# zbMATH — the first resource for mathematics

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 \begin{aligned} \dot x(t)&= x(t)[g(x(t))- p(x(t))y(t)],\\ \dot y(t)&= y(t)[-\nu+ h(x(t-\tau))], \end{aligned} 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 \begin{aligned} \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]. \end{aligned} {}.
Reviewer: V.Petrov (Plovdiv)

##### MSC:
 34K13 Periodic solutions to functional-differential equations 34C25 Periodic solutions to ordinary differential equations 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Freedman, H.I., () [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 ƒ′ (x) = −2ƒ (y −1) [1 + ƒ (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., () [14] Smith, H.L., One periodic solutions of delay integral equations modeling epidemics and population growth, () [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, (), 85-101 · Zbl 0719.34125 [17] KUANG Y., Periodic solutions in a class of delayed predator-prey systems, preprint. · Zbl 0872.34047 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.