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)
Dynamics of prey threshold harvesting and refuge. (English) Zbl 1235.92048
Summary: The dynamics of a predator-prey model with continuous threshold prey harvesting and prey refuge is studied. One central question is how harvesting and refuge could directly affect the dynamics of the ecosystem, such as the stability properties of some coexistence equilibria and periodic solutions. Theoretical and numerical methods are used to investigate the boundedness of solutions, existence of bionomic equilibria, as well as the existence and stability properties of equilibrium points and periodic solutions. Several bifurcations are also studied.

37N25Dynamical systems in biology
65C20Models (numerical methods)
Full Text: DOI
[1] Clark, C. W.: Mathematical bioeconomics: the optimal management of renewable resources, (1990) · Zbl 0712.90018
[2] Ji, L.; Wu, C.: Qualitative analysis of a predator--prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonlinear anal. RWA 11, 2285-2295 (2010) · Zbl 1203.34070 · doi:10.1016/j.nonrwa.2009.07.003
[3] Kar, T. K.: Modelling and analysis of a harvested prey--predator system incorporating a prey refuge, J. comput. Appl. math. 185, 19-33 (2006) · Zbl 1071.92041 · doi:10.1016/j.cam.2005.01.035
[4] Xiao, D.; Li, W.; Han, M.: Dynamics in a ratio-dependent predator--prey model with predator harvesting, J. math. Anal. appl. 324, 14-29 (2006) · Zbl 1122.34035 · doi:10.1016/j.jmaa.2005.11.048
[5] Asfaw, T.: Dynamics of generalized time dependent predator prey model with nonlinear harvesting, Int. J. Math. anal. 3, 1473-1485 (2009) · Zbl 1204.34061 · http://www.m-hikari.com/ijma/ijma-password-2009/ijma-password29-32-2009/index.html
[6] Leard, B.; Lewis, C.; Rebaza, J.: Dynamics of ratio-dependent predator--prey models with nonconstant harvesting, Discrete contin. Dyn. syst. Ser. S 1, 303-315 (2008) · Zbl 1151.37062 · doi:10.3934/dcdss.2008.1.303
[7] Meza, M. E. M.; Bhaya, A.; Kaszkurewiczk; Costa, M. I. S.: Threshold policies control for predator--prey systems using a control Liapunov function approach, Theor. popul. Biol. 67, 273-284 (2005) · Zbl 1072.92054 · doi:10.1016/j.tpb.2005.01.005
[8] Chen, L.; Chen, F.; Chen, L.: Qualitative analysis of a predator--prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear anal. RWA 11, 246-252 (2010) · Zbl 1186.34062 · doi:10.1016/j.nonrwa.2008.10.056
[9] Cressman, R.; Garay, J.: A predator--prey refuge system: evolutionary stability in ecological systems, Theor. popul. Biol. 76, 248-257 (2009)
[10] Liu, X.; Han, M.: Chaos and Hopf bifurcation analysis for a two species predator--prey system with prey refuge and diffusion, Nonlinear anal. Real world appl. 12, 1047-1061 (2011) · Zbl 1222.34099 · doi:10.1016/j.nonrwa.2010.08.027
[11] Hale, J.: Ordinary differential equations, (1980) · Zbl 0433.34003
[12] Kuang, N. G.; Freedman, H. I.: Uniqueness of limit cycles in gause-type models of predator--prey systems, Math. biosci. 88, 76-84 (1988) · Zbl 0642.92016 · doi:10.1016/0025-5564(88)90049-1
[13] Hale, J.: Dynamics and bifurcations, (1991) · Zbl 0745.58002
[14] Perko, L.: Differential equations and dynamical systems, (2001) · Zbl 0973.34001
[15] Hsieh, Y. H.; Hsiao, C. K.: Predator--prey model with disease infection in both populations, Math. med. Biol. 25, 247-266 (2008) · Zbl 1154.92040 · doi:10.1093/imammb/dqn017
[16] Kar, T. K.: Bioeconomic modelling of a prey predator system using differential algebraic equations, Int. J. Eng. sci. Technol. 2, 13-34 (2010)
[17] Li, Y.; Hu, M.: Stability and Hopf bifurcation analysis in a stage-structured predator--prey system with two time delays, Int. J. Appl. math. Comput. sci. 5, 165-173 (2011)