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)
Dynamic behaviour of a delayed predator-prey model with harvesting. (English) Zbl 1215.92065
Summary: We analyze the dynamics of a delayed predator-prey system in the presence of harvesting. This is a modified version of the {\it P. H. Leslie} and {\it J. C. Gower} [Biometrika 47, 219--234 (1960; Zbl 0103.12502)] and Holling-type II scheme [{\it C. S. Holling}, Mem. Ent. Sec. Can. 45, 1--60 (1965)]. The main result is given in terms of local stability, global stability, influence of harvesting and bifurcation. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by using the normal form method and center manifold theorem.

34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
65C20Models (numerical methods)
34K13Periodic solutions of functional differential equations
Full Text: DOI
[1] Aziz-Alaoui, M. A.; Okiye, M. Daher: Boundedness and global stability for a predator -- prey model with modified Leslie -- gower and Holling-type II schemes, Appl. math. Lett. 16, 1069-1074 (2003) · Zbl 1063.34044 · doi:10.1016/S0893-9659(03)90096-6
[2] Aziz-Alaoui, M. A.: Study of a Leslie -- gower-type tritrophic population, Chaos solitons fractals 14, 1275-1293 (2002) · Zbl 1031.92027 · doi:10.1016/S0960-0779(02)00079-6
[3] Beretta, E.; Kuang, Y.: Convergence results in a well known delayed predator -- prey system, J. math. Anal. appl. 204, 840-853 (1996) · Zbl 0876.92021 · doi:10.1006/jmaa.1996.0471
[4] Clark, C. W.: Mathematical bioeconomics: the optimal management of renewable resource, (1990) · Zbl 0712.90018
[5] Dai, G.; Tang, M.: Coexistence region and global dynamics of harvested predator prey system, SIAM J. Appl. math. 58, 193-210 (1998) · Zbl 0916.34034 · doi:10.1137/S0036139994275799
[6] Dubey, B.; Chemdra, P.; Sinha, P.: A resource dependent fishing model with optimal harvesting policy, J. biol. Syst. 10, 1-13 (2002) · Zbl 1109.92313 · doi:10.1142/S0218339002000494
[7] Gopalsamy, K.: Harmless delay in model systems, Bull. math. Biol. 45, 295-309 (1983) · Zbl 0514.34060
[8] Gopalsamy, K.: Delayed responses and stability in two species systems, J. austral. Math. soc. Ser. B 25, 473-500 (1984) · Zbl 0552.92016 · doi:10.1017/S0334270000004227
[9] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and application of Hopf bifurcation, (1981) · Zbl 0474.34002
[10] Holling, Cs.: The functional response of predator to prey density and its role in mimicry and population regulation, Mem. ent. Sec. can. 45, 1-60 (1965)
[11] 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
[12] Kar, T. K.; Chattopadhyay, S. K.: A focus on long run sustainability of a harvested prey -- predator system in the presence of alternative prey, Comptes renders biol. 333, 841-849 (2010)
[13] Kar, T. K.; Chaudhuri, K. S.: Harvesting in a two prey one predator fishing, A bioecon. Model. ANZIAM J 45, 443-456 (2004) · Zbl 1052.92052 · doi:10.1017/S144618110001347X
[14] Korobeinikov, A.: A Lyapunov function for Leslie -- gower predator -- prey models, Appl. math. Lett. 14, 697-699 (2002) · Zbl 0999.92036 · doi:10.1016/S0893-9659(01)80029-X
[15] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[16] Leslie, P. H.; Gower, J. C.: The properties of a stochastic model for the predator -- prey type of interaction between two species, Biometrica 47, 219-223 (1960) · Zbl 0103.12502
[17] Leslie, P. H.: Some further notes on the use of matrices in population mathematics, Biometrica 35, 213-244 (1948) · Zbl 0034.23303
[18] Letellier, C.; AguirrĂ©, L.; Maquet, J.; Aziz-Alaoui, M. A.: Should all the species of a food chain be counted to investigate the global dynamics, Chaos solitons fractals 13, 1099-1113 (2002) · Zbl 1004.92039 · doi:10.1016/S0960-0779(01)00116-3
[19] Pielou, E. C.: An introduction to mathematical ecology, (1969) · Zbl 0259.92001
[20] Ruan, S.: Absolute stability, conditional stability and bifurcation in Kolmogorov type predator-prey systems with discrete delays, Quart. appl. Math. 59, 59-173 (2001) · Zbl 1035.34084
[21] Song, X.; Chen, L.: Optimal harvesting and stability for a two species competitive system with stage structure, Math. biosci. 170, 173-186 (2001) · Zbl 1028.34049 · doi:10.1016/S0025-5564(00)00068-7
[22] Upadhyay, R. K.; Rai, V.: Why chaos is rarely observed in natural populations, Chaos solitons fractals 8, 1933-1939 (1997)
[23] Zhang, Sw.; Dong, Lz.; Chen, Ls.: The study of predator prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons fractals 23, 631-643 (2005) · Zbl 1081.34041 · doi:10.1016/j.chaos.2004.05.044