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)
The complex dynamics of a stochastic predator-prey model. (English) Zbl 1253.93142
Summary: A modified stochastic ratio-dependent Leslie-Gower predator-prey model is formulated and analyzed. For the deterministic model, we focus on the existence of equilibria, local, and global stability; for the stochastic model, by applying Itô formula and constructing Lyapunov functions, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. Based on these results, we perform a series of numerical simulations and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

93E20Optimal stochastic control (systems)
92D25Population dynamics (general)
37N25Dynamical systems in biology
60H30Applications of stochastic analysis
Full Text: DOI
[1] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0539.03004
[2] P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, no. 3-4, pp. 213-245, 1948. · Zbl 0034.23303 · doi:10.1093/biomet/35.3-4.213
[3] P. H. Leslie, “A stochastic model for studying the properties of certain biological systems by numerical methods,” Biometrika, vol. 45, pp. 16-31, 1958. · Zbl 0089.15803 · doi:10.1093/biomet/45.1-2.16
[4] H.-F. Huo and W.-T. Li, “Stable periodic solution of the discrete periodic Leslie-Gower predator-prey model,” Mathematical and Computer Modelling, vol. 40, no. 3-4, pp. 261-269, 2004. · Zbl 1067.39008 · doi:10.1016/j.mcm.2004.02.026
[5] H.-F. Huo and W.-T. Li, “Periodic solutions of delayed Leslie-Gower predator-prey models,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 591-605, 2004. · Zbl 1060.34039 · doi:10.1016/S0096-3003(03)00801-4
[6] F. Chen, “Permanence in nonautonomous multi-species predator-prey system with feedback controls,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 694-709, 2006. · Zbl 1087.92059 · doi:10.1016/j.amc.2005.04.047
[7] F. Chen and X. Cao, “Existence of almost periodic solution in a ratio-dependent Leslie system with feedback controls,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1399-1412, 2008. · Zbl 1145.34026 · doi:10.1016/j.jmaa.2007.09.075
[8] C. Ji, D. Jiang, and N. Shi, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation,” Journal of Mathematical Analysis and Applications, vol. 359, no. 2, pp. 482-498, 2009. · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[9] M. Liu and K. Wang, “Global stability of a nonlinear stochastic predator-prey system with Beddington-DeAngelis functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1114-1121, 2011. · Zbl 1221.34152 · doi:10.1016/j.cnsns.2010.06.015
[10] T. C. Gard, “Persistence in stochastic food web models,” Bulletin of Mathematical Biology, vol. 46, no. 3, pp. 357-370, 1984. · Zbl 0533.92028 · doi:10.1007/BF02462011
[11] T. C. Gard, “Stability for multispecies population models in random environments,” Nonlinear Analysis. Theory, Methods & Applications, vol. 10, no. 12, pp. 1411-1419, 1986. · Zbl 0598.92017 · doi:10.1016/0362-546X(86)90111-2
[12] R. M. May, Stability and Complexity in Model Ecosystems, vol. 6, Princeton University Press, 2001.
[13] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Limited, 1997. · Zbl 0892.60057
[14] Q. Luo and X. Mao, “Stochastic population dynamics under regime switching,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 69-84, 2007. · Zbl 1113.92052 · doi:10.1016/j.jmaa.2006.12.032
[15] X. Mao, C. Yuan, and J. Zou, “Stochastic differential delay equations of population dynamics,” Journal of Mathematical Analysis and Applications, vol. 304, no. 1, pp. 296-320, 2005. · Zbl 1062.92055 · doi:10.1016/j.jmaa.2004.09.027
[16] S. Pang, F. Deng, and X. Mao, “Asymptotic properties of stochastic population dynamics,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 15, no. 5, pp. 603-620, 2008. · Zbl 1171.34038
[17] D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525-546, 2001. · Zbl 0979.65007 · doi:10.1137/S0036144500378302