Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems

Abstract

where x(t), y(t) denote the population density of prey and predator at time t, respectively. g(e), p(e) and /z(s) are assumed to satisfy appropriate conditions. v, a positive constant, stands for the death rate of predator y in the absence of prey x. We may think of this system as herbivores (y) grazing upon vegetation (x), which take time r to recover. In view of the fact that many predator-prey systems display sustained fluctuations (for examples, see Freedman [l]), it is thus desirable to construct predator-prey models capable of producing nonconstant periodic solutions. When r = 0, (1.1) reduces to the well-known undelayed Gause-type or Kolmogorov predator-prey systems (again, see Freedman [l] and the references cited theorein). Under some assumptions, the resulting systems can have a unique limit cycle in the positive cone (see Kuang and Freedman [2] and the references cited therein). To the best of our knowledge about delayed population interaction models, the existing literature establishes the existence of a periodic solution by Hopf bifurcation argument, or reduces the delayed systems to undelayed systems of higher dimensions. Therefore, these existence results are generally local. Most of the existing results on the global existence of nonconstant periodic solutions in delayed equations deal with scalar equations with single discrete delay, such as 13-101