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)
Dynamics of a stochastic density dependent predator-prey system with Beddington-DeAngelis functional response. (English) Zbl 1232.34072
The authors study a stochastic density predator-prey system with Beddington-DeAngelis functional response. First, they prove uniqueness of the solution for the considered system (Theorem 2.1). Then, they investigate the asymptotic behavior of this system (Theorem 3.1). These results generalize some similar results given by R. Rudniki (2003), Rudniki and Pichor (2007) and T. Saha and M. Bandyopadhyay (2008). Some good simulations and numerical examples complete the paper.
MSC:
34C60Qualitative investigation and simulation of models (ODE)
34F05ODE with randomness
92D25Population dynamics (general)
34D05Asymptotic stability of ODE
References:
[1]Arditi, R.; Ginzburg, L. R.: Coupling in predator-prey dynamics: ratio-dependence, J. theoret. Biol. 139, 311-326 (1989)
[2]Arnold, L.; Horsthemke, W.; Stucki, J. W.: The influence of external real and white noise on the Lotka-Volterra model, Biom. J. 21, 451-471 (1979) · Zbl 0433.92019 · doi:10.1002/bimj.4710210507
[3]Bazykin, A. D.: Nonlinear dynamics of interacting populations, (1998)
[4]Beddington, J. R.: Mutual interference between parasites or predators and its effect on searching efficiency, J. anim. Ecol. 44, 331-340 (1975)
[5]Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator-prey system, Nonlinear anal. 32, 381-408 (1998) · Zbl 0946.34061 · doi:10.1016/S0362-546X(97)00491-4
[6]Berryman, A. A.: The origin and evolution of predator-prey theory, Ecology 73, 1530-1535 (1992)
[7]Cai, G. Q.; Lin, Y. K.: Stochastic analysis of predator-prey type ecosystems, Ecol. complex. 4, 242-249 (2007)
[8]Deangelis, D. L.; Goldstein, R. A.; O’neill, R. V.: A model for trophic interaction, Ecology 56, 881-892 (1975)
[9]Gard, T. C.: Introduction to stochastic differential equations, (1988)
[10]Has’minskii, R. Z.: Stochastic stability of differential equations, (1980)
[11]Hassell, M. P.; Varley, C. C.: New inductive population model for insect parasites and its bearing on biological control, Nature 223, 1133-1137 (1969)
[12]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
[13]Holling, C. S.: The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. entomologist 91, 293-320 (1959)
[14]Ji, C. Y.; Jiang, D. Q.; Shi, N. Z.: Analysis of a predator-prey model with modified Leslie-gower and Holling-type II schemes with stochastic perturbation, J. math. Anal. appl. 359, 482-498 (2009) · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039
[15]Ji, C. Y.; Jiang, D. Q.; Li, X. Y.: Qualitative analysis of a stochastic ratio-dependent predator-prey system, J. comput. Appl. math. 235, 1326-1341 (2011) · Zbl 1229.92076 · doi:10.1016/j.cam.2010.08.021
[16]Khasminskii, R. Z.; Klebaner, F. C.: Long term behavior of solutions of the Lotka-Volterra system under small random perturbations, Ann. appl. Probab. 11, 952-963 (2001) · Zbl 1061.34513 · doi:10.1214/aoap/1015345354
[17]Klebaner, F. C.: Introduction to stochastic calculus with applications, (1998) · Zbl 0926.60002
[18]Li, H. Y.; Takeuchi, Y.: Dynamics of the density dependent predator-prey system with beddington-deangelis functional response, J. math. Anal. appl. 374, 644-654 (2011) · Zbl 1213.34097 · doi:10.1016/j.jmaa.2010.08.029
[19]Lotka, A. J.: Elements of physical biology, (1925) · Zbl 51.0416.06
[20]Mao, X. R.: Stochastic differential equations and applications, (1997)
[21]May, R. M.: Stability and complexity in model ecosystems, (1973)
[22]Rosenzweig, M. L.; Macarthue, R. H.: Graphical representation and stability conditions of predator-prey interactions, Am. nat. 97, 205-223 (1963)
[23]Rudnicki, R.: Long-time behaviour of a stochastic prey-predator model, Stochastic process. Appl. 108, 93-107 (2003) · Zbl 1075.60539 · doi:10.1016/S0304-4149(03)00090-5
[24]Rudnicki, R.; Pichór, 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]Saha, T.; Bandyopadhyay, M.: Dynamical analysis of a delayed ratio-dependent prey-predator model within fluctuating environment, Appl. math. Comput. 196, 458-478 (2008) · Zbl 1153.34051 · doi:10.1016/j.amc.2007.06.017
[26]Strang, G.: Linear algebra and its applications, (1988)
[27]Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie d’animali conviventi, Mem. acad. Lincei 2, 31-113 (1926) · Zbl 52.0450.06
[28]Zhu, C.; Yin, G.: Asymptotic properties of hybrid diffusion systems, SIAM J. Control optim. 46, 1155-1179 (2007) · Zbl 1140.93045 · doi:10.1137/060649343