zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation. (English) Zbl 1254.34074
Summary: A two-species stochastic non-autonomous predator-prey model is investigated. Sufficient criteria for extinction, non-persistence in the mean and weak persistence in the mean are established. The critical value between persistence and extinction is obtained for each species in many cases. It is also shown that the system is globally asymptotically stable under some simple conditions. Some numerical simulations are introduced to illustrate the main results.

34D23Global stability of ODE
92D25Population dynamics (general)
34D10Stability perturbations of ODE
34F05ODE with randomness
Full Text: DOI
[1] Freedman, H. I.: Deterministic mathematical models in population ecology, (1980) · Zbl 0448.92023
[2] Vance, R. R.; Coddington, E. A.: A nonautonomous model of population growth, J. math. Biol. 27, 491-506 (1989) · Zbl 0716.92016 · doi:10.1007/BF00288430
[3] Teng, Z. D.; Yu, Y. H.: The extinction in nonautonomous prey -- predator Lotka -- Volterra systems, Acta math. Appl. sin. 15, No. 4, 401-408 (1999) · Zbl 1007.92031 · doi:10.1007/BF02684041
[4] Xu, R.; Chen, L. S.: Persistence and stability for a two-species ratio-dependent predator -- prey system with time delay in a two-patch environment, Comput. math. Appl. 40, 577-588 (2000) · Zbl 0949.92028 · doi:10.1016/S0898-1221(00)00181-4
[5] 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
[6] Teng, Z. D.; Li, Z. M.; Jiang, H. J.: Pemanence criteria in non-autonomous predator -- prey Kolmogorov systems and its applications, Dyn. syst. 19, No. 2, 171-194 (2004) · Zbl 1066.34048 · doi:10.1080/14689360410001698851
[7] Wang, K.: Permanence and global asymptotical stability of a predator -- prey model with mutual interference, Nonlinear anal. Real world appl. 12, 1062-1071 (2011) · Zbl 1213.34067 · doi:10.1016/j.nonrwa.2010.08.028
[8] Gard, T. C.: Persistence in stochastic food web models, Bull. math. Biol. 46, 357-370 (1984) · Zbl 0533.92028
[9] Gard, T. C.: Introduction to stochastic differential equations, (1988) · Zbl 0628.60064
[10] Bandyopadhyay, M.; Chattopadhyay, J.: Ratio-dependent predator -- prey model: effect of environmental fluctuation and stability, Nonlinearity 18, 913-936 (2005) · Zbl 1078.34035 · doi:10.1088/0951-7715/18/2/022
[11] May, R. M.: Stability and complexity in model ecosystems, (2001) · Zbl 1044.92047
[12] Rudnicki, R.: Long-time behaviour of a stochastic prey -- predator model, Stochast. process. Appl. 108, 93-107 (2003) · Zbl 1075.60539 · doi:10.1016/S0304-4149(03)00090-5
[13] 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
[14] Jiang, D.; Shi, N.; Li, X.: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. math. Anal. appl. 340, 588-597 (2008) · Zbl 1140.60032 · doi:10.1016/j.jmaa.2007.08.014
[15] Cheng, S. R.: Stochastic population systems, Stoch. anal. Appl. 27, 854-874 (2009) · Zbl 1180.92071 · doi:10.1080/07362990902844348
[16] Li, X.; Mao, X.: Population dynamical behavior of non-autonomous Lotka -- Volterra competitive system with random perturbation, Discrete cont. Dyn. syst. 24, 523-545 (2009) · Zbl 1161.92048 · doi:10.3934/dcds.2009.24.523
[17] Liu, M.; Wang, K.: Persistence and extinction of a stochastic single-species model under regime switching in a polluted environment, J. theoret. Biol. 264, 934-944 (2010)
[18] Liu, M.; Wang, K.; Wu, Q.: Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. math. Biol. 73, 1969-2012 (2011) · Zbl 1225.92059 · doi:10.1007/s11538-010-9569-5
[19] Liu, M.; Wang, K.: Persistence and extinction in stochastic non-autonomous logistic systems, J. math. Anal. appl. 375, 42-57 (2011) · Zbl 1214.34045 · doi:10.1016/j.jmaa.2010.09.058
[20] Liu, M.; Wang, K.; Liu, X.: Long term behaviors of stochastic single-species growth models in a polluted environment, Appl. math. Model. 35, 752-762 (2011) · Zbl 1205.60086 · doi:10.1016/j.apm.2010.07.031
[21] Liu, M.; Wang, K.; Wang, Y.: Long term behaviors of stochastic single-species growth models in a polluted environment II, Appl. math. Model. 35, 4438-4448 (2011) · Zbl 1225.60067 · doi:10.1016/j.apm.2011.03.014
[22] Ma, Z.; Cui, G.; Wang, W.: Persistence and extinction of a population in a polluted environment, Math biosci. 101, 75-97 (1990) · Zbl 0714.92027 · doi:10.1016/0025-5564(90)90103-6
[23] Wang, W.; Ma, Z.: Permanence of a nonautomonous population model, Acta math. Appl. sin. Engl. ser. 1, 86-95 (1998) · Zbl 0940.92020 · doi:10.1007/BF02677353
[24] Mao, X.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057
[25] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991) · Zbl 0734.60060
[26] Barbalat, I.: Systems dequations differentielles d’osci d’oscillations nonlineaires, Revue roumaine de mathematiques pures et appliquees 4, 267-270 (1959) · Zbl 0090.06601
[27] Higham, D. J.: An algorithmic introduction to numerical simulation of stochastic diffrential equations, SIAM rev. 43, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302