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)
Complex dynamics in a prey predator system with multiple delays. (English) Zbl 1243.92051
Summary: The complex dynamics is explored of a prey predator system with multiple delays. Holling type-II functional response is assumed for the prey dynamics. The predator dynamics is governed by a modified P.H. Leslie and J.C. Gower scheme [Biometrika 47, 219–234 (1960; Zbl 0103.12502)]. The existence of periodic solutions via Hopf-bifurcations with respect to both delays are established. An algorithm is developed for drawing two-parametric bifurcation diagrams with respect to two delays. The domain of stability with respect to τ 1 and τ 2 is thus obtained. The complex dynamical behavior of the system outside the domain of stability is evident from the exhaustive numerical simulations. Direction and stability of periodic solutions are also determined using normal form theory and center manifold arguments.
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
65C20Models (numerical methods)
37N25Dynamical systems in biology
[1]Freedman, H. I.: Deterministic mathematical models in population ecology, (1980)
[2]Kot, M.: Elements of mathematical ecology, (2001)
[3]May, R. M.: Stability and complexity in model ecosystems, (2001)
[4]Murray, J. D.: Mathematical biology I. An introduction, (2002)
[5]Cosner, C.; Angelis, D. L.; Ault, J. S.; Olson, D. B.: Effects of spatial grouping on the functional response of predators, Theor popul biol 56, 6575 (1999) · Zbl 0928.92031 · doi:10.1006/tpbi.1999.1414
[6]Hsu, S. B.; Hwang, T. W.; Kuang, Y.: Rich dynamics of ratio-dependent one prey two predators model, J math biol 43, 377-396 (2000) · Zbl 1007.34054 · doi:10.1007/s002850100100
[7]Kuang, Y.: Rich dynamics of gause-type ratio-dependent predator-prey system, Fields inst commun 21, 325-337 (1999) · Zbl 0920.92032
[8]Haque, M.: Ratio-dependent predator – prey models of interacting populations, Bull math biol 71, 430-452 (2009) · Zbl 1170.92027 · doi:10.1007/s11538-008-9368-4
[9]Leslie, P. H.; Gower, J. C.: The properties of a stochastic model for the predator – prey type of interaction between two species, Biometrica 47, 219-234 (1960) · Zbl 0103.12502
[10]Pielou, E. C.: An introduction to mathematical ecology, (1969) · Zbl 0259.92001
[11]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, 1069-1075 (2003) · Zbl 1063.34044 · doi:10.1016/S0893-9659(03)90096-6
[12]Hale, J. K.: Theory of functional differential equations, (1977)
[13]Hale, J. K.; Lunel, S. M. Verduyn: Theory of functional differential equations, (1993)
[14]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[15]Wu, J.: Theory and applications of partial functional differential equations, (1996)
[16]Macdonald, N.: Time delay in prey – predator models, Math bio 28, 321-330 (1976) · Zbl 0324.92016 · doi:10.1016/0025-5564(76)90130-9
[17]Macdonald, N.: Biological delay systems: linear stability theory, (1989)
[18]Gopalsamy, K.: Harmless delays in model systems, Bull math biol 45, 295-309 (1983) · Zbl 0514.34060
[19]Cook, K.; Grossman, Z.: Discrete delay, distributed delay and stability switches, J math anal appl 86, 592-627 (1982) · Zbl 0492.34064 · doi:10.1016/0022-247X(82)90243-8
[20]Cooke, K.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models, J math biol 39, 32-352 (1999) · Zbl 0945.92016 · doi:10.1007/s002850050194
[21]Cushing, J. M.: Integrodifferential equations and delay model in population dynamics, (1977) · Zbl 0363.92014
[22]Beretta, E.; Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependant parameters, SIAM J math anal 33, 1144-1165 (2002) · Zbl 1013.92034 · doi:10.1137/S0036141000376086
