Dynamical behaviors of a food-chain model with stage structure and time delays. (English) Zbl 1446.37083
Summary: Incorporating two delays (\(\tau_{1}\) represents the maturity of predator, \(\tau_{2}\) represents the maturity of top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing the characteristic equations, constructing a suitable Lyapunov functional, using Lyapunov-LaSalle’s principle, the comparison theorem and iterative technique, we investigate the existence of nonnegative equilibria and their stability. Some interesting findings show that the delays have great impacts on dynamical behaviors for the system: on one hand, if \(\tau_{1}\in (m_{1},m_{2})\) and \(\tau_{2}\in(m_{4}, +\infty)\), then the boundary equilibrium \(E_{2}(x^{0}, y_{1}^{0}, y_{2}^{0}, 0, 0)\) is asymptotically stable (AS), i.e., the prey species and the predator species will coexist, the top-predator species will go extinct; on the other hand, if \(\tau_{1}\in(m_{2}, +\infty)\), then the axial equilibrium \(E_{1}(k, 0, 0, 0, 0)\) is AS, i.e., all predators will go extinct. Numerical simulations are great well agreement with the theoretical results.
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