Global asymptotic stability and equiasymptotic stability for a time-varying phytoplankton-zooplankton-fish system

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In this paper, we consider a three-species system which consists of phytoplankton, zooplankton and fish. It is well known that zooplankton are harvested by other creatures besides fish, and it has been reported that zooplankton can absorb energy from other micro-organisms besides phytoplankton in recent decades. For this reason, we add the parts of extra energy source and harvesting effect in the model. Taking account of seasonal variation which can bring affects to many aspects of plankton population, we assume that some factors about the phytoplankton and the zooplankton are time-dependent. In particular, since the environment does not change regularly, we does not require these time-varied factors are periodic, which is quite different with most ecological models. The purpose of this paper is to present sufficient conditions which guarantee that the equilibrium of this three-species system is globally asymptotically stable. Moreover, we will show that the equilibrium is equiasymptotically stable under relatively weaker conditions.