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)
Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge. (English) Zbl 1186.34062

From the introduction: The aim of this paper is to consider instability and global stability properties of the equilibria and the existence and uniqueness of limit cycles of the following model with Holling type II functional response incorporating a constant prey refuge m

x ˙=αx1-x k-β(x-m)y 1+a(x-m),y ˙=-dy+cβ(x-m)y 1+a(x-m),(1)

where x,y denote prey and predator population respectively at any time t,d,k,α,β,a,c,m are positive constants.

MSC:
34C60Qualitative investigation and simulation of models (ODE)
34C05Location of integral curves, singular points, limit cycles (ODE)
34D20Stability of ODE
92D25Population dynamics (general)
References:
[1]Holling, C. S.: The functional response of predators to prey density and its role in mimicry and population regulations, Memoirs of the entomological society of Canada 45, 3-60 (1965)
[2]González-Olivares, E.; Ramos-Jiliberto, R.: Dynamic consequences of prey refuges in a simple model system: more prey,fewer predators and enhanced stability, Ecological modelling 166, 135-146 (2003)
[3]Hassel, M. P.: The dynamics of arthropod predator–prey systems, (1978)
[4]Kar, T. K.: Stability analysis of a prey–predator model incorporating a prey refuge, Communications in nonlinear science and numerical simulation 10, 681-691 (2005)
[5]Huang, Y. J.; Chen, F. D.; Li, Z.: Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge, Applied mathematics and computation 182, 672-683 (2006) · Zbl 1102.92056 · doi:10.1016/j.amc.2006.04.030
[6]Mcnair, J. N.: The effects of refuges on predator–prey interactions: A reconsideration, Theoretical population biology 29, 38-63 (1986) · Zbl 0594.92017 · doi:10.1016/0040-5809(86)90004-3
[7]Mcnair, J. N.: Stability effects of prey refuges with entry–exit dynamics, Journal of theoretical biology 125, 449-464 (1987)
[8]Ko, W.; Ryu, K.: Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a prey refuge, Journal of differential equations 231, 534-550 (2006)
[9]Kar, T. K.: Modelling and analysis of a harvested prey–predator system incorporating a prey refuge, Journal of computational and applied mathematics 185, 19-33 (2006) · Zbl 1071.92041 · doi:10.1016/j.cam.2005.01.035
[10]Collings, J. B.: Bifurcation and stability analysis of a temperature-dependent mite predator–prey interaction model incorporating a prey refuge, Bulletin of mathematical biology 57, 63-76 (1995) · Zbl 0810.92024
[11]Sih, A.: Prey refuges and predator–prey stability, Theoretical population biology 31, 1-12 (1987)
[12]Krivan, V.: Effects of optimal antipredator behavior of prey on predator–prey dynamics: the role of refuges, Theoretical population biology 53, 131-142 (1998) · Zbl 0945.92021 · doi:10.1006/tpbi.1998.1351
[13]Zhang, Z. F.; Ding, T. R.; Huang, W. Z.; Dong, Z. X.: Qualitative theory of differential equations, (1985)