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)
Positive periodic solutions for a class of neutral delay gause-type predator-prey system. (English) Zbl 1181.34089
By using the continuation theorem from coincidence degree theory, the authors establish an existence theorem for positive periodic solutions of the following neutral delay Gause-type predator-prey system $$\cases x^{\prime}(t)=x(t)[r(t)-a(t)x(t-\sigma_1)-\rho x^{\prime}(t-\sigma_2)]-\phi(t,x(t))y(t-\tau_1(t)),\\ y^{\prime}(t)=y(t)[-d(t)+h(t,x(t-\tau_2(t))]. \endcases$$ The technique is standard, and the key point is an a priori estimate of the bound for the solutions of the system.

34K60Qualitative investigation and simulation of models
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34K40Neutral functional-differential equations
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[1] Ardito, A.; Ricciardi, P.: Lyapunov functions for a generalized gause-type model. J. math. Biol. 33, 816-823 (1995) · Zbl 0831.92023
[2] Cui, J.; Takeuchi, Y.; Permanence: Extinction and periodic solution of predator--prey system with beddington-deangelis functional response. J. math. Anal. appl. 317, 464-474 (2006) · Zbl 1102.34033
[3] Ding, X.; Jiang, J.: Positive periodic solutions in delayed gause-type predator--prey systems. J. math. Anal. appl. 339, 1220-1230 (2008) · Zbl 1137.34033
[4] Ding, X.; Jiang, J.: Multiple periodic solutions in delayed gause-type ratio-dependent predator--prey systems with non-monotonic numerical responses. Math. comput. Modelling 47, 1323-1331 (2008) · Zbl 1145.34332
[5] H.I. Freedman, Deterministic mathematical models in population ecology, in: HIFR Conf., 2nd ed., Edmonton, 1980 · Zbl 0448.92023
[6] Hesaaraki, M.; Moghadas, S. M.: Existence of limit cycles for predator--prey systems with a class of functional responses. Ecol. modell 142, 1-9 (2001) · Zbl 0977.92019
[7] Hu, X.; Liu, G.; Yan, J.: Existence of multiple positive periodic solutions of delayed predator--prey models with functional responses. Comput. math. Appl. 52, 1453-1462 (2006) · Zbl 1128.92047
[8] King, A. A.; Schaffer, W. M.: The rainbow Bridge: Hamiltonian limits and resonance in predator--prey dynamics. J. math. Biol. 39, 439-469 (1999) · Zbl 0986.92037
[9] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[10] Kuang, Y.: On neutral delay logistic gause-type predator--prey systems. Dyn. stab. Syst. 6, 173-189 (1991) · Zbl 0728.92016
[11] Liu, X.; Chen, L.: Complex dynamics of Holling-type II Lotka-Volterra predator--prey system with impulsive perturbations in the predator. Chaos solitons fractals 16, 311-320 (2003) · Zbl 1085.34529
[12] Moghadas, S. M.; Alexander, M. E.: Dynamics of a generalized gause-type predator--prey model with a seasonal functional response. Chaos solitons fractals 23, 55-65 (2005) · Zbl 1058.92049
[13] Moghadas, S. M.; Alexander, M. E.; Corbett, B. D.: A non-standard numerical scheme for a generalized gause-type predator--prey model. Physica D 188, 134-151 (2004) · Zbl 1043.92040
[14] Murray, J. D.: Mathematical biology. (1989) · Zbl 0682.92001
[15] Sikder, A.; Roy, A. B.: Persistence of a generalized gause-type two prey-two predator pair linked by competition mathematical biosciences. 122, 1-23 (1994) · Zbl 0816.92019
[16] Sugie, J.; Kohno, R.; Miyazaki, R.: On a predator--prey system of Holling type. Proc. amer. Math. soc. 125, 2041-2050 (1997) · Zbl 0868.34023
[17] Xu, R.; Chaplain, M. A. J.: Persistence and global stability in a delayed gause-type predator--prey system without dominating instantaneous negative feedbacks. J. math. Anal. appl. 265, 148-162 (2002) · Zbl 1013.34074
[18] Xu, R.; Ma, Z.: Stability and Hopf bifurcation in a ratio-dependent predator--prey system with stage structure. Chaos solitons fractals 38, 669-684 (2008) · Zbl 1146.34323
[19] Yang, S. J.; Shi, B.: Periodic solution for a three-stage-structured predator--prey system with time delay. J. math. Anal. appl. 341, 287-294 (2008) · Zbl 1144.34048
[20] Zhang, Z.; Hou, Z.; Wang, L.: Multiplicity of positive periodic solutions to a generalized delayed predator--prey system with stocking. Nonlinear anal. 68, 2608-2622 (2008) · Zbl 1146.34050
[21] Zhao, C. J.: On a periodic predator--prey system with time delays. J. math. Anal. appl. 331, 978-985 (2007) · Zbl 1140.34423
[22] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator--prey systems. Nonlinear anal. 28, 1373-1394 (1997) · Zbl 0872.34047
[23] Freedman, H. I.; Wu, J.: Periodic solutions of single species models with periodic delay. SIAM J. Math. anal. Appl. 23, 689-701 (1992) · Zbl 0764.92016
[24] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031