, 331-341 (1975)
[4] Deangelis, D. L.; Goldsten, R. A.; Neill, R.: A model for trophic interaction, Ecology 56, 881-892 (1975)
[5] 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)
[6] Liu, S.; Beretta, E.: A stage-structured predator -- prey model of beddingtoncdeangelis type, SIAM J appl math 66, 1101-1129 (2006) · Zbl 1110.34059 · doi:10.1137/050630003
[7] 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 · doi:10.1016/j.jmaa.2007.12.038
[8] Zhao, M.; Lv, S.: Chaos in a three-species food chain model with a beddington -- deangelis functional response, Chaos soliton fract 40, 2305-2316 (2009) · Zbl 1198.37139 · doi:10.1016/j.chaos.2007.10.025
[9] 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)
[10] 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 · doi:10.1016/j.cnsns.2007.10.013
[11] 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 · doi:10.1016/j.na.2009.09.003
[12] Nie, H.; Wu, J.: Coexistence of an unstirred chemostat model with beddington -- deangelis functional response and inhibitor, Nonlinear anal real world appl 11, 3639-3652 (2010) · Zbl 1203.35128 · doi:10.1016/j.nonrwa.2010.01.010
[13] Wang, W.; Chen, L. S.: A predator -- prey system with stage-structure for predator, Comput math appl 33, 83-91 (1997)
[14] Wang, W.; Mulone, G.; Salemi, F.; Salone, V.: Permanence and stability of a stage-structured predator -- prey model, J math anal appl 262, 499-528 (2001) · Zbl 0997.34069 · doi:10.1006/jmaa.2001.7543
[15] Xiao, Y. N.; Chen, L. S.: Effects of toxicants on a stage-structured population growth model, Appl math comput 123, 63-73 (2001) · Zbl 1017.92044 · doi:10.1016/S0096-3003(00)00057-6
[16] Xu, R.; Chaplain, M. A. J.; Davidson, F. A.: Persistence and global stability of a ratio-dependent predator -- prey model with stage structure, Appl math comput 158, 729-744 (2004) · Zbl 1058.92053 · doi:10.1016/j.amc.2003.10.012
[17] Zhang, L.; Zhang, C.: Rich dynamic of a stage-structured prey -- predator model with cannibalism and periodic attacking rate, Commun nonlinear sci numer simulat 15, 4029-4040 (2010) · Zbl 1222.37105 · doi:10.1016/j.cnsns.2010.02.009
[18] Gard, T. C.: Persistence in stochastic food web models, Bull math biol 46, 357-370 (1984) · Zbl 0533.92028
[19] Gard, T. C.: Stability for multispecies population models in random environments, Nonlinear anal 10, 1411-1419 (1986) · Zbl 0598.92017 · doi:10.1016/0362-546X(86)90111-2
[20] May, R. M.: Stability and complexity in model ecosystems, (2001) · Zbl 1044.92047
[21] Braumann, C. A.: Variable effert harvesting models in random environments: generalization to density-dependent noise intensities, Math biosci 177& 178, 229-245 (2002) · Zbl 1003.92027
[22] Mao, X.; Yuan, C.; Zou, J.: Stochastic differential delay equations of population dynamics, J math anal appl 304, 296-320 (2005) · Zbl 1062.92055 · doi:10.1016/j.jmaa.2004.09.027
[23] Luo, Q.; Mao, X.: Stochastic population dynamics under regime switching, J math anal appl 334, 69-84 (2007) · Zbl 1113.92052 · doi:10.1016/j.jmaa.2006.12.032
[24] Rudnicki, R.; Pichor, K.: Influence of stochastic perturbation on prey -- predator systems, Math biosci, 206 108-119 (2007) · Zbl 1124.92055 · doi:10.1016/j.mbs.2006.03.006
[25] 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 · doi:10.3934/dcds.2009.24.523
[26] Zhu, C.; Yin, G.: On hybrid competitive Lotka -- Volterra ecosystems, Nonlinear anal 71, e1370-e1379 (2009)
[27] Liu, M.; Wang, K.: Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J theoret biol 264, 934-944 (2010)
[28] Liu, M.; Wang, K.: Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II, J theoret biol 267, 283-291 (2010)
[29] Liu, M.; Wang, K.; Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull math biol (2010) · Zbl 1225.92059
[30] Liu, M.; Wang, K.: Extinction and permanence in a stochastic nonautonomous population system, Appl math lett 23, 1464-1467 (2010) · Zbl 1206.34079 · doi:10.1016/j.aml.2010.08.012
[31] Liu, M.; Wang, K.: Persistence and extinction in stochastic non-autonomous logistic systems, J math anal appl 375, 443-457 (2011) · Zbl 1214.34045 · doi:10.1016/j.jmaa.2010.09.058
[32] Liu, M.; Wang, K.: Global stability of a nonlinear stochastic predator- prey system with beddington -- deangelis functional response, Commun nonlinear sci numer simulat 16, 1114-1121 (2011) · Zbl 1221.34152 · doi:10.1016/j.cnsns.2010.06.015
[33] Gourley, S. A.; Kuang, Y.: A stage structured predator -- prey model and its dependence on through-stage delay and death rate, J math biol 49, 188-200 (2004) · Zbl 1055.92043 · doi:10.1007/s00285-004-0278-2
[34] Mao, X.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057
[35] Mao, X.: A note on the lasalle-type theorems for stochastic differential delay equations, J math anal appl 268, 125-142 (2002) · Zbl 0996.60064 · doi:10.1006/jmaa.2001.7803
[36] Higham, D. J.: An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev 43, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302