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)
Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response. (English) Zbl 1063.39013
For the system $$\align N_1(k+1) &= N_1(k)\exp \{b_1(k)- a_1(k)N_1(k-\tau_1)- \alpha_1(k) Z(k)N_2(k)/N_1(k)\},\\ N_2(k+1) &= N_2(k)\exp\{-b_2(k)+ \alpha_2(k)Z(k-\tau_2)\},\endalign$$ with the abbreviation $Z(k)=N_1^2 (k)/(N_1^2(k)+m^2N^2_2(k))$ sufficient conditions are given such that all positive solutions are asymptotically uniformly bounded and uniformly bounded away from zero.

MSC:
39A12Discrete version of topics in analysis
92D25Population dynamics (general)
39A20Generalized difference equations
39A11Stability of difference equations (MSC2000)
WorldCat.org
Full Text: DOI
References:
[1] Kocic, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. (1993) · Zbl 0787.39001
[2] Agarwal, R. P.: Difference equations and inequalities: theory, methods and applications. Monogr. textbooks pure appl. Math. 228 (2000)
[3] Arditi, R.; Perrin, N.; Saiah, H.: Functional response and heterogeneities: an experiment test with cladocerans. Oikos 60, 69-75 (1991)
[4] Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator -- prey systems. Nonlinear anal. TMA 32, 381-408 (1998) · Zbl 0946.34061
[5] Berryman, A. A.: The origins and evolution of predator -- prey theory. Ecology 73, 1530-1535 (1992)
[6] Fan, M.; Wang, K.: Periodicity in a delayed ratio-dependent predator -- prey system. J. math. Anal. appl. 262, 179-190 (2001) · Zbl 0994.34058
[7] Fan, M.; Wang, K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator -- prey system. Math. comput. Modelling 35, 951-961 (2002) · Zbl 1050.39022
[8] Fan, Y. H.; Li, W. T.; Wang, L. L.: Periodic solutions of delayed ratio-dependent predator -- prey models with monotonic or nonmonotonic functional response. Nonlinear anal. RWA 5, 247-263 (2004) · Zbl 1069.34098
[9] Freedman, H. I.: Deterministic mathematical models in population ecology. (1980) · Zbl 0448.92023
[10] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics. (1992) · Zbl 0752.34039
[11] Hanski, I.: The functional response of predator: worries bout scale. Tree 6, 141-142 (1991)
[12] Holling, C. S.: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. entomol. Sec. can. 45, 1-60 (1965)
[13] Hsu, S. B.; Hwang, T. W.; Kuang, Y.: Global analysis of the michaelis -- menten type ratio-dependent predator -- prey system. J. math. Biol. 42, 489-506 (2001) · Zbl 0984.92035
[14] Hsu, S. B.; Hwang, T. W.; Kuang, Y.: Rich dynamics of a ratio-dependent one-prey two-predators model. J. math. Biol. 43, 377-396 (2001) · Zbl 1007.34054
[15] Jost, C.; Arino, O.; Arditi, R.: About deterministic extinction in ratio-dependent predator -- prey models. Bull. math. Biol. 61, 19-32 (1999) · Zbl 1323.92173
[16] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[17] Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator -- prey system. J. math. Biol. 36, 389-406 (1998) · Zbl 0895.92032
[18] W.T. Li, Y.H. Fan, S.G. Ruan, Periodic solutions in a delayed predator -- prey model with nonmonotonic functional response, submitted for publication
[19] May, R. M.: Complexity and stability in model ecosystems. (1973)
[20] May, R. M.: Biological populations obeying difference equations: stable points, stable cycles and chaos. J. theory biol. 51, 511-524 (1975)
[21] Murry, J. D.: Mathematical biology. (1989)
[22] Rosenzweig, M. L.: Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science 171, 385-387 (1971)
[23] Ruan, S.; Xiao, D.: Global analysis in a predator -- prey system with nonmonotonic functional response. SIAM J. Appl. math. 61, 1445-1472 (2001) · Zbl 0986.34045
[24] Takeuchi, Y.: Global dynamical properties of Lotka -- Volterra systems. (1996) · Zbl 0844.34006
[25] Wang, L. L.; Li, W. T.: Existence of periodic solutions of a delayed predator -- prey system with functional response. International J. Math. math. Sci. 1, 55-63 (2002) · Zbl 1075.34067
[26] Wang, L. L.; Li, W. T.: Existence and global stability of positive periodic solutions of a predator -- prey system with delays. Appl. math. Comput. 146, 167-185 (2003) · Zbl 1029.92025
[27] Wang, L. L.; Li, W. T.: Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator -- prey model with Holling type functional response. J. comput. Appl. math. 162, 341-357 (2004) · Zbl 1076.34085
[28] L.L. Wang, W.T. Li, Periodic solutions and stability for a delayed discrete ratio-dependent predator -- prey system with Holling type functional response, Discrete Dyn. Nat. Soc., in press · Zbl 1073.39008
[29] Xiao, D.; Ruan, S.: Global dynamics of a ratio-dependent predator -- prey system. J. math. Biol. 43, 268-290 (2001) · Zbl 1007.34031
[30] Xiao, D.; Zhang, Z.: On the uniqueness and nonexistence of limit cycles for predator -- prey systems. Nonlinearity 16, 1185-1201 (2003) · Zbl 1042.34060
[31] Zhu, H. P.; Campebell, S.; Wolkowicz, G.: Bifurcation analysis of a predator -- prey system with nonmonotonic functional response. SIAM J. Appl. math. 63, 636-682 (2002) · Zbl 1036.34049
[32] Wang, L.; Wang, M. Q.: Ordinary difference equations. (1989) · Zbl 0732.65076