[23]Hastings, A.: Delays in recruitment at different trophic levels: effect on stability, J math biol 21, 35-44 (1984) · Zbl 0547.92014 · doi:10.1007/BF00275221
[24]Ruan, S.: The effect of delays on stability and persistence in plankton models, Nonlinear anal: theory methods appl 24, 575-585 (1995) · Zbl 0830.34067 · doi:10.1016/0362-546X(95)93092-I
[25]Ruan, S.; Wei, J.: On the zeros of transcendental function with applications to stability of delay differential equations with two zeros, Dyn cont disc impulsive syst ser A: math anal 10, 863-874 (2003) · Zbl 1068.34072
[26]Aiello, W. G.; Freedman, H. I.: A time-delay model of single species growth with stage structure, Math biosci 101, 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[27]Aiello, W. G.; Freedman, H. I.; Wu, J.: Analysis of a model representing stage-structured population growth with state dependent time delay, SIAM J appl math 52, 855-869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048
[28]Bellman, R.; Cooke, K. L.: Differential-difference equations, (1963)
[29]Nindjin, A. F.; Aziz-Alaoui, M. A.; Cadivel, M.: Analysis of a predator – prey model with modified Leslie – gower and Holling-type II schemes with time delay, Nonlinear anal. 7, 1104-1118 (2006) · Zbl 1104.92065 · doi:10.1016/j.nonrwa.2005.10.003
[30]Gazi, N. H.; Bandyopadhyay, M.: Effect of time delay on a detritus-based ecosystem, Int J math math sci, 1-28 (2006) · Zbl 1127.92042 · doi:10.1155/IJMMS/2006/25619
[31]Hu, G.; Li, W.; Yan, X.: Hopf bifurcations in a predator – prey system with multiple delays, Chaos solitons fract 42, 12731285 (2009) · Zbl 1198.34143 · doi:10.1016/j.chaos.2009.03.075
[32]He, X.: Stability and delays in a predator – prey system, J math anal appl 198, 355-370 (1996) · Zbl 0873.34062 · doi:10.1006/jmaa.1996.0087
[33]Yuan, S.; Song, Y.: Bifurcation and stability analysis for a delayed Leslie gower predator – prey system, IMA J appl math 74, 574-603 (2009) · Zbl 1201.34132 · doi:10.1093/imamat/hxp013
[34]Gakkhar, S.; Sahani, S. K.; Negi, K.: Effects of seasonal growth on delayed prey – predator model, Chaos solitons fract 39, 230-239 (2009) · Zbl 1197.34134 · doi:10.1016/j.chaos.2007.01.141
[35]Gakkhar, S.; Negi, K.; Sahani, S. K.: Effects of seasonal growth on ratio dependent delayed prey – predator system, Commun nonlinear sci numer simulat 14, 850-862 (2009) · Zbl 1221.34187 · doi:10.1016/j.cnsns.2007.10.013
[36]Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)
[37]Song, Y.; Wei, J.: Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos, Chaos solitons fract 22, 75-91 (2004) · Zbl 1112.37303 · doi:10.1016/j.chaos.2003.12.075
[38]Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039
[39]Li, X.; Ruan, S.; Wei, J.: Stability and bifurcation in delay-differential equations with two delays, J math anal appl 236, 254-280 (1999) · Zbl 0946.34066 · doi:10.1006/jmaa.1999.6418
[40]Faria, T.: Stability and bifurcation for a delayed predator – prey model and the effect of diffusion, J math anal appl 254, 433-463 (2001) · Zbl 0973.35034 · doi:10.1006/jmaa.2000.7182
[41]Song, Y.; Han, M.; Peng, Y.: Stability and Hopf bifurcations in a competitive lotkavolterra system with two delays, Chaos solitons fract 22, 1139-1148 (2004) · Zbl 1067.34075 · doi:10.1016/j.chaos.2004.03.026
[42]Wu, S.; Ren, G.: A note on delay-independent stability of a predator – prey model, J sound vibr 275, 17-25 (2004)
[43]Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays, Physica D 130, 255-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3