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)
Dynamic of a non-autonomous predator-prey system with infinite delay and diffusion. (English) Zbl 1155.34329
Summary: In the present paper, a nonlinear non-autonomous predator-prey dispersion model with continuous delay is studied. Sufficient conditions which guarantee the existence of a periodic positive solution are obtained by using Gaines and Mawhin’s continuation theorem of coincidence degree theory. Moreover, globally asymptotically stability of the system is also obtained by means of a suitable Lyapunov functional. The applications show that these criteria are easily verified.
MSC:
34D05Asymptotic stability of ODE
34D23Global stability of ODE
92D25Population dynamics (general)
References:
[1]Cohen, M.; Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. Syst. man cybern. 13, 815-826 (1983) · Zbl 0553.92009
[2]Xu, Rui; Chaplain, M. A. J.; Davidson, F. A.: Periodic solutions for a delayed predator–prey model of prey dispersal in two-patch environments, Nonlinear anal. RWA 5, 183-206 (2004) · Zbl 1066.92059 · doi:10.1016/S1468-1218(03)00032-4
[3]Rui Xu, Zhien Ma, The effect of dispersal on the permanence of a predator–prey system with time delay, Nonlinear Anal. RWA (in press) · Zbl 1142.34055 · doi:10.1016/j.nonrwa.2006.11.004
[4]Xiaoquan Ding, Fangfang Wang, Positive periodic solution for a semi-ratio-dependent predator–prey system with diffusion and time delays, Nonlinear Anal. RWA (in press) · Zbl 1166.34044 · doi:10.1142/S179352450800028X
[5]Fengde Chen, Xiangdong Xie, Jinlin Shi, Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. Comput. Appl. Math. (in press)
[6]Mengfan; Wang, Ke: Global periodic solutions of a generalized n-species gilpin-ayala competition model, Comput. math. Appl. 40, 1141-1151 (2000) · Zbl 0954.92027 · doi:10.1016/S0898-1221(00)00228-5
[7]Fan, Meng; Wang, Qian; Zou, Xinfu: Dynamics of a non-autonomous ratio-dependent predator–prey system, Proc. roy. Soc. Edinburgh A 133, 97-118 (2003) · Zbl 1032.34044 · doi:10.1017/S0308210500002304
[8]Skellem, J. D.: Random dispersal in theoretical population, Biometika 38, 196-216 (1951)
[9]Chen, T. P.; Rong, L. B.: Delay independent stability analysis of Cohen–Grossberg neural networks, Phys. lett. A 317, 436-449 (2003) · Zbl 1030.92002 · doi:10.1016/j.physleta.2003.08.066
[10]Lu, W.; Chen, T.: New conditions for global stability of Cohen–Grossberg neural networks, Neural comput. 15, 1173-1189 (2003) · Zbl 1086.68573 · doi:10.1162/089976603765202703
[11]Song, Q.; Cao, J.: Stability analysis of Cohen–Grossberg neural network with both time varying and continuously distributed delays, J. comput. Appl. math. 197, No. 1.1, 188-203 (2006) · Zbl 1108.34060 · doi:10.1016/j.cam.2005.10.029
[12]Yuan, K.; Cao, J.: An analysis of global asymptotic stability of delayed Cohen–Grossberg neural networks via non-smooth analysis, IEEE trans. Circuits syst. I 52, 1854-1861 (2005)
[13]Wang, L.: Stability of Cohen–Grossberg neural networks with distributed delays, Appl. math. Comput. 160, 93-110 (2005) · Zbl 1069.34113 · doi:10.1016/j.amc.2003.09.014
[14]Cao, J.; Liang, J.: Boundedness and stability for Cohen–Grossberg neural network with time- varying delays, J. math. Anal. appl. 296, 665-685 (2004) · Zbl 1044.92001 · doi:10.1016/j.jmaa.2004.04.039
[15]Cao, J.; Song, Q.: Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity 19, 1601-1617 (2006) · Zbl 1118.37038 · doi:10.1088/0951-7715/19/7/008
[16]Yang, Z. C.; Xu, D. Y.: Impulsive effects on stability of Cohen–Grossberg neural networks with variable delays, Appl. math. Comput. 177, 63-78 (2006) · Zbl 1103.34067 · doi:10.1016/j.amc.2005.10.032
[17]Chen, Z.; Jiong, R.: Global stability analysis of impulsive Cohen–Grossberg neural networks with delay, Phys. lett. A 345, 101-111 (2005)
[18]Zhu, Hongguang; Wang, Ke; Li, Xiaojian: Existence and global stability of positive periodic solutions for predator–prey system with infinite delay and diffusion, Nonlinear anal. RWA 8, 872-886 (2007) · Zbl 1144.34049 · doi:10.1016/j.nonrwa.2006.03.011
[19]Guo, S. J.; Huang, L. H.: Stability analysis of Cohen–Grossberg neural networks, IEEE trans. Neural netw. 17, 106-117 (2006)
[20]Chen, Y.: Global asymptotic stability of delayed Cohen–Grossberg neural networks, IEEE trans. Circuits syst. I 53, 351-357 (2006)
[21]Xia, Y. H.; Cao, J.: Almost periodic solutions for an ecological model with infinite delays, Proc. edinb. Math. soc. 501, 229-249 (2007) · Zbl 1130.34044 · doi:10.1017/S0013091504001233
[22]Xia, Y. H.; Cao, J.: Global attractivity of a periodic ecological model with m-predators and n-preys by ”pure-delay type” system, Comput. math. Appl. 52, No. 6–7, 829-852 (2006) · Zbl 1135.34038 · doi:10.1016/j.camwa.2006.06.002
[23]Xia, Y. H.; Cheng, S. S.: Quasi-uniformly asymptotic stability and existence of almost periodic solutions of difference equations with applications in population dynamic systems, J. difference equ. Appl. 14, No. 1, 59-81 (2008) · Zbl 1141.39012 · doi:10.1080/10236190701470407
[24]Zhang, Y.; Sun, J. T.: Stability of impulsive neural networks with delays, Phys. lett. A 348, No. 1–2, 44-50 (2005) · Zbl 1195.93122 · doi:10.1016/j.physleta.2005.08.030
[25]Feng, C.; Plamondon, R.: Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, IEEE trans. Neural netw. 14, 1560-1564 (2003)
[26]Lasalle, J. P.: The stability of dynamical system, (1976)
[27]J. Mawhin, Topological degree method in nonlinear boundary value problems, NSFCBMS Regional Conf. Series in math., Amer. Math. Soc., Providence, RI, 1979
[28]Goplsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039