# 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)
Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems. (English) Zbl 0872.34047
The authors consider the following delayed Gause-type predator-prey system \align \dot x(t)&= x(t)[g(x(t))- p(x(t))y(t)],\\ \dot y(t)&= y(t)[-\nu+ h(x(t-\tau))], \endalign where $x(t),y(t)$ denote the population density of prey and predator at time $t$, respectively, the positive constant $\nu$ stands for the death rate of predator $y$ in the absence of prey $x$, and $\tau$ is the recover-time. The main goal of the paper is to establish global existence of nonconstant periodic solutions. This is done by proving the existence of nontrivial fixed points of an appropriate map. In the last section of the paper the authors apply their results to the following delayed Lotka-Volterra predator-prey system \align \dot x(t)&= x(t)\Biggl[ \gamma-ax(t)- \frac{by(t)} {1+cx(t)}\Biggr],\\ \dot y(t)&= y(t)\Biggl[ -\nu+ \frac{dx(t-\tau)} {1+cx(t-\tau)}\Biggr]. \endalign{}.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34C25 Periodic solutions of ODE 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Freedman, H.I.: Deterministic mathematical models in population ecology. (1980) · Zbl 0448.92023 [2] Kuang, Y.; Freedman, H.I.: Uniqueness of limit cycle in gause-type predator-prey systems. Math. biosci. 88, 67-84 (1988) · Zbl 0642.92016 [3] Chow, S.N.: Existence of periodic solutions of autonomous functional differential equations. J. diff. Eqns 15, 350-378 (1974) · Zbl 0295.34055 [4] Chow, S.N.; Hale, J.K.: Periodic solutions of autonomous equations. J. math. Anal. appl. 66, 495-506 (1978) · Zbl 0397.34091 [5] Grafton, R.B.: A periodicity theorem for autonomous functional differential equations. J. diff. Eqns 6, 87-109 (1969) · Zbl 0175.38503 [6] Jones, G.S.: The existence of periodic solutions of f’ (x) = -2f (y -1) [1 + f (x) ]. J. math. Anal. appl. 5, 435-450 (1962) · Zbl 0106.29504 [7] Kuang, Y.; Smith, H.L.: Slowly oscillating periodic solutions of autonomous state-dependent delay equations. Nonlinear analysis 19, 855-872 (1992) · Zbl 0774.34054 [8] KUANG Y. & SMITH H.L., Periodic solutions of autonomous state-dependent threshold-delay equations, preprint. · Zbl 0774.34054 [9] Nussbaum, R.: Periodic solutions of some nonlinear autonomous functional differential equations. Annali mat. Pura appl. 101, 263-306 (1974) · Zbl 0323.34061 [10] Nussbaum, R.: Periodic solutions of some nonlinear autonomous functional differential equations II. J. diff. Eqns 14, 360-394 (1973) · Zbl 0311.34087 [11] Hadeler, K.P.; Tomiuk, J.: Periodic solutions of difference-differential equations. Arch. ration. Mech. analysis 65, 87-95 (1977) · Zbl 0426.34058 [12] Mallet-Paret, J.; Nussbaum, R.: Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation. Annali mat. Pura appl. 145, 33-128 (1986) · Zbl 0617.34071 [13] Hale, J.K.: Theory of functional differential equations. (1977) · Zbl 0352.34001 [14] Smith, H.L.: One periodic solutions of delay integral equations modeling epidemics and population growth. Ph.d. thesis (1976) [15] Walther, H.O.: Existence of a nonconstant periodic solution of a nonlinear autonomous functional differential equation representing the growth of a single species population. J. math. Biol. 1, 227-240 (1975) · Zbl 0299.34102 [16] Táboas, P.: Periodic solutions of a planar delay equation. Proc. R. Soc. edinb. 116A, 85-101 (1990) · Zbl 0719.34125 [17] KUANG Y., Periodic solutions in a class of delayed predator-prey systems, preprint. · Zbl 1009.34063 [18] Browder, F.E.: A further generalization of the Schauder fixed point theorem. Duke math. J. 32, 575-578 (1965) · Zbl 0137.32601 [19] Wolfgang, A.: Some periodicity criteria for functional differential equations. Manuscripta math. 23, 295-318 (1978) · Zbl 0367.34049 [20] Leung, A.: Periodic solutions for a prey-predator delay equation. J. diff. Eqns 26, 391-403 (1979) · Zbl 0365.34078