| This paper consists of three parts:Firstly, we study a food-chain system with Holling III type functional response. By applyingthe coincidence degree theorey, sufficient conditions which guarantee the existence of at leastone positive periodic solution of the model are obtained.Secondly, we consider the stability property of the positive equilibrium of a kind oftwo-prey one-predator model. We first show by an example that the deduction of M. F.Elettreby (Two-prey one-predator model, Chaos, Soliton&Fractals,2009,39(5):2018-2027) isnot strict. Next, with the help of a suitable Lyapunov function, we are able to show that once thepositive equilibrium of the system exists, it is globally asymptotically stable. Our result improveand supplement the corresponding main result of M. F. Elettreby.Finally, we consider a two-prey one-predator system with constant delays. First, we discussthe existence of positive equilibrium. Next, by applying the iterative method, a set of sufficientconditions which guarantee the globally attractivity of the positive equilibrium of the system areobtained. Our result generalize the main result of M. F. Elettreby’s to the delay case. |
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