Periodic Solution for a Delayed Three-species Food-chain System with Holling Type-ii Functional Response

A delayed three-species periodic food-chain system with Holling type-II functional response is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions is derived for the existence of positive periodic solutions to the system. 1. Introduction. A rather characteristic behavior of population dynamics is the often-observed oscillatory phenomenon of the population densities. There are three typical approaches for modelling such a behavior: (i) introducing more species into the model and considering the higher-dimensional systems (like predator-prey interactions, see May [8]); (ii) assuming that the per capita growth function is time dependent and periodic in time; (iii) taking into account the time-delay effect on the population dynamics (Smith and Kuang [9], Zhao et al. [12]). In most of the models considered so far, it has been assumed that all biological and environmental parameters are constants in time. However, any biological or environmental parameters are naturally subject to fluctuation in time. The effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (such as seasonal effects of weather, food supplies, mating habits, etc.); on the other hand, it is generally recognized that some kinds of time delays are inevitable in population interactions (see [1, 2] and the references cited therein). Time delay due to gestation is a common example because, generally, the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. The effect of time delays on the asymptotic behavior of populations has been studied by a number of authors (see, e.g., [3, 11]). Therefore, more realistic models of population interactions should take into account the seasonality of the changing environment and the effect of time delays.