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)
Periodic solutions of delayed ratio-dependent predator--prey models with monotonic or nonmonotonic functional responses. (English) Zbl 1069.34098
The authors consider the following delayed ratio-dependent predator-prey system $$\aligned x(t) &= x(t)\Biggl[a(t)- b(t) \int^t_{-\infty} K(t- s) x(s)\,ds\Biggr]- e(t) g\Biggl({x(t)\over y(t)}\Biggr),\\ y(t) &= y(t)\Biggl[e(t) g\Biggl({x(t)- \tau(t)\over y(t)- \tau(t)}\Biggr)- d(t)\Biggr],\endaligned\tag{$*$}$$ where $x(t)$ and $y(t)$ represent the predator and prey densities, respectively, $a(t)$, $b(t)$, $c(t)$, $d(t)$, $e(t)$ and $\tau(t)$ are positive periodic continuous functions with period $\omega> 0$, $\omega$ is a positive real constant. $K(s): \bbfR^+\to \bbfR^+$ is a measurable, $\omega$-periodic, normalized function such that $\int^{+\infty}_0 K(s)\,ds= 1$. By using the continuation theorem of the coincidence degree theory [see {\it R. E. Gaines} and {\it J. L. Mawhin}, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)], the authors establish two main theorems on the existence of at least one positive $\omega$-periodic solution of system $(*)$ when the functional response function $g$ is monotonic or nonmonotonic. As corollaries, some applications are listed.

MSC:
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
WorldCat.org
Full Text: DOI
References:
[1] Andrews, J. F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. bioeng. 10, 707-723 (1986)
[2] Arditi, R.; Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544-1551 (1992)
[3] Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator--prey systems. Nonlinear anal. TMA 32, 381-408 (1998) · Zbl 0946.34061
[4] Berryman, A. A.: The origins and evolution of predator--prey theory. Ecology 75, 1530-1535 (1992)
[5] Bush, A. W.; Cook, A. E.: The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. theoret. Biol. 63, 385-395 (1976)
[6] Caperson, J.: Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology 50, 188-192 (1969)
[7] Fan, M.; Wang, K.: Periodicity in a delayed ratio-dependent predator--prey system. J. math. Anal. appl. 262, 179-190 (2001) · Zbl 0994.34058
[8] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations. (1977) · Zbl 0339.47031
[9] Gutierrez, A. P.: The physiological basis of ratio-dependent predator--prey theorya metabolic pool model of Nicholson’s blowflies as an example. Ecology 73, 1552-1563 (1992)
[10] Hsu, S. B.; Hwang, T. W.; Kuang, Y.: Global analysis of the michaelis--menten type ratio-dependent predator--prey system. J. math. Biol. 42, 489-506 (2001) · Zbl 0984.92035
[11] H.F. Huo, W.T. Li, Periodic solutions of a ratio-dependent food chain model with delays, Taiwanese J. Math., in press. · Zbl 1064.34045
[12] H.F. Huo, W.T. Li, Periodic solution of a periodic two-species competition model with delays, Internat. J. Appl. Math., in press. · Zbl 1043.34074
[13] Jost, C.; Arino, O.; Arditi, R.: About deterministic extinction in ratio-dependent predator--prey models. Bull. math. Biol. 61, 19-32 (1999) · Zbl 1323.92173
[14] Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator--prey system. J. math. Biol. 36, 389-406 (1998) · Zbl 0895.92032
[15] Li, Y.: Periodic solutions of a periodic delay predator--prey system. Proc. amer. Math. soc. 127, 1331-1335 (1999) · Zbl 0917.34057
[16] Li, Y.; Kuang, Y.: Periodic solutions of periodic delay Lotka--Volterra equations and systems. J. math. Anal. appl. 255, 260-280 (2001) · Zbl 1024.34062
[17] Ruan, S.; Xiao, D.: Global analysis in a predator--prey system with nonmonotonic functional response. SIAM J. Appl. math. 61, 1445-1472 (2001) · Zbl 0986.34045
[18] Sokol, W.; Howell, J. A.: Kinetics of phenol oxidation by washed cells. Biotechnol. bioeng. 23, 2039-2049 (1980)
[19] L.L. Wang, W.T. Li, Existence and global stability of positive periodic solutions of a predator--prey system with delays, Appl. Math. Comput., in press. · Zbl 1029.92025
[20] Xiao, D.; Ruan, S.: Multiple bifurcations in a delayed predator--prey system with nonmonotonic functional response. J. differential equations 176, 494-510 (2001) · Zbl 1003.34064
[21] Zhao, T.; Kuang, Y.; Smith, H. L.: Global existence of periodic solutions in a class of delayed gause-type predator--prey systems. Nonlinear anal. TMA 28, 1373-1394 (1997) · Zbl 0872.34047