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 a competitive Lotka-Volterra system with three delays. (English) Zbl 1238.34148
The authors study the following three-species Lotka-Volterra type competition system with three discrete time delays $$ \aligned \dot x_1(t)=&x_1(t)[r_1-a_{11}x_1(t)-a_{13}x_3(t-\tau_3)],\\ \dot x_2(t)=&x_2(t)[r_2-a_{21}x_1(t-\tau_1)-a_{22}x_2(t)],\\ \dot x_3(t)=&x_3(t)[r_3-a_{32}x_2(t-\tau_2)] \endaligned\tag1$$ with initial conditions $$x_i(t)=\phi_i(t)\geq 0, \ t\in [-\tau, 0), \ \phi_i(0)>0, \ i=1,2,3,$$ here $\tau=\tau_1+\tau_2+\tau_3$. In system (1), $x_i(t)$ represents the density of the $i$th species at time $t$, respectively, $i=1,2,3$; $\tau_i$ is the feedback time delay of the species $x_i (i=1,2,3)$ to the growth of the species itself; $r_i$ is the intrinsic growth rate of the $i$th species and $r_i/a_{ii}$ is the carrying capacity of the $i$th species, $a_{13}, a_{21}$ and $a_{32}$ are competition coefficients. By choosing $\tau$ as a bifurcation parameter, it is shown that system (1) undergoes a Hopf bifurcation at the positive equilibrium as $\tau$ crosses some critical values. The formulae determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form theory and center manifold theorem.

34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34K20Stability theory of functional-differential equations
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K19Invariant manifolds (functional-differential equations)
Full Text: DOI
[1] 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
[2] Cushing, J. M.: Periodic time-dependent predator -- prey systems, SIAMJ. appl. Math. 32, 82-95 (1997) · Zbl 0348.34031 · doi:10.1137/0132006
[3] Faria, T.: Stability and bifurcation for a delay 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
[4] Hale, J.; Lunel, S. V.: Introduction to functional differential equations, (1993) · Zbl 0787.34002
[5] Hassard, B.; Kazarino, D.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981) · Zbl 0474.34002
[6] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[7] Song, Y.; Han, M.; Peng, Y.: Stability and Hopf bifurcation in a competitive Lotka -- Volterra system with two delays, Chaos solitons fract. 22, 1139-1148 (2004) · Zbl 1067.34075 · doi:10.1016/j.chaos.2004.03.026
[8] Song, Y.; Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator -- prey system, J. math. Anal. appl. 301, 1-21 (2005) · Zbl 1067.34076 · doi:10.1016/j.jmaa.2004.06.056
[9] Wu, J.: Theory and applications of partial functional differential equations, (1996) · Zbl 0870.35116
[10] Yan, X.; Li, W.: Bifurcation and global periodic solutions in a delayed facultative mutualism system, Physica D 227, 51-69 (2007) · Zbl 1123.34055 · doi:10.1016/j.physd.2006.12.007
[11] Yan, X.; Li, W.: Hopf bifurcation and global periodic solutions in a delayed predator -- prey system, Appl. math. Comput. 177, 427-445 (2006) · Zbl 1090.92052 · doi:10.1016/j.amc.2005.11.020
[12] Yan, X.; Zhang, C.: Hopf bifurcation in a delayed Lotka -- Volterra predator -- prey system, Nonlinar anal. 9, 114-127 (2008) · Zbl 1149.34048 · doi:10.1016/j.nonrwa.2006.09.007
[13] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator -- prey system, Nonlinar anal. 28, 1373-1394 (1997) · Zbl 0872.34047 · doi:10.1016/0362-546X(95)00230-S
[14] Gopalsamy, K.; Weng, P.: Global attractivity in a competitive system with feedback controls, Comput. math. Appl. 45, 665-676 (2003) · Zbl 1059.93111 · doi:10.1016/S0898-1221(03)00026-9
[15] Saito, Y.: The necessary and sufficient condition for global stability of a Lotka -- Volterra cooperative or competitive system with delays, J. math. Anal. appl. 268, 109-124 (2002) · Zbl 1012.34072 · doi:10.1006/jmaa.2001.7801
[16] Jin, Z.; Ma, Z.: Uniform persistence of n-dimensional Lotka -- Volterra competitive systems with finite delay, Adv. top. Biomath., 91-95 (1998) · Zbl 0984.92037
[17] Lu, Z.; Takeuchi, Y.: Permanence and global attractivity for competitive Lotka -- Volterra system with delay, Nonlinear anal. TMA 22, 847-856 (1994) · Zbl 0809.92025 · doi:10.1016/0362-546X(94)90053-1
[18] Tang, X. H.; Zou, X. F.: Global attractivity of non-autonomous Lotka -- Volterra competitive system without instantaneous negative feedback, J. differ. Equ. 19, 2502-2535 (2003) · Zbl 1035.34085
[19] Tang, X. H.; Zou, X. F.: 32-type criteria for global attractivity of Lotka -- Volterra competitive system without instantaneous negative feedback, J. differ. Equ. 186, 420-439 (2002) · Zbl 1028.34070 · doi:10.1016/S0022-0396(02)00011-6