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)
Periodicity in a delayed ratio-dependent predator-prey system. (English) Zbl 0994.34058
The authors consider the following nonautonomous ratio-dependent predator-prey system $$\align \frac{dx}{dt}&= x\Biggl[a(t)-b(t)\int_{-\infty}^tk(t-u)x(u) du \Biggr]-\frac{c(t)xy}{my+x},\\ \frac{dy}{dt}&= y\Biggl[-d(t)+\frac{f(t)x(t-\tau(t))}{my(t-\tau(t))+x(t-\tau(t))} \Biggr],\endalign$$ where $x(t)$ and $y(t)$ stand for the prey’s and the predator’s density at time $t$, respectively; $m$ is a positive constant and $a\in C(\bbfR,\bbfR)$, $b, c, d,f,\tau\in C(\bbfR, \bbfR^+)$ are $\omega$-periodic functions. By using Gaines and Mawhin’s coincidence degree theory, some sufficient conditions which guarantee the existence of positive periodic solutions are obtained.

34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34C25Periodic solutions of ODE
Full Text: DOI
[1] Arditi, R.; Ginzburg, L. R.: Coupling in predator--prey dynamics: ratio-dependence. J. theor. Biol 139, 311-326 (1989)
[2] Arditi, R.; Ginzburg, L. R.; Akcakaga, H. R.: Variation in plankton densities among lakes: A case for ratio-dependent models. Amer. naturalist 138, 1287-1296 (1991)
[3] Arditi, R.; Perrin, N.; Saiah, H.: Functional response and heterogeneities: an experimental test with cladocerans. Oikos 60, 69-75 (1991)
[4] Arditi, R.; Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544-1551 (1992)
[5] Beretta, E.; Kuang, Y.: Global analyses in some delayed ratio-dependent predator--prey systems. Nonlinear anal. 32, 381-408 (1998) · Zbl 0946.34061
[6] Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. Lecture notes in biomathematics 20 (1977) · Zbl 0363.92014
[7] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031
[8] Ginzburg, L. R.; Akcakaya, H. R.: Consequences of ratio-dependent predation for steady state properties of ecosystems. Ecology 73, 1536-1543 (1992)
[9] Gutierrez, A. P.: The physiological basis of ratio-dependent predator--prey theory: A metabolic pool model of Nicholson’s blowflies as an example. Ecology 73, 1552-1563 (1992)
[10] Hanski, I.: The functional response of predator: worries about scale. Tree 6, 141-142 (1991)
[11] Hsu, S. B.; Huang, T. W.: Global stability for a class of predator--prey systems. SIAM J. Appl. math. 55, 763-783 (1995) · Zbl 0832.34035
[12] Kuang, Y.: Delay differential equations with application in population dynamics. Mathematics in science and engineering 191 (1993) · Zbl 0777.34002
[13] Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator--prey system. J. math. Biol 36, 389-406 (1998) · Zbl 0895.92032
[14] Li, Y. K.: Periodic solutions of a periodic delay predator--prey system. Proc. amer. Math. soc 127, 1331-1335 (1999) · Zbl 0917.34057
[15] Sáez, E.; González-Olivares, E.: Dynamics of a predator--prey model. SIAM J. Appl. math. 59, 1867-1878 (1999) · Zbl 0934.92027